To subtract 2 fractions with like denominators, just subtract the numerators (example: 3/10-2/10=1/10). If you have different MORE
In arithmetic, long division
is a standard division algorithm suitable for dividing simple or complex multidigit numbers that is simple enough to perform by hand. It breaks down a division problem into a series of easier steps. As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a result called the quotient. It enables computations involving arbitrarily large numbers to be performed by following a series of simple steps. The abbreviated form of long division is called short division, which is almost always used instead of long division when the divisor has only one digit.
Inexpensive calculators and computers have become the most common way to solve division problems, eliminating a traditional mathematical exercise, and decreasing the educational opportunity to show how to do so by paper and pencil techniques. (Internally, those devices use one of a variety of division algorithms). In the United States, long division has been especially targeted for de-emphasis, or even elimination from the school curriculum, by reform mathematics, though traditionally introduced in the 4th or 5th grades.
In English-speaking countries, long division does not use the slash (/) or obelus (÷) signs, instead displaying the dividend, divisor, and (once it is found) quotient in a tableau.
The process is begun by dividing the left-most digit of the dividend by the divisor. The quotient (rounded down to an integer) becomes the first digit of the result, and the remainder is calculated (this step is notated as a subtraction). This remainder carries forward when the process is repeated on the following digit of the dividend (notated as 'bringing down' the next digit to the remainder). When all digits have been processed and no remainder is left, the process is complete.
An example is shown below, representing the division of 500 by 4 (with a result of 125).
In the above example, the first step is to find the shortest sequence of digits starting from the left end of the dividend, 500, that the divisor 4 goes into at least once; this shortest sequence in this example is simply the first digit, 5. The largest number that the divisor 4 can be multiplied by without exceeding 5 is 1, so the digit 1 is put above the 5 to start constructing the quotient. Next, the 1 is multiplied by the divisor 4, to obtain the largest whole number (4 in this case) that is a multiple of the divisor 4 without exceeding the 5; this product of 1 times 4 is 4, so 4 is placed underneath the 5. Next the 4 under the 5 is subtracted from the 5 to get the remainder, 1, which is placed under the 4 under the 5. This remainder 1 is necessarily smaller than the divisor 4. Next the first as-yet unused digit in the dividend, in this case the first digit 0 after the 5, is copied directly underneath itself and next to the remainder 1, to form the number 10. At this point the process is repeated enough times to reach a stopping point: The largest number by which the divisor 4 can be multiplied without exceeding 10 is 2, so 2 is written above the 0 that is next to the 5 – that is, directly above the last digit in the 10. Then the latest entry to the quotient, 2, is multiplied by the divisor 4 to get 8, which is the largest multiple of 4 that does not exceed 10; so 8 is written below 10, and the subtraction 10 minus 8 is performed to get the remainder 2, which is placed below the 8. This remainder 2 is necessarily smaller than the divisor 4. The next digit of the dividend (the last 0 in 500) is copied directly below itself and next to the remainder 2, to form 20. Then the largest number by which the divisor 4 can be multiplied without exceeding 20 is ascertained; this number is 5, so 5 is placed above the last dividend digit that was brought down (i.e., above the rightmost 0 in 500). Then this new quotient digit 5 is multiplied by the divisor 4 to get 20, which is written at the bottom below the existing 20. Then 20 is subtracted from 20, yielding 0, which is written below the 20. We know we are done now because two things are true: there are no more digits to bring down from the dividend, and the last subtraction result was 0.
If the last remainder when we ran out of dividend digits had been something other than 0, there would have been two possible courses of action. (1) We could just stop there and say that the dividend divided by the divisor is the quotient written at the top with the remainder written at the bottom; equivalently we could write the answer as the quotient followed by a fraction that is the remainder divided by the divisor. Or, (2) we could extend the dividend by writing it as, say, 500.000... and continue the process (using a decimal point in the quotient directly above the decimal point in the dividend), in order to get a decimal answer, as in the following example.
In this example, the decimal part of the result is calculated by continuing the process beyond the units digit, "bringing down" zeros as being the decimal part of the dividend.
This example also illustrates that, at the beginning of the process, a step that produces a zero can be omitted. Since the first digit 1 is less than the divisor 4, the first step is instead performed on the first two digits 12. Similarly, if the divisor were 13, one would perform the first step on 127 rather than 12 or 1.
A divisor of any number of digits can be used. In this example, 37 is to be divided into 1260257. First the problem is set up as follows:
Digits of the number 1260257 are taken until a number greater than 37 occurs. So 1 and 12 are less than 37, but 126 is greater. Next, the greatest multiple of 37 less than 126 is computed. So 3 × 37 = 111 < 126, but 4 × 37 > 126. This is written underneath the 126 and the multiple of 37 is written on the top where the solution will appear:
Note carefully which columns these digits are written into - the 3 is put in the same column as the 6 in the dividend 1260257.
The 111 is then subtracted from the above line, ignoring all digits to the right:
Now digits are copied down from the dividend and appended to the result of 15 until a number greater than 37 is obtained. 150 is greater so only the 0 is copied:
The process repeats: the greatest multiple of 37 less than 150 is subtracted. This is 148 = 4 × 37, so a 4 is added to the solution line. Then the result of the subtraction is extended by digits taken from the dividend:
Notice that two digits had to be used to extend 2, as 22 < 37.
This is repeated until 37 divides the last line exactly:
For non-decimal currencies (such as the British £sd system before 1971) and measures (such as avoirdupois) mixed mode
division must be used. Consider dividing 50 miles 600 yards into 37 pieces:
Each of the four columns is worked in turn. Starting with the miles: 50/37 = 1 remainder 13. No further division is possible, so perform a long multiplication by 1,760 to convert miles to yards, the result is 22,880 yards. Carry this to the top of the yards column and add it to the 600 yards in the dividend giving 23,480. Long division of 23,480 / 37 now proceeds as normal yielding 634 with remainder 22. The remainder is multiplied by 3 to get feet and carried up to the feet column. Long division of the feet gives 1 remainder 29 which is then multiplied by twelve to get 348 inches. Long division continues with the final remainder of 15 inches being shown on the result line.
The same method and layout is used for binary, octal and hexadecimal. An address range of 0xf412df divided into 0x12 parts is:
Binary is of course trivial because each digit in the result can only be 1 or 0:
When the quotient is not an integer and the division process is extended beyond the decimal point, one of two things can happen. (1) The process can terminate, which means that a remainder of 0 is reached; or (2) a remainder could be reached that is identical to a previous remainder that occurred after the decimal points were written. In the latter case, continuing the process would be pointless, because from that point onward the same sequence of digits would appear in the quotient over and over. So a bar is drawn over the repeating sequence to indicate that it repeats forever.
China, Japan and India use the same notation as English-speakers. Elsewhere, the same general principles are used, but the figures are often arranged differently.
In Latin America (except Mexico, Colombia, Venezuela and Brazil), the calculation is almost exactly the same, but is written down differently as shown below with the same two examples used above. Usually the quotient is written under a bar drawn under the divisor. A long vertical line is sometimes drawn to the right of the calculations.
In Mexico, the US notation is used, except that only the result of the subtraction is annotated and the calculation is done mentally, as shown below:
In Brazil, Venezuela and Colombia, the European notation (see below) is used, except that the quotient is not separated by a vertical line, as shown below:
Same procedure applies in Mexico, only the result of the subtraction is annotated and the calculation is done mentally.
In Spain, Italy, France, Portugal, Romania, Turkey, Greece, Belgium, and Russia, the divisor is to the right of the dividend, and separated by a vertical bar. The division also occurs in the column, but the quotient (result) is written below the divider, and separated by the horizontal line.
In France, a long vertical bar separates the dividend and subsequent subtractions from the quotient and divisor, as in the example below of 6359 divided by 17, which is 374 with a remainder of 1.
Decimal numbers are not divided directly, the dividend and divisor are multiplied by a power of ten so that the division involves two whole numbers. Therefore, if one were dividing 12,7 by 0,4 (commas being used instead of decimal points), the dividend and divisor would first be changed to 127 and 4, and then the division would proceed as above.
In Germany, the notation of a normal equation is used for dividend, divisor and quotient (cf. first section of Latin American countries above, where it's done virtually the same way):
The same notation is adopted in Denmark, Norway, Macedonia, Poland, Croatia, Slovenia, Hungary, Czech Republic, Slovakia, , Vietnam and in Serbia.
In the Netherlands, the following notation is used:
Long division of integers can easily be extended to include non-integer dividends, as long as they are rational. This is because every rational number has a recurring decimal expansion. The procedure can also be extended to include divisors which have a finite or terminating decimal expansion (i.e. decimal fractions). In this case the procedure involves multiplying the divisor and dividend by the appropriate power of ten so that the new divisor is an integer – taking advantage of the fact that a
) ÷ (cb
) – and then proceeding as above.
A generalised version of this method called polynomial long division is also used for dividing polynomials (sometimes using a shorthand version called synthetic division).
In arithmetic, subtraction
is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with. Subtraction is denoted by a minus sign in infix notation, in contrast to the use of the plus sign for addition.
Since subtraction is not a commutative operator, the two operands are named. The traditional names for the parts of the formula
) − subtrahend
) = difference
Subtraction is used to model four related processes:
In mathematics, it is often useful to view or even define subtraction as a kind of addition, the addition of the additive inverse. We can view 7 − 3 = 4 as the sum of two terms: 7 and -3. This perspective allows us to apply to subtraction all of the familiar rules and nomenclature of addition. Subtraction is not associative or commutative—in fact, it is anticommutative and left-associative—but addition of signed numbers is both.
Imagine a line segment of length b
with the left end labeled a
and the right end labeled c
. Starting from a
, it takes b
steps to the right to reach c
. This movement to the right is modeled mathematically by addition:
, it takes b
steps to the left
to get back to a
. This movement to the left is modeled by subtraction:
Now, a line segment labeled with the numbers 1, 2, and 3. From position 3, it takes no steps to the left to stay at 3, so 3 − 0 = 3. It takes 2 steps to the left to get to position 1, so 3 − 2 = 1. This picture is inadequate to describe what would happen after going 3 steps to the left of position 3. To represent such an operation, the line must be extended.
To subtract arbitrary natural numbers, one begins with a line containing every natural number (0, 1, 2, 3, 4, 5, 6, ...). From 3, it takes 3 steps to the left to get to 0, so 3 − 3 = 0. But 3 − 4 is still invalid since it again leaves the line. The natural numbers are not a useful context for subtraction.
The solution is to consider the integer number line (..., −3, −2, −1, 0, 1, 2, 3, ...). From 3, it takes 4 steps to the left to get to −1:
There are some cases where subtraction as a separate operation becomes problematic. For example, 3 − (−2) (i.e. subtract −2 from 3) is not immediately obvious from either a natural number view or a number line view, because it is not immediately clear what it means to move −2 steps to the left or to take away −2 apples. One solution is to view subtraction as addition of signed numbers. Extra minus signs simply denote additive inversion. Then we have 3 − (−2) = 3 + 2 = 5. This also helps to keep the ring of integers "simple" by avoiding the introduction of "new" operators such as subtraction. Ordinarily a ring only has two operations defined on it; in the case of the integers, these are addition and multiplication. A ring already has the concept of additive inverses, but it does not have any notion of a separate subtraction operation, so the use of signed addition as subtraction allows us to apply the ring axioms to subtraction without needing to prove anything.
There are various algorithms for subtraction, and they differ in their suitability for various applications. A number of methods are adapted to hand calculation; for example, when making change, no actual subtraction is performed, but rather the change-maker counts forward.
For machine calculation, the method of complements is preferred, whereby the subtraction is replaced by an addition in a modular arithmetic.
Methods used to teach subtraction to elementary school varies from country to country, and within a country, different methods are in fashion at different times. In what is, in the U.S., called traditional mathematics, a specific process is taught to students at the end of the 1st year or during the 2nd year for use with multi-digit whole numbers, and is extended in either the fourth or fifth grade to include decimal representations of fractional numbers.
Some American schools currently teach a method of subtraction using borrowing and a system of markings called crutches]
[. Although a method of borrowing had been known and published in textbooks prior, apparently the crutches are the invention of William A. Brownell who used them in a study in November 1937]
[. This system caught on rapidly, displacing the other methods of subtraction in use in America at that time.
Some European schools employ a method of subtraction called the Austrian method, also known as the additions method. There is no borrowing in this method. There are also crutches (markings to aid memory), which vary by country]
Both these methods break up the subtraction as a process of one digit subtractions by place value. Starting with a least significant digit, a subtraction of subtrahend:
where each si
is a digit, proceeds by writing down m1
, and so forth, as long as si
does not exceed mi
. Otherwise, mi
is increased by 10 and some other digit is modified to correct for this increase. The American method corrects by attempting to decrease the minuend digit mi+1
by one (or continuing the borrow leftwards until there is a non-zero digit from which to borrow). The European method corrects by increasing the subtrahend digit si+1
704 − 512. The minuend is 704, the subtrahend is 512. The minuend digits are m3
= 7, m2
= 0 and m1
= 4. The subtrahend digits are s3
= 5, s2
= 1 and s1
= 2. Beginning at the one's place, 4 is not less than 2 so the difference 2 is written down in the result's one place. In the ten's place, 0 is less than 1, so the 0 is increased to 10, and the difference with 1, which is 9, is written down in the ten's place. The American method corrects for the increase of ten by reducing the digit in the minuend's hundreds place by one. That is, the 7 is struck through and replaced by a 6. The subtraction then proceeds in the hundreds place, where 6 is not less than 5, so the difference is written down in the result's hundred's place. We are now done, the result is 192.
The Austrian method does not reduce the 7 to 6. Rather it increases the subtrahend hundred's digit by one. A small mark is made near or below this digit (depending on the school). Then the subtraction proceeds by asking what number when increased by 1, and 5 is added to it, makes 7. The answer is 1, and is written down in the result's hundred's place.
There is an additional subtlety in that the student always employs a mental subtraction table in the American method. The Austrian method often encourages the student to mentally use the addition table in reverse. In the example above, rather than adding 1 to 5, getting 6, and subtracting that from 7, the student is asked to consider what number, when increased by 1, and 5 is added to it, makes 7.
When subtracting two numbers with units, they must have the same unit. In most cases the difference will have the same unit as the original numbers. One exception is when subtracting two numbers with percentage as unit. In this case, the difference will have percentage points as unit.
is the simplified portion of arithmetic which includes the operations of addition, subtraction, multiplication, and division.
Elementary arithmetic starts with the natural numbers and the written symbols (digits) which represent them. The process for combining a pair of these numbers with the four basic operations traditionally relies on memorized results for small values of numbers, including the contents of a multiplication table to assist with multiplication and division.
Elementary arithmetic also includes fractions and negative numbers, which can be represented on a number line.
The abacus is an early mechanical device for performing elementary arithmetic, which is still used in many parts of Asia. Modern calculating tools which perform elementary arithmetic operations include cash registers, electronic calculators, and computers.
Digits are the entire set of symbols used to represent numbers. In a particular numeral system, a single digit represents a different amount than any other digit, although the symbols in the same numeral system might vary between cultures.
In modern usage, the Arabic numerals are the most common set of symbols, and the most frequently used form of these digits is the Western style. Each single digit matches the following amounts:
, zero. Used in the absence of objects to be counted. For example, a different way of saying "there are no sticks here", is to say "the number of sticks here is 0".
, one. Applied to a single item. For example, here is one stick: I
, two. Applied to a pair of items. Here are two sticks: I I
, three. Applied to three items. Here are three sticks: I I I
, four. Applied to four items. Here are four sticks: I I I I
, five. Applied to five items. Here are five sticks: I I I I I
, six. Applied to six items. Here are six sticks: I I I I I I
, seven. Applied to seven items. Here are seven sticks: I I I I I I I
, eight. Applied to eight items. Here are eight sticks: I I I I I I I I
, nine. Applied to nine items. Here are nine sticks: I I I I I I I I I
Any numeral system defines the value of all numbers which contain more than one digit, most often by addition of the value for adjacent digits. The Hindu–Arabic numeral system includes positional notation to determine the value for any numeral. In this type of system, the increase in value for an additional digit includes one or more multiplications with the radix value and the result is added to the value of an adjacent digit. With Arabic numerals, the radix value of ten produces a value of twenty-one (equal to 2×10 + 1) for the numeral "21". An additional multiplication with the radix value occurs for each additional digit, so the numeral "201" represents a value of two-hundred-and-one (equal to 2×10×10 + 0×10 + 1).
The elementary level of study typically includes understanding the value of individual whole numbers using Arabic numerals with a maximum of seven digits, and performing the four basic operations using Arabic numerals with a maximum of four digits each.
When two numbers are added together, the result is called a sum. The two numbers being added together are called addends.
Suppose you have two bags, one bag holding five apples and a second bag holding three apples. Grabbing a third, empty bag, move all the apples from the first and second bags into the third bag. The third bag now holds eight apples. This illustrates the combination of three apples and five apples is eight apples; or more generally: "three plus five is eight" or "three plus five equals eight" or "eight is the sum of three and five". Numbers are abstract, and the addition of a group of three things to a group of five things will yield a group of eight things. Addition is a regrouping: two sets of objects which were counted separately are put into a single group and counted together: the count of the new group is the "sum" of the separate counts of the two original groups.
This operation of combining
is only one of several possible meanings that the mathematical operation of addition can have. Other meanings for addition include:
Symbolically, addition is represented by the "plus sign": +. So the statement "three plus five equals eight" can be written symbolically as . The order in which two numbers are added does not matter, so . This is the commutative property of addition.
To add a pair of digits using the table, find the intersection of the row of the first digit with the column of the second digit: the row and the column intersect at a square containing the sum of the two digits. Some pairs of digits add up to two-digit numbers, with the tens-digit always being a 1. In the addition algorithm the tens-digit of the sum of a pair of digits is called the "carry digit".
For simplicity, consider only numbers with three digits or less. To add a pair of numbers (written in Arabic numerals), write the second number under the first one, so that digits line up in columns: the rightmost column will contain the ones-digit of the second number under the ones-digit of the first number. This rightmost column is the ones-column. The column immediately to its left is the tens-column. The tens-column will have the tens-digit of the second number (if it has one) under the tens-digit of the first number (if it has one). The column immediately to the left of the tens-column is the hundreds-column. The hundreds-column will line up the hundreds-digit of the second number (if there is one) under the hundreds-digit of the first number (if there is one).
After the second number has been written down under the first one so that digits line up in their correct columns, draw a line under the second (bottom) number. Start with the ones-column: the ones-column should contain a pair of digits: the ones-digit of the first number and, under it, the ones-digit of the second number. Find the sum of these two digits: write this sum under the line and in the ones-column. If the sum has two digits, then write down only the ones-digit of the sum. Write the "carry digit" above the top digit of the next column: in this case the next column is the tens-column, so write a 1 above the tens-digit of the first number.
If both first and second number each have only one digit then their sum is given in the addition table, and the addition algorithm is unnecessary.
Then comes the tens-column. The tens-column might contain two digits: the tens-digit of the first number and the tens-digit of the second number. If one of the numbers has a missing tens-digit then the tens-digit for this number can be considered to be a 0. Add the tens-digits of the two numbers. Then, if there is a carry digit, add it to this sum. If the sum was 18 then adding the carry digit to it will yield 19. If the sum of the tens-digits (plus carry digit, if there is one) is less than ten then write it in the tens-column under the line. If the sum has two digits then write its last digit in the tens-column under the line, and carry its first digit (which should be a 1) over to the next column: in this case the hundreds-column.
If none of the two numbers has a hundreds-digit then if there is no carry digit then the addition algorithm has finished. If there is a carry digit (carried over from the tens-column) then write it in the hundreds-column under the line, and the algorithm is finished. When the algorithm finishes, the number under the line is the sum of the two numbers.
If at least one of the numbers has a hundreds-digit then if one of the numbers has a missing hundreds-digit then write a 0 digit in its place. Add the two hundreds-digits, and to their sum add the carry digit if there is one. Then write the sum of the hundreds-column under the line, also in the hundreds column. If the sum has two digits then write down the last digit of the sum in the hundreds-column and write the carry digit to its left: on the thousands-column.
Say one wants to find the sum of the numbers 653 and 274. Write the second number under the first one, with digits aligned in columns, like so:
Then draw a line under the second number and put a plus sign. The addition starts with the ones-column. The ones-digit of the first number is 3 and of the second number is 4. The sum of three and four is seven, so write a 7 in the ones-column under the line:
Next, the tens-column. The tens-digit of the first number is 5, and the tens-digit of the second number is 7, and five plus seven is twelve: 12, which has two digits, so write its last digit, 2, in the tens-column under the line, and write the carry digit on the hundreds-column above the first number:
Next, the hundreds-column. The hundreds-digit of the first number is 6, while the hundreds-digit of the second number is 2. The sum of six and two is eight, but there is a carry digit, which added to eight is equal to nine. Write the 9 under the line in the hundreds-column:
No digits (and no columns) have been left unadded, so the algorithm finishes, and
The result of the addition of one to a number is the successor
of that number. Examples:
the successor of zero is one,
the successor of one is two,
the successor of two is three,
the successor of ten is eleven.
Every natural number has a successor.
The predecessor of the successor of a number is the number itself. For example, five is the successor of four therefore four is the predecessor of five. Every natural number except zero has a predecessor.
If a number is the successor of another number, then the first number is said to be larger than
the other number. If a number is larger than another number, and if the other number is larger than a third number, then the first number is also larger than the third number. Example: five is larger than four, and four is larger than three, therefore five is larger than three. But six is larger than five, therefore six is also larger than three. But seven is larger than six, therefore seven is also larger than three ... therefore eight is larger than three ... therefore nine is larger than three, etc.
If two non-zero natural numbers are added together, then their sum is larger than either one of them. Example: three plus five equals eight, therefore eight is larger than three () and eight is larger than five (). The symbol for "larger than" is >.
If a number is larger than another one, then the other is smaller than
the first one. Examples: three is smaller than eight () and five is smaller than eight (). The symbol for smaller than is <. A number cannot be at the same time larger and smaller than another number. Neither can a number be at the same time larger than and equal to another number. Given a pair of natural numbers, one and only one of the following cases must be true:
To count a group of objects means to assign a natural number to each one of the objects, as if it were a label for that object, such that a natural number is never assigned to an object unless its predecessor was already assigned to another object, with the exception that zero is not assigned to any object: the smallest natural number to be assigned is one, and the largest natural number assigned depends on the size of the group. It is called the count
and it is equal to the number of objects in that group.
The process of counting a group is the following:
When the counting is finished, the last value of the count will be the final count. This count is equal to the number of objects in the group.
Often, when counting objects, one does not keep track of what numerical label corresponds to which object: one only keeps track of the subgroup of objects which have already been labeled, so as to be able to identify unlabeled objects necessary for Step 2. However, if one is counting persons, then one can ask the persons who are being counted to each keep track of the number which the person's self has been assigned. After the count has finished it is possible to ask the group of persons to file up in a line, in order of increasing numerical label. What the persons would do during the process of lining up would be something like this: each pair of persons who are unsure of their positions in the line ask each other what their numbers are: the person whose number is smaller should stand on the left side and the one with the larger number on the right side of the other person. Thus, pairs of persons compare their numbers and their positions, and commute their positions as necessary, and through repetition of such conditional commutations they become ordered.
Subtraction is the mathematical operation which describes a reduced quantity. The result of this operation is the difference
between two numbers. As with addition, subtraction can have a number of interpretations, such as:
As with addition, there are other possible interpretations, such as motion
Symbolically, the minus sign ("−") represents the subtraction operation. So the statement "five minus three equals two" is also written as . In elementary arithmetic, subtraction uses smaller positive numbers for all values to produce simpler solutions.
Unlike addition, subtraction is not commutative, so the order of numbers in the operation will change the result. Therefore, each number is provided a different distinguishing name. The first number (5 in the previous example) is formally defined as the minuend
and the second number (3 in the previous example) as the subtrahend
. The value of the minuend is larger than the value of the subtrahend so that the result is a positive number, but a smaller value of the minuend will result in negative numbers.
There are several methods to accomplish subtraction. The method which is in the United States of America referred to as traditional mathematics taught elementary school students to subtract using methods suitable for hand calculation. The particular method used varies from country from country, and within a country, different methods are in fashion at different times. Reform mathematics is distinguished generally by the lack of preference for any specific technique, replaced by guiding 2nd-grade students to invent their own methods of computation, such as using properties of negative numbers in the case of TERC.
American schools currently teach a method of subtraction using borrowing and a system of markings called crutches. Although a method of borrowing had been known and published in textbooks prior, apparently the crutches are the invention of William A. Browell, who used them in a study in November 1937 . This system caught on rapidly, displacing the other methods of subtraction in use in America at that time.
Students in some European countries are taught, and some older Americans employ, a method of subtraction called the Austrian method, also known as the additions method. There is no borrowing in this method. There are also crutches (markings to aid the memory) which [probably] vary according to country.
In the method of borrowing, a subtraction such as will accomplish the ones-place subtraction of 9 from 6 by borrowing a 10 from 80 and adding it to the 6. The problem is thus transformed into effectively. This is indicated by striking through the 8, writing a small 7 above it, and writing a small 1 above the 6. These markings are called crutches
. The 9 is then subtracted from 16, leaving 7, and the 30 from the 70, leaving 40, or 47 as the result.
In the additions method, a 10 is borrowed to make the 6 into 16, in preparation for the subtraction of 9, just as in the borrowing method. However, the 10 is not taken by reducing the minuend, rather one augments the subtrahend. Effectively, the problem is transformed into . Typically a crutch of a small one is marked just below the subtrahend digit as a reminder. Then the operations proceed: 9 from 16 is 7; and 40 (that is, ) from 80 is 40, or 47 as the result.
The additions method seem to be taught in two variations, which differ only in psychology. Continuing the example of , the first variation attempts to subtract 9 from 6, and then 9 from 16, borrowing a 10 by marking near the digit of the subtrahend in the next column. The second variation attempts to find a digit which, when added to 9, gives 6, and recognizing that is not possible, gives 16, and carrying the 10 of the 16 as a one marking near the same digit as in the first method. The markings are the same; it is just a matter of preference as to how one explains its appearance.
As a final caution, the borrowing method gets a bit complicated in cases such as , where a borrow cannot be made immediately, and must be obtained by reaching across several columns. In this case, the minuend is effectively rewritten as , by taking a 100 from the hundreds, making ten 10s from it, and immediately borrowing that down to nine 10s in the tens column and finally placing a 10 in the ones column.
When two numbers are multiplied together, the result is called a product
. The two numbers being multiplied together are called factors
Suppose there are five red bags, each one containing three apples. Now grabbing an empty green bag, move all the apples from all five red bags into the green bag. Now the green bag will have fifteen apples.
Thus the product of five and three is fifteen.
This can also be stated as "five times three is fifteen" or "five times three equals fifteen" or "fifteen is the product of five and three". Multiplication can be seen to be a form of repeated addition: the first factor indicates how many times the second factor should be added onto itself; the final sum being the product.
Symbolically, multiplication is represented by the multiplication sign
: ×. So the statement "five times three equals fifteen" can be written symbolically as
In some countries, and in more advanced arithmetic, other multiplication signs are used, e.g. . In some situations, especially in algebra, where numbers can be symbolized with letters, the multiplication symbol may be omitted; e.g. xy
means . The order in which two numbers are multiplied does not matter, so that, for example, three times four equals four times three. This is the commutative property of multiplication.
To multiply a pair of digits using the table, find the intersection of the row of the first digit with the column of the second digit: the row and the column intersect at a square containing the product of the two digits. Most pairs of digits produce two-digit numbers. In the multiplication algorithm the tens-digit of the product of a pair of digits is called the "carry digit".
Consider a multiplication where one of the factors has only one digit, whereas the other factor has an arbitrary quantity of digits. Write down the multi-digit factor, then write the single-digit factor under the last digit of the multi-digit factor. Draw a horizontal line under the single-digit factor. Henceforth, the single-digit factor will be called the "multiplier" and the multi-digit factor will be called the "multiplicand".
Suppose for simplicity that the multiplicand has three digits. The first digit is the hundreds-digit, the middle digit is the tens-digit, and the last, rightmost, digit is the ones-digit. The multiplier only has a ones-digit. The ones-digits of the multiplicand and multiplier form a column: the ones-column.
Start with the ones-column: the ones-column should contain a pair of digits: the ones-digit of the multiplicand and, under it, the ones-digit of the multiplier. Find the product of these two digits: write this product under the line and in the ones-column. If the product has two digits, then write down only the ones-digit of the product. Write the "carry digit" as a superscript of the yet-unwritten digit in the next column and under the line: in this case the next column is the tens-column, so write the carry digit as the superscript of the yet-unwritten tens-digit of the product (under the line).
If both first and second number each have only one digit then their product is given in the multiplication table, and the multiplication algorithm is unnecessary.
Then comes the tens-column. The tens-column so far contains only one digit: the tens-digit of the multiplicand (though it might contain a carry digit under the line). Find the product of the multiplier and the tens-digits of the multiplicand. Then, if there is a carry digit (superscripted, under the line and in the tens-column), add it to this product. If the resulting sum is less than ten then write it in the tens-column under the line. If the sum has two digits then write its last digit in the tens-column under the line, and carry its first digit over to the next column: in this case the hundreds column.
If the multiplicand does not have a hundreds-digit then if there is no carry digit then the multiplication algorithm has finished. If there is a carry digit (carried over from the tens-column) then write it in the hundreds-column under the line, and the algorithm is finished. When the algorithm finishes, the number under the line is the product of the two numbers.
If the multiplicand has a hundreds-digit, find the product of the multiplier and the hundreds-digit of the multiplicand, and to this product add the carry digit if there is one. Then write the resulting sum of the hundreds-column under the line, also in the hundreds column. If the sum has two digits then write down the last digit of the sum in the hundreds-column and write the carry digit to its left: on the thousands-column.
Say one wants to find the product of the numbers 3 and 729. Write the single-digit multiplier under the multi-digit multiplicand, with the multiplier under the ones-digit of the multiplicand, like so:
Then draw a line under the multiplier and put a multiplication symbol. Multiplication starts with the ones-column. The ones-digit of the multiplicand is 9 and the multiplier is 3. The product of 3 and 9 is 27, so write a 7 in the ones-column under the line, and write the carry-digit 2 as a superscript of the yet-unwritten tens-digit of the product under the line:
Next, the tens-column. The tens-digit of the multiplicand is 2, the multiplier is 3, and three times two is six. Add the carry-digit, 2, to the product, 6, to obtain 8. Eight has only one digit: no carry-digit, so write in the tens-column under the line. You can erase the two now.
Next, the hundreds-column. The hundreds-digit of the multiplicand is 7, while the multiplier is 3. The product of 3 and 7 is 21, and there is no previous carry-digit (carried over from the tens-column). The product 21 has two digits: write its last digit in the hundreds-column under the line, then carry its first digit over to the thousands-column. Since the multiplicand has no thousands-digit, then write this carry-digit in the thousands-column under the line (not superscripted):
No digits of the multiplicand have been left unmultiplied, so the algorithm finishes, and
Given a pair of factors, each one having two or more digits, write both factors down, one under the other one, so that digits line up in columns.
For simplicity consider a pair of three-digits numbers. Write the last digit of the second number under the last digit of the first number, forming the ones-column. Immediately to the left of the ones-column will be the tens-column: the top of this column will have the second digit of the first number, and below it will be the second digit of the second number. Immediately to the left of the tens-column will be the hundreds-column: the top of this column will have the first digit of the first number and below it will be the first digit of the second number. After having written down both factors, draw a line under the second factor.
The multiplication will consist of two parts. The first part will consist of several multiplications involving one-digit multipliers. The operation of each one of such multiplications was already described in the previous multiplication algorithm, so this algorithm will not describe each one individually, but will only describe how the several multiplications with one-digit multipliers shall be coordinated. The second part will add up all the subproducts of the first part, and the resulting sum will be the product.
. Let the first factor be called the multiplicand. Let each digit of the second factor be called a multiplier. Let the ones-digit of the second factor be called the "ones-multiplier". Let the tens-digit of the second factor be called the "tens-multiplier". Let the hundreds-digit of the second factor be called the "hundreds-multiplier".
Start with the ones-column. Find the product of the ones-multiplier and the multiplicand and write it down in a row under the line, aligning the digits of the product in the previously-defined columns. If the product has four digits, then the first digit will be the beginning of the thousands-column. Let this product be called the "ones-row".
Then the tens-column. Find the product of the tens-multiplier and the multiplicand and write it down in a row—call it the "tens-row"—under the ones-row, but shifted one column to the left
. That is, the ones-digit of the tens-row will be in the tens-column of the ones-row; the tens-digit of the tens-row will be under the hundreds-digit of the ones-row; the hundreds-digit of the tens-row will be under the thousands-digit of the ones-row. If the tens-row has four digits, then the first digit will be the beginning of the ten-thousands-column.
Next, the hundreds-column. Find the product of the hundreds-multiplier and the multiplicand and write it down in a row—call it the "hundreds-row"—under the tens-row, but shifted one more column to the left. That is, the ones-digit of the hundreds-row will be in the hundreds-column; the tens-digit of the hundreds-row will be in the thousands-column; the hundreds-digit of the hundreds-row will be in the ten-thousands-column. If the hundreds-row has four digits, then the first digit will be the beginning of the hundred-thousands-column.
After having down the ones-row, tens-row, and hundreds-row, draw a horizontal line under the hundreds-row. The multiplications are over.
. Now the multiplication has a pair of lines. The first one under the pair of factors, and the second one under the three rows of subproducts. Under the second line there will be six columns, which from right to left are the following: ones-column, tens-column, hundreds-column, thousands-column, ten-thousands-column, and hundred-thousands-column.
Between the first and second lines, the ones-column will contain only one digit, located in the ones-row: it is the ones-digit of the ones-row. Copy this digit by rewriting it in the ones-column under the second line.
Between the first and second lines, the tens-column will contain a pair of digits located in the ones-row and the tens-row: the tens-digit of the ones-row and the ones-digit of the tens-row. Add these digits up and if the sum has just one digit then write this digit in the tens-column under the second line. If the sum has two digits then the first digit is a carry-digit: write the last digit down in the tens-column under the second line and carry the first digit over to the hundreds-column, writing it as a superscript to the yet-unwritten hundreds-digit under the second line.
Between the first and second lines, the hundreds-column will contain three digits: the hundreds-digit of the ones-row, the tens-digit of the tens-row, and the ones-digit of the hundreds-row. Find the sum of these three digits, then if there is a carry-digit from the tens-column (written in superscript under the second line in the hundreds-column) then add this carry-digit as well. If the resulting sum has one digit then write it down under the second line in the hundreds-column; if it has two digits then write the last digit down under the line in the hundreds-column, and carry the first digit over to the thousands-column, writing it as a superscript to the yet-unwritten thousands-digit under the line.
Between the first and second lines, the thousands-column will contain either two or three digits: the hundreds-digit of the tens-row, the tens-digit of the hundreds-row, and (possibly) the thousands-digit of the ones-row. Find the sum of these digits, then if there is a carry-digit from the hundreds-column (written in superscript under the second line in the thousands-column) then add this carry-digit as well. If the resulting sum has one digit then write it down under the second line in the thousands-column; if it has two digits then write the last digit down under the line in the thousands-column, and carry the first digit over to the ten-thousands-column, writing it as a superscript to the yet-unwritten ten-thousands-digit under the line.
Between the first and second lines, the ten-thousands-column will contain either one or two digits: the hundreds-digit of the hundreds-column and (possibly) the thousands-digit of the tens-column. Find the sum of these digits (if the one in the tens-row is missing think of it as a 0), and if there is a carry-digit from the thousands-column (written in superscript under the second line in the ten-thousands-column) then add this carry-digit as well. If the resulting sum has one digit then write it down under the second line in the ten-thousands-column; if it has two digits then write the last digit down under the line in the ten-thousands-column, and carry the first digit over to the hundred-thousands-column, writing it as a superscript to the yet-unwritten hundred-thousands digit under the line. However, if the hundreds-row has no thousands-digit then do not write this carry-digit as a superscript, but in normal size, in the position of the hundred-thousands-digit under the second line, and the multiplication algorithm is over.
If the hundreds-row does have a thousands-digit, then add to it the carry-digit from the previous row (if there is no carry-digit then think of it as a 0) and write the single-digit sum in the hundred-thousands-column under the second line.
The number under the second line is the sought-after product of the pair of factors above the first line.
Let our objective be to find the product of 789 and 345. Write the 345 under the 789 in three columns, and draw a horizontal line under them:
. Start with the ones-column. The multiplicand is 789 and the ones-multiplier is 5. Perform the multiplication in a row under the line:
Then the tens-column. The multiplicand is 789 and the tens-multiplier is 4. Perform the multiplication in the tens-row, under the previous subproduct in the ones-row, but shifted one column to the left:
Next, the hundreds-column. The multiplicand is once again 789, and the hundreds-multiplier is 3. Perform the multiplication in the hundreds-row, under the previous subproduct in the tens-row, but shifted one (more) column to the left. Then draw a horizontal line under the hundreds-row:
Now add the subproducts between the first and second lines, but ignoring any superscripted carry-digits located between the first and second lines.
The answer is
In mathematics, especially in elementary arithmetic, division
is an arithmetic operation which is the inverse of multiplication.
Specifically, if c
is not zero, then a
divided by b
In the above expression, a
is called the dividend
Division by zero (i.e. where the divisor is zero) is not defined.
Division is most often shown by placing the dividend
over the divisor
with a horizontal line, also called a vinculum, between them. For example, a
divided by b
This can be read out loud as "a
divided by b
" or "a
". A way to express division all on one line is to write the dividend
, then a slash, then the divisor
, like this:
This is the usual way to specify division in most computer programming languages since it can easily be typed as a simple sequence of characters.
A handwritten or typographical variation, which is halfway between these two forms, uses a solidus (fraction slash) but elevates the dividend, and lowers the divisor:
Any of these forms can be used to display a fraction. A common fraction
is a division expression where both dividend and divisor are integers (although typically called the numerator
), and there is no implication that the division needs to be evaluated further.
A more basic way to show division is to use the obelus (or division sign) in this manner:
This form is infrequent except in basic arithmetic. The obelus is also used alone to represent the division operation itself, for instance, as a label on a key of a calculator.
In some non-English-speaking cultures, "a
divided by b
" is written a
. However, in English usage the colon is restricted to expressing the related concept of ratios (then "a
is to b
With a knowledge of multiplication tables, two integers can be divided on paper using the method of long division. If the dividend has a fractional part (expressed as a decimal fraction), one can continue the algorithm past the ones place as far as desired. If the divisor has a decimal fractional part, one can restate the problem by moving the decimal to the right in both numbers until the divisor has no fraction.
To divide by a fraction, multiply by the reciprocal (reversing the position of the top and bottom parts) of that fraction.
Local standards usually define the educational methods and content included in the elementary level of instruction. In the United States and Canada, controversial subjects include the amount of calculator usage compared to manual computation and the broader debate between traditional mathematics and reform mathematics.
In the United States, the 1989 NCTM standards led to curricula which de-emphasized or omitted much of what was considered to be elementary arithmetic in elementary school, and replaced it with emphasis on topics traditionally studied in college such as algebra, statistics and problem solving, and non-standard computation methods unfamiliar to most adults.
An Egyptian fraction
is the sum of distinct unit fractions, such as
. That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from each other. The value of an expression of this type is a positive rational number a
; for instance the Egyptian fraction above sums to 43/48. Every positive rational number can be represented by an Egyptian fraction. Sums of this type, and similar sums also including 2/3 and 3/4 as summands, were used as a serious notation for rational numbers by the ancient Egyptians, and continued to be used by other civilizations into medieval times. In modern mathematical notation, Egyptian fractions have been superseded by vulgar fractions and decimal notation. However, Egyptian fractions continue to be an object of study in modern number theory and recreational mathematics, as well as in modern historical studies of ancient mathematics.
Egyptian fraction notation was developed in the Middle Kingdom of Egypt, altering the Old Kingdom's Eye of Horus numeration system. Five early texts in which Egyptian fractions appear were the Egyptian Mathematical Leather Roll, the Moscow Mathematical Papyrus, the Reisner Papyrus, the Kahun Papyrus and the Akhmim Wooden Tablet. A later text, the Rhind Mathematical Papyrus, introduced improved ways of writing Egyptian fractions. The Rhind papyrus was written by Ahmes and dates from the Second Intermediate Period; it includes a n
table of Egyptian fraction expansions for rational numbers 2/, as well as 84 word problems. Solutions to each problem were written out in scribal shorthand, with the final answers of all 84 problems being expressed in Egyptian fraction notation. 2/n
tables similar to the one on the Rhind papyrus also appear on some of the other texts. However, as the Kahun Papyrus shows, vulgar fractions were also used by scribes within their calculations.
To write the unit fractions used in their Egyptian fraction notation, in hieroglyph script, the Egyptians placed the hieroglyph
, "[one] among" or possibly re
, mouth) above a number to represent the reciprocal of that number. Similarly in hieratic script they drew a line over the letter representing the number. For example:
The Egyptians had special symbols for 1/2, 2/3, and 3/4 that were used to reduce the size of numbers greater than 1/2 when such numbers were converted to an Egyptian fraction series. The remaining number after subtracting one of these special fractions was written using as a sum of distinct unit fractions according to the usual Egyptian fraction notation.
The Egyptians also used an alternative notation modified from the Old Kingdom and based on the parts of the Eye of Horus to denote a special set of fractions of the form 1/2k
= 1, 2, ..., 6) and sums of these numbers, which are necessarily dyadic rational numbers. These "Horus-Eye fractions" were used in the Middle Kingdom in conjunction with the later notation for Egyptian fractions to subdivide a hekat, the primary ancient Egyptian volume measure for grain, bread, and other small quantities of volume, as described in the Akhmim Wooden Tablet. If any remainder was left after expressing a quantity in Eye of Horus fractions of a hekat, the remainder was written using the usual Egyptian fraction notation as multiples of a ro
, a unit equal to 1/320 of a hekat.
Modern historians of mathematics have studied the Rhind papyrus and other ancient sources in an attempt to discover the methods the Egyptians used in calculating with Egyptian fractions. In particular, study in this area has concentrated on understanding the tables of expansions for numbers of the form 2/n
in the Rhind papyrus. Although these expansions can generally be described as algebraic identities, the methods used by the Egyptians may not correspond directly to these identities. Additionally, the expansions in the table do not match any single identity; rather, different identities match the expansions for prime and for composite denominators, and more than one identity fits the numbers of each type:
Egyptian fraction notation continued to be used in Greek times and into the Middle Ages (Struik 1967), despite complaints as early as Ptolemy's Almagest about the clumsiness of the notation compared to alternatives such as the Babylonian base-60 notation. An important text of medieval mathematics, the Liber Abaci
(1202) of Leonardo of Pisa (more commonly known as Fibonacci), provides some insight into the uses of Egyptian fractions in the Middle Ages, and introduces topics that continue to be important in modern mathematical study of these series.
The primary subject of the Liber Abaci
is calculations involving decimal and vulgar fraction notation, which eventually replaced Egyptian fractions. Fibonacci himself used a complex notation for fractions involving a combination of a mixed radix notation with sums of fractions. Many of the calculations throughout Fibonacci's book involve numbers represented as Egyptian fractions, and one section of this book (Sigler 2002, chapter II.7) provides a list of methods for conversion of vulgar fractions to Egyptian fractions. If the number is not already a unit fraction, the first method in this list is to attempt to split the numerator into a sum of divisors of the denominator; this is possible whenever the denominator is a practical number, and Liber Abaci
includes tables of expansions of this type for the practical numbers 6, 8, 12, 20, 24, 60, and 100.
The next several methods involve algebraic identities such as
For instance, Fibonacci represents the fraction
by splitting the numerator into a sum of two numbers, each of which divides one plus the denominator:
Fibonacci applies the algebraic identity above to each these two parts, producing the expansion
Fibonacci describes similar methods for denominators that are two or three less than a number with many factors.
In the rare case that these other methods all fail, Fibonacci suggests a greedy algorithm for computing Egyptian fractions, in which one repeatedly chooses the unit fraction with the smallest denominator that is no larger than the remaining fraction to be expanded: that is, in more modern notation, we replace a fraction x
by the expansion
represents the ceiling function.
Fibonacci suggests switching to another method after the first such expansion, but he also gives examples in which this greedy expansion was iterated until a complete Egyptian fraction expansion was constructed:
As later mathematicians showed, each greedy expansion reduces the numerator of the remaining fraction to be expanded, so this method always terminates with a finite expansion. However, compared to ancient Egyptian expansions or to more modern methods, this method may produce expansions that are quite long, with large denominators, and Fibonacci himself noted the awkwardness of the expansions produced by this method. For instance, the greedy method expands
while other methods lead to the much better expansion
Sylvester's sequence 2, 3, 7, 43, 1807, ... can be viewed as generated by an infinite greedy expansion of this type for the number one, where at each step we choose the denominator
, and sometimes Fibonacci's greedy algorithm is attributed to Sylvester.
After his description of the greedy algorithm, Fibonacci suggests yet another method, expanding a fraction
by searching for a number c
having many divisors, with
, and expanding
as a sum of divisors of
, similar to the method proposed by Hultsch and Bruins to explain some of the expansions in the Rhind papyrus.
Modern number theorists have studied many different problems related to Egyptian fractions, including problems of bounding the length or maximum denominator in Egyptian fraction representations, finding expansions of certain special forms or in which the denominators are all of some special type, the termination of various methods for Egyptian fraction expansion, and showing that expansions exist for any sufficiently dense set of sufficiently smooth numbers.
Some notable problems remain unsolved with regard to Egyptian fractions, despite considerable effort by mathematicians.
Guy (2004) describes these problems in more detail and lists numerous additional open problems.
In mathematics, especially in elementary arithmetic, division
(÷) is an arithmetic operation. Specifically, if b
is not zero, then a
divided by b
In the expression a ÷ b = c, a
is called the dividend
and the result c
is called the quotient
Conceptually, division describes two distinct but related settings. Partitioning
involves taking a set of size a
and forming b
groups that are equal in size. The size of each group formed, c
, is the quotient of a
division involves taking a set of size a
and forming groups of size c
. The number of groups of this size that can be formed, b
, is the quotient of a
Teaching division usually leads to the concept of fractions being introduced to students. Unlike addition, subtraction, and multiplication, the set of all integers is not closed under division. Dividing two integers may result in a remainder. To complete the division of the remainder, the number system is extended to include fractions or rational numbers as they are more generally called.
Division is often shown in algebra and science by placing the dividend
over the divisor
with a horizontal line, also called a vinculum or fraction bar, between them. For example, a
divided by b
This can be read out loud as "a divided by b", "a by b" or "a over b". A way to express division all on one line is to write the dividend
(or numerator), then a slash, then the divisor
(or denominator), like this:
This is the usual way to specify division in most computer programming languages since it can easily be typed as a simple sequence of ASCII characters.
A typographical variation halfway between these two forms uses a solidus (fraction slash) but elevates the dividend, and lowers the divisor:
Any of these forms can be used to display a fraction. A fraction is a division expression where both dividend and divisor are integers (although typically called the numerator
), and there is no implication that the division must be evaluated further. A second way to show division is to use the obelus (or division sign), common in arithmetic, in this manner:
This form is infrequent except in elementary arithmetic. ISO 80000-2-9.6 states it should not be used. The obelus is also used alone to represent the division operation itself, as for instance as a label on a key of a calculator.
In some non-English-speaking cultures, "a divided by b" is written a
. This notation was introduced in 1631 by William Oughtred in his Clavis Mathematicae
and later popularized by Gottfried Wilhelm Leibniz. However, in English usage the colon is restricted to expressing the related concept of ratios (then "a is to b").
In elementary mathematics the notation
is used to denote a
divided by b
. This notation was first introduced by Michael Stifel in Arithmetica integra
, published in 1544.
Division is often introduced through the notion of "sharing out" a set of objects, for example a pile of sweets, into a number of equal portions. Distributing the objects several at a time in each round of sharing to each portion leads to the idea of "chunking", i.e., division by repeated subtraction.
More systematic and more efficient (but also more formalised and more rule-based, and more removed from an overall holistic picture of what division is achieving), a person who knows the multiplication tables can divide two integers using pencil and paper using the method of short division, if the divisor is simple. Long division is used for larger integer divisors. If the dividend has a fractional part (expressed as a decimal fraction), one can continue the algorithm past the ones place as far as desired. If the divisor has a fractional part, we can restate the problem by moving the decimal to the right in both numbers until the divisor has no fraction.
A person can calculate division with an abacus by repeatedly placing the dividend on the abacus, and then subtracting the divisor the offset of each digit in the result, counting the number of divisions possible at each offset.
A person can use logarithm tables to divide two numbers, by subtracting the two numbers' logarithms, then looking up the antilogarithm of the result.
A person can calculate division with a slide rule by aligning the divisor on the C scale with the dividend on the D scale. The quotient can be found on the D scale where it is aligned with the left index on the C scale. The user is responsible, however, for mentally keeping track of the decimal point.
Modern computers compute division by methods that are faster than long division: see Division algorithm.
In modular arithmetic, some numbers have a multiplicative inverse with respect to the modulus. We can calculate division by multiplication in such a case. This approach is useful in computers that do not have a fast division instruction.
The division algorithm is a mathematical theorem that precisely expresses the outcome of the usual process of division of integers. In particular, the theorem asserts that integers called the quotient q
and remainder r
always exist and that they are uniquely determined by the dividend a
and divisor d
, with d
≠ 0. Formally, the theorem is stated as follows: There exist unique integers q
such that a
and 0 ≤ r
< | d
|, where | d
| denotes the absolute value of d
Division of integers is not closed. Apart from division by zero being undefined, the quotient is not an integer unless the dividend is an integer multiple of the divisor. For example 26 cannot be divided by 11 to give an integer. Such a case uses one of five approaches:
Dividing integers in a computer program requires special care. Some programming languages, such as C, treat integer division as in case 5 above, so the answer is an integer. Other languages, such as MATLAB and every computer algebra system return a rational number as the answer, as in case 3 above. These languages provide also functions to get the results of the other cases, either directly of from the result of case 3.
Names and symbols used for integer division include div, /, \, and %. Definitions vary regarding integer division when the dividend or the divisor is negative: rounding may be toward zero (so called T-division) or toward −∞ (F-division); rarer styles can occur – see Modulo operation for the details.
Divisibility rules can sometimes be used to quickly determine whether one integer divides exactly into another.
The result of dividing two rational numbers is another rational number when the divisor is not 0. We may define division of two rational numbers p
All four quantities are integers, and only p
may be 0. This definition ensures that division is the inverse operation of multiplication.
Division of two real numbers results in another real number when the divisor is not 0. It is defined such a
if and only if a
Division of any number by zero (where the divisor is zero) is undefined. This is because zero multiplied by any finite number always results in a product of zero. Entry of such an expression into most calculators produces an error message.
Dividing two complex numbers results in another complex number when the divisor is not 0, defined thus:
All four quantities are real numbers. r
may not both be 0.
Division for complex numbers expressed in polar form is simpler than the definition above:
Again all four quantities are real numbers. r
may not be 0.
One can define the division operation for polynomials in one variable over a field. Then, as in the case of integers, one has a remainder. See Euclidean division of polynomials, and, for hand-written computation, polynomial long division or synthetic division.
One can define a division operation for matrices. The usual way to do this is to define , where denotes the inverse of B
, but it is far more common to write out explicitly to avoid confusion.
Because matrix multiplication is not commutative, one can also define a left division or so-called backslash-division
as . For this to be well defined, need not exist, however does need to exist. To avoid confusion, division as defined by is sometimes called right division
in this context.
Note that with left and right division defined this way, is in general not the same as and nor is the same as , but and .
To avoid problems when and/or do not exist, division can also be defined as multiplication with the pseudoinverse, i.e., and , where and denote the pseudoinverse of A
In abstract algebras such as matrix algebras and quaternion algebras, fractions such as
are typically defined as
is presumed an invertible element (i.e., there exists a multiplicative inverse
is the multiplicative identity). In an integral domain where such elements may not exist, division
can still be performed on equations of the form
by left or right cancellation, respectively. More generally "division" in the sense of "cancellation" can be done in any ring with the aforementioned cancellation properties. If such a ring is finite, then by an application of the pigeonhole principle, every nonzero element of the ring is invertible, so division
by any nonzero element is possible in such a ring. To learn about when algebras
(in the technical sense) have a division operation, refer to the page on division algebras. In particular Bott periodicity can be used to show that any real normed division algebra must be isomorphic to either the real numbers R
, the complex numbers C
, the quaternions H
, or the octonions O
The derivative of the quotient of two functions is given by the quotient rule:
There is no general method to integrate the quotient of two functions.
A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/n. Examples are 1/1, 1/2, 1/3, 1/4 etc.
Multiplying any two unit fractions results in a product that is another unit fraction:
However, adding, subtracting, or dividing two unit fractions produces a result that is generally not a unit fraction:
Unit fractions play an important role in modular arithmetic, as they may be used to reduce modular division to the calculation of greatest common divisors. Specifically, suppose that we wish to perform divisions by a value x, modulo y. In order for division by x to be well defined modulo y, x and y must be relatively prime. Then, by using the extended Euclidean algorithm for greatest common divisors we may find a and b such that
from which it follows that
Thus, to divide by x (modulo y) we need merely instead multiply by a.
Any positive rational number can be written as the sum of unit fractions, in multiple ways. For example,
The ancient Egyptians used sums of distinct unit fractions in their notation for more general rational numbers, and so such sums are often called Egyptian fractions. There is still interest today in analyzing the methods used by the ancients to choose among the possible representations for a fractional number, and to calculate with such representations. The topic of Egyptian fractions has also seen interest in modern number theory; for instance, the Erdős–Graham conjecture and the Erdős–Straus conjecture concern sums of unit fractions, as does the definition of Ore's harmonic numbers.
In geometric group theory, triangle groups are classified into Euclidean, spherical, and hyperbolic cases according to whether an associated sum of unit fractions is equal to one, greater than one, or less than one respectively.
Many well-known infinite series have terms that are unit fractions. These include:
The Hilbert matrix is the matrix with elements
It has the unusual property that all elements in its inverse matrix are integers. Similarly, Richardson (2001) defined a matrix with elements
where Fi denotes the ith Fibonacci number. He calls this matrix the Filbert matrix and it has the same property of having an integer inverse.
Two fractions are called adjacent if their difference is a unit fraction.
In a uniform distribution on a discrete space, all probabilities are equal unit fractions. Due to the principle of indifference, probabilities of this form arise frequently in statistical calculations. Additionally, Zipf's law states that, for many observed phenomena involving the selection of items from an ordered sequence, the probability that the nth item is selected is proportional to the unit fraction 1/n.
The energy levels of photons that can be absorbed or emitted by a hydrogen atom are, according to the Rydberg formula, proportional to the differences of two unit fractions. An explanation for this phenomenon is provided by the Bohr model, according to which the energy levels of electron orbitals in a hydrogen atom are inversely proportional to square unit fractions, and the energy of a photon is quantized to the difference between two levels.
Arthur Eddington argued that the fine structure constant was a unit fraction, first 1/136 then 1/137. This contention has been falsified, given that current estimates of the fine structure constant are (to 6 significant digits) 1/137.036.
is a mathematical operation that represents the total amount of objects together in a collection. It is signified by the plus sign (+). For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two apples together, which is a total of 5 apples. Therefore, 3 + 2 = 5. Besides counting fruits, addition can also represent combining other physical and abstract quantities using different kinds of objects: negative numbers, fractions, irrational numbers, vectors, decimals, functions, matrices and more.
Addition follows several important patterns. It is commutative, meaning that order does not matter, and it is associative, meaning that when one adds more than two numbers, order in which addition is performed does not matter (see Summation
). Repeated addition of 1 is the same as counting; addition of 0 does not change a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication. All of these rules can be proven, starting with the addition of natural numbers and generalizing up through the real numbers and beyond. General binary operations that continue these patterns are studied in abstract algebra.
Performing addition is one of the simplest numerical tasks. Addition of very small numbers is accessible to toddlers; the most basic task, 1 + 1, can be performed by infants as young as five months and even some animals. In primary education, students are taught to add numbers in the decimal system, starting with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancient abacus to the modern computer, where research on the most efficient implementations of addition continues to this day.
Addition is written using the plus sign "+" between the terms; that is, in infix notation. The result is expressed with an equals sign. For example,
There are also situations where addition is "understood" even though no symbol appears:
The sum of a series of related numbers can be expressed through capital sigma notation, which compactly denotes iteration. For example,
The numbers or the objects to be added in general addition are called the terms
, the addends
, or the summands
; this terminology carries over to the summation of multiple terms. This is to be distinguished from factors
, which are multiplied. Some authors call the first addend the augend
. In fact, during the Renaissance, many authors did not consider the first addend an "addend" at all. Today, due to the commutative property of addition, "augend" is rarely used, and both terms are generally called addends.
All of this terminology derives from Latin. "Addition" and "add" are English words derived from the Latin verb addere
, which is in turn a compound of ad
"to" and dare
"to give", from the Proto-Indo-European root "to give"; thus to add
is to give to
. Using the gerundive suffix -nd
results in "addend", "thing to be added". Likewise from augere
"to increase", one gets "augend", "thing to be increased".
"Sum" and "summand" derive from the Latin noun summa
"the highest, the top" and associated verb summare
. This is appropriate not only because the sum of two positive numbers is greater than either, but because it was once common to add upward, contrary to the modern practice of adding downward, so that a sum was literally higher than the addends. Addere
date back at least to Boethius, if not to earlier Roman writers such as Vitruvius and Frontinus; Boethius also used several other terms for the addition operation. The later Middle English terms "adden" and "adding" were popularized by Chaucer.
Addition is used to model countless physical processes. Even for the simple case of adding natural numbers, there are many possible interpretations and even more visual representations.
Possibly the most fundamental interpretation of addition lies in combining sets:
This interpretation is easy to visualize, with little danger of ambiguity. It is also useful in higher mathematics; for the rigorous definition it inspires, see Natural numbers
below. However, it is not obvious how one should extend this version of addition to include fractional numbers or negative numbers.
One possible fix is to consider collections of objects that can be easily divided, such as pies or, still better, segmented rods. Rather than just combining collections of segments, rods can be joined end-to-end, which illustrates another conception of addition: adding not the rods but the lengths of the rods.
A second interpretation of addition comes from extending an initial length by a given length:
The sum a
can be interpreted as a binary operation that combines a
, in an algebraic sense, or it can be interpreted as the addition of b
more units to a
. Under the latter interpretation, the parts of a sum a
play asymmetric roles, and the operation a
is viewed as applying the unary operation +b
. Instead of calling both a
addends, it is more appropriate to call a
in this case, since a
plays a passive role. The unary view is also useful when discussing subtraction, because each unary addition operation has an inverse unary subtraction operation, and vice versa.
Addition is commutative, meaning that one can reverse the terms in a sum left-to-right, and the result is the same as the last one. Symbolically, if a
are any two numbers, then
The fact that addition is commutative is known as the "commutative law of addition". This phrase suggests that there are other commutative laws: for example, there is a commutative law of multiplication. However, many binary operations are not commutative, such as subtraction and division, so it is misleading to speak of an unqualified "commutative law".
A somewhat subtler property of addition is associativity, which comes up when one tries to define repeated addition. Should the expression
be defined to mean (a
) + c
)? That addition is associative tells us that the choice of definition is irrelevant. For any three numbers a
, and c
, it is true that
For example, (1 + 2) + 3 = 3 + 3 = 6 = 1 + 5 = 1 + (2 + 3). Not all operations are associative, so in expressions with other operations like subtraction, it is important to specify the order of operations.
When adding zero to any number, the quantity does not change; zero is the identity element for addition, also known as the additive identity. In symbols, for any a
This law was first identified in Brahmagupta's Brahmasphutasiddhanta
in 628 AD, although he wrote it as three separate laws, depending on whether a
is negative, positive, or zero itself, and he used words rather than algebraic symbols. Later Indian mathematicians refined the concept; around the year 830, Mahavira wrote, "zero becomes the same as what is added to it", corresponding to the unary statement 0 + a
. In the 12th century, Bhaskara wrote, "In the addition of cipher, or subtraction of it, the quantity, positive or negative, remains the same", corresponding to the unary statement a
+ 0 = a
In the context of integers, addition of one also plays a special role: for any integer a
, the integer (a
+ 1) is the least integer greater than a
, also known as the successor of a
. Because of this succession, the value of some a
can also be seen as the
successor of a
, making addition iterated succession.
To numerically add physical quantities with units, they must first be expressed with common units. For example, if a measure of 5 feet is extended by 2 inches, the sum is 62 inches, since 60 inches is synonymous with 5 feet. On the other hand, it is usually meaningless to try to add 3 meters and 4 square meters, since those units are incomparable; this sort of consideration is fundamental in dimensional analysis.
Studies on mathematical development starting around the 1980s have exploited the phenomenon of habituation: infants look longer at situations that are unexpected. A seminal experiment by Karen Wynn in 1992 involving Mickey Mouse dolls manipulated behind a screen demonstrated that five-month-old infants expect
1 + 1 to be 2, and they are comparatively surprised when a physical situation seems to imply that 1 + 1 is either 1 or 3. This finding has since been affirmed by a variety of laboratories using different methodologies. Another 1992 experiment with older toddlers, between 18 to 35 months, exploited their development of motor control by allowing them to retrieve ping-pong balls from a box; the youngest responded well for small numbers, while older subjects were able to compute sums up to 5.
Even some nonhuman animals show a limited ability to add, particularly primates. In a 1995 experiment imitating Wynn's 1992 result (but using eggplants instead of dolls), rhesus macaques and cottontop tamarins performed similarly to human infants. More dramatically, after being taught the meanings of the Arabic numerals 0 through 4, one chimpanzee was able to compute the sum of two numerals without further training.
Typically, children first master counting. When given a problem that requires that two items and three items be combined, young children model the situation with physical objects, often fingers or a drawing, and then count the total. As they gain experience, they learn or discover the strategy of "counting-on": asked to find two plus three, children count three past two, saying "three, four, five
" (usually ticking off fingers), and arriving at five. This strategy seems almost universal; children can easily pick it up from peers or teachers. Most discover it independently. With additional experience, children learn to add more quickly by exploiting the commutativity of addition by counting up from the larger number, in this case starting with three and counting "four, five
." Eventually children begin to recall certain addition facts ("number bonds"), either through experience or rote memorization. Once some facts are committed to memory, children begin to derive unknown facts from known ones. For example, a child asked to add six and seven may know that 6+6=12 and then reason that 6+7 is one more, or 13. Such derived facts can be found very quickly and most elementary school student eventually rely on a mixture of memorized and derived facts to add fluently.
The prerequisite to addition in the decimal system is the fluent recall or derivation of the 100 single-digit "addition facts". One could memorize all the facts by rote, but pattern-based strategies are more enlightening and, for most people, more efficient:
As students grow older, they commit more facts to memory, and learn to derive other facts rapidly and fluently. Many students never commit all the facts to memory, but can still find any basic fact quickly.
The standard algorithm for adding multidigit numbers is to align the addends vertically and add the columns, starting from the ones column on the right. If a column exceeds ten, the extra digit is "carried" into the next column. An alternate strategy starts adding from the most significant digit on the left; this route makes carrying a little clumsier, but it is faster at getting a rough estimate of the sum. There are many other alternative methods.
Analog computers work directly with physical quantities, so their addition mechanisms depend on the form of the addends. A mechanical adder might represent two addends as the positions of sliding blocks, in which case they can be added with an averaging lever. If the addends are the rotation speeds of two shafts, they can be added with a differential. A hydraulic adder can add the pressures in two chambers by exploiting Newton's second law to balance forces on an assembly of pistons. The most common situation for a general-purpose analog computer is to add two voltages (referenced to ground); this can be accomplished roughly with a resistor network, but a better design exploits an operational amplifier.
Addition is also fundamental to the operation of digital computers, where the efficiency of addition, in particular the carry mechanism, is an important limitation to overall performance.
Blaise Pascal invented the mechanical calculator in 1642, it was the first operational adding machine. It made use of an ingenious gravity-assisted carry mechanism. It was the only operational mechanical calculator in the 17th century and the earliest automatic, digital computers. Pascal's calculator was limited by its carry mechanism which forced its wheels to only turn one way, so it could add but, to subtract, the operator had to use of the method of complements which required as many steps as an addition. Pascal was followed by Giovanni Poleni who built the second functional mechanical calculator in 1709, a calculating clock, which was made of wood and which could, once setup, multiply two numbers automatically.
Adders execute integer addition in electronic digital computers, usually using binary arithmetic. The simplest architecture is the ripple carry adder, which follows the standard multi-digit algorithm. One slight improvement is the carry skip design, again following human intuition; one does not perform all the carries in computing 999 + 1, but one bypasses the group of 9s and skips to the answer.
Since they compute digits one at a time, the above methods are too slow for most modern purposes. In modern digital computers, integer addition is typically the fastest arithmetic instruction, yet it has the largest impact on performance, since it underlies all the floating-point operations as well as such basic tasks as address generation during memory access and fetching instructions during branching. To increase speed, modern designs calculate digits in parallel; these schemes go by such names as carry select, carry lookahead, and the Ling pseudocarry. Almost all modern implementations are, in fact, hybrids of these last three designs.
Unlike addition on paper, addition on a computer often changes the addends. On the ancient abacus and adding board, both addends are destroyed, leaving only the sum. The influence of the abacus on mathematical thinking was strong enough that early Latin texts often claimed that in the process of adding "a number to a number", both numbers vanish. In modern times, the ADD instruction of a microprocessor replaces the augend with the sum but preserves the addend. In a high-level programming language, evaluating a
does not change either a
; if the goal is to replace a
with the sum this must be explicitly requested, typically with the statement a
. Some languages such as C or C++ allow this to be abbreviated as a
To prove the usual properties of addition, one must first define
addition for the context in question. Addition is first defined on the natural numbers. In set theory, addition is then extended to progressively larger sets that include the natural numbers: the integers, the rational numbers, and the real numbers. (In mathematics education, positive fractions are added before negative numbers are even considered; this is also the historical route)
There are two popular ways to define the sum of two natural numbers a
. If one defines natural numbers to be the cardinalities of finite sets, (the cardinality of a set is the number of elements in the set), then it is appropriate to define their sum as follows:
is the union of A
. An alternate version of this definition allows A
to possibly overlap and then takes their disjoint union, a mechanism that allows common elements to be separated out and therefore counted twice.
The other popular definition is recursive:
Again, there are minor variations upon this definition in the literature. Taken literally, the above definition is an application of the Recursion Theorem on the poset N
2. On the other hand, some sources prefer to use a restricted Recursion Theorem that applies only to the set of natural numbers. One then considers a
to be temporarily "fixed", applies recursion on b
to define a function "a
+ ", and pastes these unary operations for all a
together to form the full binary operation.
This recursive formulation of addition was developed by Dedekind as early as 1854, and he would expand upon it in the following decades. He proved the associative and commutative properties, among others, through mathematical induction; for examples of such inductive proofs, see Addition of natural numbers
The simplest conception of an integer is that it consists of an absolute value (which is a natural number) and a sign (generally either positive or negative). The integer zero is a special third case, being neither positive nor negative. The corresponding definition of addition must proceed by cases:
Although this definition can be useful for concrete problems, it is far too complicated to produce elegant general proofs; there are too many cases to consider.
A much more convenient conception of the integers is the Grothendieck group construction. The essential observation is that every integer can be expressed (not uniquely) as the difference of two natural numbers, so we may as well define
an integer as the difference of two natural numbers. Addition is then defined to be compatible with subtraction:
Addition of rational numbers can be computed using the least common denominator, but a conceptually simpler definition involves only integer addition and multiplication:
The commutativity and associativity of rational addition is an easy consequence of the laws of integer arithmetic. For a more rigorous and general discussion, see field of fractions
A common construction of the set of real numbers is the Dedekind completion of the set of rational numbers. A real number is defined to be a Dedekind cut of rationals: a non-empty set of rationals that is closed downward and has no greatest element. The sum of real numbers a
is defined element by element:
This definition was first published, in a slightly modified form, by Richard Dedekind in 1872. The commutativity and associativity of real addition are immediate; defining the real number 0 to be the set of negative rationals, it is easily seen to be the additive identity. Probably the trickiest part of this construction pertaining to addition is the definition of additive inverses.
Unfortunately, dealing with multiplication of Dedekind cuts is a case-by-case nightmare similar to the addition of signed integers. Another approach is the metric completion of the rational numbers. A real number is essentially defined to be the a limit of a Cauchy sequence of rationals, lim an
. Addition is defined term by term:
This definition was first published by Georg Cantor, also in 1872, although his formalism was slightly different. One must prove that this operation is well-defined, dealing with co-Cauchy sequences. Once that task is done, all the properties of real addition follow immediately from the properties of rational numbers. Furthermore, the other arithmetic operations, including multiplication, have straightforward, analogous definitions.
There are many binary operations that can be viewed as generalizations of the addition operation on the real numbers. The field of abstract algebra is centrally concerned with such generalized operations, and they also appear in set theory and category theory.
In linear algebra, a vector space is an algebraic structure that allows for adding any two vectors and for scaling vectors. A familiar vector space is the set of all ordered pairs of real numbers; the ordered pair (a
) is interpreted as a vector from the origin in the Euclidean plane to the point (a
) in the plane. The sum of two vectors is obtained by adding their individual coordinates:
This addition operation is central to classical mechanics, in which vectors are interpreted as forces.
In modular arithmetic, the set of integers modulo 12 has twelve elements; it inherits an addition operation from the integers that is central to musical set theory. The set of integers modulo 2 has just two elements; the addition operation it inherits is known in Boolean logic as the "exclusive or" function. In geometry, the sum of two angle measures is often taken to be their sum as real numbers modulo 2π. This amounts to an addition operation on the circle, which in turn generalizes to addition operations on many-dimensional tori.
The general theory of abstract algebra allows an "addition" operation to be any associative and commutative operation on a set. Basic algebraic structures with such an addition operation include commutative monoids and abelian groups.
A far-reaching generalization of addition of natural numbers is the addition of ordinal numbers and cardinal numbers in set theory. These give two different generalizations of addition of natural numbers to the transfinite. Unlike most addition operations, addition of ordinal numbers is not commutative. Addition of cardinal numbers, however, is a commutative operation closely related to the disjoint union operation.
In category theory, disjoint union is seen as a particular case of the coproduct operation, and general coproducts are perhaps the most abstract of all the generalizations of addition. Some coproducts, such as Direct sum
and Wedge sum
, are named to evoke their connection with addition.
can be thought of as a kind of addition—that is, the addition of an additive inverse. Subtraction is itself a sort of inverse to addition, in that adding x
and subtracting x
are inverse functions.
Given a set with an addition operation, one cannot always define a corresponding subtraction operation on that set; the set of natural numbers is a simple example. On the other hand, a subtraction operation uniquely determines an addition operation, an additive inverse operation, and an additive identity; for this reason, an additive group can be described as a set that is closed under subtraction.
can be thought of as repeated addition. If a single term x
appears in a sum n
times, then the sum is the product of n
. If n
is not a natural number, the product may still make sense; for example, multiplication by −1 yields the additive inverse of a number.
In the real and complex numbers, addition and multiplication can be interchanged by the exponential function:
This identity allows multiplication to be carried out by consulting a table of logarithms and computing addition by hand; it also enables multiplication on a slide rule. The formula is still a good first-order approximation in the broad context of Lie groups, where it relates multiplication of infinitesimal group elements with addition of vectors in the associated Lie algebra.
There are even more generalizations of multiplication than addition. In general, multiplication operations always distribute over addition; this requirement is formalized in the definition of a ring. In some contexts, such as the integers, distributivity over addition and the existence of a multiplicative identity is enough to uniquely determine the multiplication operation. The distributive property also provides information about addition; by expanding the product (1 + 1)(a
) in both ways, one concludes that addition is forced to be commutative. For this reason, ring addition is commutative in general.
is an arithmetic operation remotely related to addition. Since a
−1), division is right distributive over addition: (a
) / c
. However, division is not left distributive over addition; 1/ (2 + 2) is not the same as 1/2 + 1/2.
The maximum operation
)" is a binary operation similar to addition. In fact, if two nonnegative numbers a
are of different orders of magnitude, then their sum is approximately equal to their maximum. This approximation is extremely useful in the applications of mathematics, for example in truncating Taylor series. However, it presents a perpetual difficulty in numerical analysis, essentially since "max" is not invertible. If b
is much greater than a
, then a straightforward calculation of (a
) − b
can accumulate an unacceptable round-off error, perhaps even returning zero. See also Loss of significance
The approximation becomes exact in a kind of infinite limit; if either a
is an infinite cardinal number, their cardinal sum is exactly equal to the greater of the two. Accordingly, there is no subtraction operation for infinite cardinals.
Maximization is commutative and associative, like addition. Furthermore, since addition preserves the ordering of real numbers, addition distributes over "max" in the same way that multiplication distributes over addition:
For these reasons, in tropical geometry one replaces multiplication with addition and addition with maximization. In this context, addition is called "tropical multiplication", maximization is called "tropical addition", and the tropical "additive identity" is negative infinity. Some authors prefer to replace addition with minimization; then the additive identity is positive infinity.
Tying these observations together, tropical addition is approximately related to regular addition through the logarithm:
which becomes more accurate as the base of the logarithm increases. The approximation can be made exact by extracting a constant h
, named by analogy with Planck's constant from quantum mechanics, and taking the "classical limit" as h
tends to zero:
In this sense, the maximum operation is a dequantized
version of addition.
, also known as the successor operation, is the addition of 1 to a number.
describes the addition of arbitrarily many numbers, usually more than just two. It includes the idea of the sum of a single number, which is itself, and the empty sum, which is zero. An infinite summation is a delicate procedure known as a series.
a finite set is equivalent to summing 1 over the set.
is a kind of "summation" over a continuum, or more precisely and generally, over a differentiable manifold. Integration over a zero-dimensional manifold reduces to summation.
combine multiplication and summation; they are sums in which each term has a multiplier, usually a real or complex number. Linear combinations are especially useful in contexts where straightforward addition would violate some normalization rule, such as mixing of strategies in game theory or superposition of states in quantum mechanics.
is used to add two independent random variables defined by distribution functions. Its usual definition combines integration, subtraction, and multiplication. In general, convolution is useful as a kind of domain-side addition; by contrast, vector addition is a kind of range-side addition.
Mathematical analysis is a branch of mathematics that includes the theories of differentiation, integration, measure, limits, infinite series, and analytic functions. These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry. However, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).
Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy. Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids. In India, the 12th century mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolle's theorem. Division
Elementary arithmetic is the simplified portion of arithmetic which includes the operations of addition, subtraction, multiplication, and division.
Elementary arithmetic starts with the natural numbers and the written symbols (digits) which represent them. The process for combining a pair of these numbers with the four basic operations traditionally relies on memorized results for small values of numbers, including the contents of a multiplication table to assist with multiplication and division. Fraction
Quote notation is a number system for representing rational numbers which was designed to be attractive for use in computer architecture. In a typical computer architecture, the representation and manipulation of rational numbers is a complex topic. In Quote notation, arithmetic operations take particularly simple, consistent forms, and can produce exact answers with no roundoff error.
Quote notation’s arithmetic algorithms work with a typical right-to-left direction, in which the addition, subtraction, and multiplication algorithms have the same complexity for natural numbers, and division is easier than a typical division algorithm. Mathematics
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers ai are called the coefficients or terms of the continued fraction.
Continued fractions have a number of remarkable properties related to the Euclidean algorithm for integers or real numbers. Every rational number pq has two closely related expressions as a finite continued fraction, whose coefficients ai can be determined by applying the Euclidean algorithm to (p, q). The numerical value of an infinite continued fraction will be irrational; it is defined from its infinite sequence of integers as the limit of a sequence of values for finite continued fractions. Each finite continued fraction of the sequence is obtained by using a finite prefix of the infinite continued fraction's defining sequence of integers. Moreover, every irrational number α is the value of a unique infinite continued fraction, whose coefficients can be found using the non-terminating version of the Euclidean algorithm applied to the incommensurable values α and 1. This way of expressing real numbers (rational and irrational) is called their continued fraction representation.