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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.
and
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* ÷ *b* = (*ca*) ÷ (*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
are *minuend* (*c*) − *subtrahend* (*b*) = *difference* (*a*).
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:
From *c*, 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:
from minuend
where each *s*_{i} and *m*_{i} is a digit, proceeds by writing down *m*_{1} − *s*_{1}, *m*_{2} − *s*_{2}, and so forth, as long as *s*_{i} does not exceed *m*_{i}. Otherwise, *m*_{i} 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 *m*_{i+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 *s*_{i+1} by one.
**Example:** 704 − 512. The minuend is 704, the subtrahend is 512. The minuend digits are *m*_{3} = 7, *m*_{2} = 0 and *m*_{1} = 4. The subtrahend digits are *s*_{3} = 5, *s*_{2} = 1 and *s*_{1} = 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.

Addition

+

**Subtraction**

**−**

Multiplication

×

Division

÷

Addition

+

Multiplication

×

Division

÷

0, 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".

1, one. Applied to a single item. For example, here is one stick: I

2, two. Applied to a pair of items. Here are two sticks: I I

3, three. Applied to three items. Here are three sticks: I I I

4, four. Applied to four items. Here are four sticks: I I I I

5, five. Applied to five items. Here are five sticks: I I I I I

6, six. Applied to six items. Here are six sticks: I I I I I I

7, seven. Applied to seven items. Here are seven sticks: I I I I I I I

8, eight. Applied to eight items. Here are eight sticks: I I I I I I I I

9, 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

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

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

Addition

+

Subtraction

−

Multiplication

×

Division

÷

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*/*b*; 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
(*er*, "[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 1/2*k* (for *k* = 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*/*y* by the expansion
where 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: and
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 instead of , 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 , replacing by , 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* times *c* equals *a*, written:
where *b* is not zero, then *a* divided by *b* equals *c*, written:
For instance,
since
In the expression a ÷ b = c, *a* is called the **dividend** or **numerator**, *b* the **divisor** or **denominator** 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* and *b*. *Quotative* 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* and *c*.
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* is written
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 *denominator*), 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* : *b*. 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 or 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* and *r* such that *a* = *qd* + *r* 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*/*q* and *r*/*s* by
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*/*b* = *c* if and only if *a* = *cb* and *b* ≠ 0.
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* and *s* 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* or *slash-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* and *B*.
In abstract algebras such as matrix algebras and quaternion algebras, fractions such as are typically defined as or where is presumed an invertible element (i.e., there exists a multiplicative inverse such that where is the multiplicative identity). In an integral domain where such elements may not exist, *division* can still be performed on equations of the form or 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.

Addition

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**Division**

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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
or equivalently
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 *F*_{i} denotes the *i*th 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 *n*th 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.

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

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**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.

**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.

**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.

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 a_{i} 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 *p*/*q* has two closely related expressions as a finite continued fraction, whose coefficients a_{i} 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*.

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