In math the answer to the equation -5-10 is -15. Thanks for texting!
In mathematics, an equation
is a formula of the form A
, where A
are expressions containing one or several variables called unknowns
, and "=" denotes the equality binary relation. Although written in the form of proposition, an equation is not a statement, but a problem consisting in finding the values, called solutions
, that, when substituted to the unknowns, yields equal values of expressions A
. For example, 2 is the unique solution
of the equation x
+ 2 = 4, in which the unknown
. Historically, equations arose from the mathematical discipline of algebra, but later become ubiquitous. An equation may not be confused with identities which are presented with the same notation but have a different semantic: for example 2 + 2 = 4 and x
are identities (which implies they are necessarily true) in arithmetic, and do not constitute any values-finding problem, even if include variables.
may also refer to a relation between some variables that is expressed by the equality of some expressions of their values. For example the equation
of the unit circle is x
2 + y
2 = 1, which means that a point belongs to the circle if and only if its coordinates are related by this equation. Most physical laws are expressed by equations. One of the most popular ones is Einstein's equation 2mc
The = symbol was invented by Robert Recorde (1510–1558), who considered that nothing could be more equal than parallel straight lines with the same length.
Centuries ago, the word "equation" frequently meant what we now usually call "correction" or "adjustment".]
[ This meaning is still occasionally found, especially in names which were originally given long ago. The "equation of time", for example, is a correction that must be applied to the reading of a sundial in order to obtain mean time, as would be shown by a clock.
Equations often contain other variables than the unknowns. These other variables that are supposed to be known
are usually called constants
. Usually, the unknowns are denoted by letters at the end of the alphabet, x
, …, while coefficients are denoted by letters at the beginning, a
, … . For example, the general quadratic equation is usually written ax
2 + bx
= 0. The process of finding the solutions, or in case of parameters, expressing the unknowns in terms of the parameters is called solving the equation. Such expressions of the solutions in terms of the parameters are also called solutions
A system of equations is a set of simultaneous equations
, usually in several unknowns, for which the common solutions are sought. Thus a solution to the system
is a set of one value for each unknown, which is a solution to each equation in the system. For example, the system
has the unique solution x
= −1, y
The analogy often presented is a weighing scale, balance, seesaw, or the like.
Each side of the balance corresponds to each side of the equation. Different quantities can be placed on each side, if they are equal the balance corresponds to an equality (equation), if not then an inequality.
In the illustration, x
are all different quantities (in this case real numbers) represented as circular weights, each of x
has a different weight. Addition corresponds to adding weight, subtraction corresponds to removing weight from what is already placed on. The total weight on each side is the same.
Equations can be classified according to the types of operations and quantities involved. Important types include:
is a statement resembling an equation which is true for all possible values of the variable(s) it contains. Many identities are known, especially in trigonometry. Probably the best known example is:
, which is true for all values of θ
In the process of solving an equation, it is often useful to combine it with an identity to produce an equation which is more easily soluble. For example, to solve the equation:
use the identity:
so the above equation becomes:
Two equations or two systems of equations are equivalent
if they have the same set of solutions. The following operations transform an equation or a system into an equivalent one:
If some functions is be applied to both sides of an equation, the resulting equation has the solutions of the initial equation among its solutions, but may have further solutions called extraneous solutions. If the function is not defined everywhere, (like 1/x
that is not defined for x
= 0) some solutions may be lost. Thus caution must be exercised when applying such a transformation to an equation. For example, the equation
has the solution
Raising both sides to the exponent of 2 (which means, applying the function
to both sides of the equation) changes our equation into
, which not only has the previous solution but also introduces the extraneous solution,
Above transformations are the basis of most elementary methods for equation solving and some less elementary ones, like Gaussian elimination
In mathematics, a quadratic equation
is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the form
represents a variable or an unknown, and a
, and c
are constants with a
not equal to 0. (If a
= 0, the equation is a linear equation.) Dividing the above equation by a
gives the simplified monic form
The constants a
, and c
are called, respectively, the quadratic coefficient, the linear coefficient and the constant or free term. Quadratic equations can be solved by factoring (or "factorising" in British English), completing the square, using the quadratic formula, and graphing.
A quadratic equation with real or complex coefficients has two solutions, called roots
. These two solutions may or may not be distinct, and they may or may not be real.
A quadratic equation in
form can always be expressed as a product
. In many cases, it is possible, by simple inspection, to determine values of p
that will allow writing the left side of the quadratic equation in factored form. If the quadratic equation is written in factored form, then the "Zero Factor Property" states that the quadratic equation is satisfied if px
= 0 or rx
= 0. Solving these two linear equations provides the roots of the quadratic.
For most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed.:202–207 If one is given a quadratic equation in monic form,
, one would seek to find two numbers that add up to b
and whose product is c
("Vieta's Rule"). The more general case where a
does not equal 1 can require a considerable effort in trial and error guess-and-check, assuming that it can be factored at all by inspection.
Except for special cases such as where b
= 0 or c
= 0, factoring by inspection only works for quadratic equations that have rational roots. This means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection.:207
Completing the square makes use of the algebraic identity
which represents a well-defined algorithm that can be used to solve any quadratic equation.:207 Starting with a quadratic equation in standard form,
We illustrate use of this algorithm by solving
Completing the square can be used to derive a general formula for solving quadratic equations, the quadratic formula.
Dividing the quadratic equation
, which is allowed because a
is non-zero, gives
The quadratic equation is now in a form to which the method of completing the square can be applied. To "complete the square" is to add a constant to both sides of the equation such that the left hand side becomes a complete square:
The right side can be written as a single fraction with common denominator
. This gives
Taking the square root of both sides yields
The plus-minus symbol "±" indicates that both
are solutions of the quadratic equation.
Some sources, particularly older ones, use alternative parameterizations such as
, where b
has a magnitude one half of the more common one. These result in slightly different forms for the roots and discriminant, but are otherwise equivalent.
In the above formula, the expression underneath the square root sign is called the discriminant
of the quadratic equation, and is often represented using an upper case D
or an upper case Greek delta:
A quadratic equation with real
coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots. There are three cases:
Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative.
is the quadratic function. The graph of any quadratic function has the same general shape, which is called a parabola. The location and size of the parabola, and how it opens, depends on the values of a
, and c
. As shown in Figure 1, if a
> 0, the parabola has a minimum point and opens upward. If a
< 0, the parabola has a maximum point and opens downward. The extreme point of the parabola, whether minimum or maximum, corresponds to its vertex. The x-coordinate
of the vertex will be located at
, and the y-coordinate
of the vertex may be found by substituting this x-value
into the function. The y-intercept
is located at the point (0,c
The solutions of the quadratic equation
correspond to the roots of the function
, since they are the values of x
. As shown in Figure 2, if a
, and c
are real numbers and the domain of f
is the set of real numbers, then the roots of f
are exactly the x
-coordinates of the points where the graph touches the x
-axis. As shown in Figure 3, if the discriminant is positive, the graph touches the -axisx
at two points; if zero, the graph touches at one point; and if negative, the graph does not touch the x
is a factor of the polynomial
if and only if r
is a root of the quadratic equation
It follows from the quadratic formula that
In the special case
where the quadratic has only one distinct root (i.e.
the discriminant is zero), the quadratic polynomial can be factored as
For most of the twentieth century, graphing was rarely mentioned as a method for solving quadratic equations in high school or college algebra texts. Students learned to solve quadratic equations by factoring, completing the square, and applying the quadratic formula. Recently, graphing calculators have become more common in schools and graphical methods have started to appear in textbooks, but they are generally not highly emphasized. Graphical methods can only produce an approximate solution in the case of irrational roots.
Being able to use a graphing calculator to solve a quadratic equation requires the ability to produce a graph of y
), the ability to scale the graph appropriately to the dimensions of the graphing surface, and the recognition that when f
) = 0, x
is a solution to the equation. The skills required to solve a quadratic equation on a calculator are in fact applicable to finding the real roots of any arbitrary function.
Since an arbitrary function may cross the x-axis at multiple points, graphing calculators generally require one to identify the desired root by positioning a cursor at a "guessed" value for the root. (Some graphing calculators require bracketing the root on both sides of the zero.) The calculator then proceeds, by an iterative algorithm, to refine the estimated position of the root to the limit of calculator accuracy.
Although the quadratic formula provides what in principle should be an exact solution, it does not, from a numerical analysis standpoint, provide a completely stable method for evaluating the roots of a quadratic equation. If the two roots of the quadratic equation vary greatly in absolute magnitude, b
will be very close in magnitude to
, and the subtraction of two nearly equal numbers will cause loss of significance or catastrophic cancellation. A second form of cancellation can occur between the terms
of the discriminant, which can lead to loss of up to half of correct significant figures.
See the Floating-point implementation section for a description of how loss of significance issues would be handled in a carefully written computer program.
Babylonian mathematicians, as early as 2000 BC (displayed on Old Babylonian clay tablets) could solve problems relating the areas and sides of rectangles. In modern notation, the problems typically involved solving a pair of simultaneous equations of the form:
which are equivalent to the equation::86
The steps given by Babylonian scribes for solving the above rectangle problem were as follows:
Note that step (5) is essentially equivalent to calculating
There is evidence dating this algorithm as far back as the Ur III dynasty.
In the Sulba Sutras in ancient India circa 8th century BC quadratic equations of the form
were explored using geometric methods. Babylonian mathematicians from circa 400 BC and Chinese mathematicians from circa 200 BC used geometric methods of dissection to solve quadratic equations with positive roots, but do not appear to have had a general formula.
Euclid, the Greek mathematician, produced a more abstract geometrical method around 300 BC. Pythagoras and Euclid used a strictly geometric approach, and found a general procedure to solve the quadratic equation. In his work Arithmetica
, the Greek mathematician Diophantus solved the quadratic equation, but giving only one root, even when both roots were positive.
In 628 AD, Brahmagupta, an Indian mathematician, gave the first explicit (although still not completely general) solution of the quadratic equation
This is equivalent to:
The Bakhshali Manuscript
written in India in the 7th century AD contained an algebraic formula for solving quadratic equations, as well as quadratic indeterminate equations (originally of type
Muhammad ibn Musa al-Khwarizmi (Persia, 9th century), inspired by Brahmagupta, developed a set of formulas that worked for positive solutions. Al-Khwarizmi goes further in providing a full solution to the general quadratic equation, accepting one or two numerical answers for every quadratic equation, while providing geometric proofs in the process. He also described the method of completing the square and recognized that the discriminant must be positive,:230 which was proven by his contemporary 'Abd al-Hamīd ibn Turk (Central Asia, 9th century) who gave geometric figures to prove that if the discriminant is negative, a quadratic equation has no solution.:234 While al-Khwarizmi himself did not accept negative solutions, later Islamic mathematicians that succeeded him accepted negative solutions,:191 as well as irrational numbers as solutions. Abū Kāmil Shujā ibn Aslam (Egypt, 10th century) in particular was the first to accept irrational numbers (often in the form of a square root, cube root or fourth root) as solutions to quadratic equations or as coefficients in an equation.
The Indian mathematician Sridhara, who flourished in the 9th and 10th centuries AC provided the modern solution of the quadratic equation.
The Jewish mathematician Abraham bar Hiyya Ha-Nasi (12th century, Spain) authored the first European book to include the full solution to the general quadratic equation. His solution was largely based on Al-Khwarizmi's work. The writing of the Chinese mathematician Yang Hui (1238-1298 AD) represents the first in which quadratic equations with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi.
By 1545 Gerolamo Cardano compiled the works related to the quadratic equations. The quadratic formula covering all cases was first obtained by Simon Stevin in 1594. In 1637 René Descartes published La Géométrie
containing the quadratic formula in the form we know today. The first appearance of the general solution in the modern mathematical literature appeared in a 1896 paper by Henry Heaton.
A number of alternative derivations of the quadratic formula can be found in the literature which either (a) are simpler than the standard completing the square method, (b) represent interesting applications of other frequently used techniques in algebra, or (c) offer insight into other areas of mathematics.
The great majority of algebra texts published over the last several decades teach completing the square using the sequence presented earlier: (1) divide each side by a
, (2) rearrange, (3) then add the square of one-half of b/a
However, as pointed out by Larry Hoehn in 1975, completing the square can accomplished by a different sequence that leads to a simpler sequence of intermediate terms: (1) multiply each side by 4a
, (2) rearrange, (3) then add
In other words, the quadratic formula can be derived as follows:
This actually represents an ancient derivation of the quadratic formula, and was known to the Hindus at least as far back as 1025 A.D. Compared with the derivation in standard usage, this alternate derivation is shorter, involves fewer computations with literal coefficients, avoids fractions until the last step, has simpler expressions, and uses simpler math. As Hoehn states, "it is easier 'to add the square of b'
than it is 'to add the square of half the coefficient of the x
An alternative way of deriving the quadratic formula is via the method of Lagrange resolvents, which is an early part of Galois theory. This method can be generalized to give the roots of cubic polynomials and quartic polynomials, and leads to Galois theory, which allows one to understand the solution of algebraic equations of any degree in terms of the symmetry group of their roots, the Galois group.
This approach focuses on the roots
more than on rearranging the original equation. Given a monic quadratic polynomial
assume that it factors as
Since the order of multiplication does not matter, one can switch
and the values of p
will not change: one says that p
are symmetric polynomials in
. In fact, they are the elementary symmetric polynomials – any symmetric polynomial in
can be expressed in terms of
The Galois theory approach to analyzing and solving polynomials is: given the coefficients of a polynomial, which are symmetric functions in the roots, can one "break the symmetry" and recover the roots? Thus solving a polynomial of degree n
is related to the ways of rearranging ("permuting") n
terms, which is called the symmetric group on n
letters, and denoted
For the quadratic polynomial, the only way to rearrange two terms is to swap them ("transpose" them), and thus solving a quadratic polynomial is simple.
To find the roots
consider their sum and difference:
These are called the Lagrange resolvents
of the polynomial; notice that one of these depends on the order of the roots, which is the key point. One can recover the roots from the resolvents by inverting the above equations:
Thus, solving for the resolvents gives the original roots.
Formally, the resolvents are called the discrete Fourier transform (DFT) of order 2, and the transform can be expressed by the matrix
with inverse matrix
The transform matrix is also called the DFT matrix or Vandermonde matrix.
is a symmetric function in
so it can be expressed in terms of p
and in fact
as noted above. But
is not symmetric, since switching
(formally, this is termed a group action of the symmetric group of the roots). Since
is not symmetric, it cannot be expressed in terms of the polynomials p
, as these are symmetric in the roots and thus so is any polynomial expression involving them. However, changing the order of the roots only changes
by a factor of
and thus the square is
symmetric in the roots, and thus expressible in terms of p
Using the equation
If one takes the positive root, breaking symmetry, one obtains:
Thus the roots are
which is the quadratic formula. Substituting
yields the usual form for when a quadratic is not monic. The resolvents can be recognized as
being the vertex, and
is the discriminant (of a monic polynomial).
A similar but more complicated method works for cubic equations, where one has three resolvents and a quadratic equation (the "resolving polynomial") relating
which one can solve by the quadratic equation, and similarly for a quartic (degree 4) equation, whose resolving polynomial is a cubic, which can in turn be solved. However, the same method for a quintic equation yields a polynomial of degree 24, which does not simplify the problem, and in fact solutions to quintic equations in general cannot be expressed using only roots.
A careful floating point computer implementation combines several strategies to produce a robust result. Assuming the discriminant, , is positive and b
is nonzero, the computation would be as follows:
Here sgn denotes the sign function, where sgn(b) is 1 if b is positive and −1 if b is negative. This avoids cancellation problems between b and the square root of the discriminant by ensuring that only numbers of the same sign are added.
To illustrate the instability of the standard quadratic formula versus
this variant formula, consider a quadratic equation with roots
. To sixteen significant figures, roughly corresponding to double-precision accuracy on a computer, the monic quadratic equation with these roots may be written as:
Using the standard quadratic formula and maintaining sixteen significant figures at each step, the standard quadratic formula yields
Note how cancellation has resulted in
being computed to only eight significant digits of accuracy. The variant formula presented here, however, yields the following:
Note the retention of all significant digits for
Note that while the above formulation avoids catastrophic cancellation between b
, there remains a form of cancellation between the terms
of the discriminant, which can still lead to loss of up to half of correct significant figures. The discriminant
needs to be computed in arithmetic of twice the precision of the result to avoid this (e.g. quad precision if the final result is to be accurate to full double precision). This can be in the form of a fused multiply-add operation.
To illustrate this, consider the following quadratic equation, adapted from Kahan (2004):
This equation has
and has roots
However, when computed using IEEE 754 double-precision arithmetic corresponding to 15 to 17 significant digits of accuracy,
is rounded to 0.0, and the computed roots are
which are both false after the eighth significant digit. This is despite the fact that superficially, the problem seems to require only eleven significant digits of accuracy for its solution.
Vieta's formulas give a simple relation between the roots of a polynomial and its coefficients. In the case of the quadratic polynomial, they take the following form:
These results follow immediately from the relation:
which can be compared term by term with
The first formula above yields a convenient expression when graphing a quadratic function. Since the graph is symmetric with respect to a vertical line through the vertex, when there are two real roots the vertex’s x-coordinate is located at the average of the roots (or intercepts). Thus the x-coordinate of the vertex is given by the expression
The y-coordinate can be obtained by substituting the above result into the given quadratic equation, giving
As a practical matter, Vieta's formulas provide a useful method for finding the roots of a quadratic in the case where one root is much smaller than the other. If |x 2
| << |x 1
|, then x 1
+ x 2
≈ x 1
, and we have the estimate:
The second Vieta's formula then provides:
These formulas are much easier to evaluate than the quadratic formula under the condition of one large and one small root, because the quadratic formula evaluates the small root as the difference of two very nearly equal numbers (the case of large b
), which causes round-off error in a numerical evaluation. Figure 5 shows the difference between (i) a direct evaluation using the quadratic formula (accurate when the roots are near each other in value) and (ii) an evaluation based upon the above approximation of Vieta's formulas (accurate when the roots are widely spaced). As the linear coefficient b
increases, initially the quadratic formula is accurate, and the approximate formula improves in accuracy, leading to a smaller difference between the methods as b
increases. However, at some point the quadratic formula begins to lose accuracy because of round off error, while the approximate method continues to improve. Consequently the difference between the methods begins to increase as the quadratic formula becomes worse and worse.
This situation arises commonly in amplifier design, where widely separated roots are desired to ensure a stable operation (see step response).
In the days before calculators, people would use mathematical tables—lists of numbers showing the results of calculation with varying arguments—to simplify and speed up computation. Tables of logarithms and trigonometric functions were common in math and science textbooks. Specialized tables were published for applications such as astronomy, celestial navigation and statistics. Methods of numerical approximation existed, called prosthaphaeresis, that offered shortcuts around time-consuming operations such as multiplication and taking powers and roots. Astronomers, especially, were concerned with methods that could speed up the long series of computations involved in celestial mechanics calculations.
It is within this context that we may understand the development of means of solving quadratic equations by the aid of trigonometric substitution. Consider the following alternate form of the quadratic equation,
where the sign of the ± symbol is chosen so that a
may both be positive. By substituting
and then multiplying through by
Introducing functions of
and rearranging, we obtain
where the subscripts n
correspond, respectively, to the use of a negative or positive sign in equation 
. Substituting the two values of
found from equations 
gives the required roots of 
. Complex roots occur in the solution based on equation 
if the absolute value of
exceeds unity. The amount of effort involved in solving quadratic equations using this mixed trigonometric and logarithmic table look-up strategy was two-thirds the effort using logarithmic tables alone. Calculating complex roots would require using a different trigonometric form.
The quadratic equation may be solved geometrically in a number of ways. One way is via Lill's method. The three coefficients a
are drawn with right angles between them as in SA, AB, and BC in Figure 6. A circle is drawn with the start and end point SC as a diameter. If this cuts the middle line AB of the three then the equation has a solution, and the solutions are given by negative of the distance along this line from A divided by the first coefficient a
or SA. If a
is 1 the coefficients may be read off directly. Thus the solutions in the diagram are −AX1/SA and −AX2/SA.
The formula and its derivation remain correct if the coefficients a
are complex numbers, or more generally members of any field whose characteristic is not 2. (In a field of characteristic 2, the element 2a
is zero and it is impossible to divide by it.)
in the formula should be understood as "either of the two elements whose square is
, if such elements exist". In some fields, some elements have no square roots and some have two; only zero has just one square root, except in fields of characteristic 2. Note that even if a field does not contain a square root of some number, there is always a quadratic extension field which does, so the quadratic formula will always make sense as a formula in that extension field.
In a field of characteristic 2, the quadratic formula, which relies on 2 being a unit, does not hold. Consider the monic quadratic polynomial
over a field of characteristic 2. If
, then the solution reduces to extracting a square root, so the solution is
and note that there is only one root since
See quadratic residue for more information about extracting square roots in finite fields.
In the case that
, there are two distinct roots, but if the polynomial is irreducible, they cannot be expressed in terms of square roots of numbers in the coefficient field. Instead, define the 2-root R
) of c
to be a root of the polynomial
an element of the splitting field of that polynomial. One verifies that R
) + 1 is also a root. In terms of the 2-root operation, the two roots of the (non-monic) quadratic
For example, let a
denote a multiplicative generator of the group of units of
the Galois field of order four (thus a
+ 1 are roots of
+ 1 is the unique solution of the quadratic equation
On the other hand, the polynomial
is irreducible over
but it splits over
where it has the two roots ab
, where b
is a root of
This is a special case of Artin-Schreier theory.
In number theory, Euler's totient
or phi function
) is an arithmetic function that counts the totatives of n
, that is, the positive integers less than or equal to n
that are relatively prime to n
. Thus if n
is a positive integer, then φ(n
) is the number of integers k
in the range 1 ≤ k
for which gcd(n
) = 1. The totient function is a multiplicative function, meaning that if two numbers m
are relatively prime (to each other), then φ(mn
) = φ(m
For example let n
= 9. Then gcd(9, 3) = gcd(9, 6) = 3 and gcd(9, 9) = 9. The other six numbers in the range 1 ≤ k
≤ 9, that is, 1, 2, 4, 5, 7 and 8, are relatively prime to 9. Therefore, φ(9) = 6. As another example, φ(1) = 1 since gcd(1, 1) = 1.
The totient function is important mainly because it gives the order of the n
multiplicative group of integers modulo (the group of units of the ring
). See Euler's theorem.
The totient function also plays a key role in the definition of the RSA encryption system.
Leonhard Euler introduced the function in 1760. The standard notation φ(n
) is from Gauss' 1801 treatise Disquisitiones Arithmeticae
. Thus, it is usually called Euler's phi function
or simply the phi function
In 1879 J. J. Sylvester coined the term totient
for this function, so it is also referred to as the totient function
, the Euler totient
, or Euler's totient
. Jordan's totient is a generalization of Euler's.
is defined as n
), i.e., the number of positive integers less than or equal to n
that are divisible by at least one prime that also divides n
There are several formulae for the totient.
where the product is over the distinct prime numbers dividing n. (The notation is described in the article Arithmetical function.)
The proof of Euler's product formula depends on two important facts.
This means that if gcd(m
) = 1, then φ(mn
) = φ(m
(Sketch of proof: let A
be the sets of residue classes modulo-and-coprime-to m
respectively; then there is a bijection between A
, by the Chinese remainder theorem.)
That is, if p
is prime and k
≥ 1 then
is a prime number the only possible values of gcd(pk
) are 1, p
2, ..., pk
, and the only way for gcd(pk
) to not equal 1 is for m
to be a multiple of p
. The multiples of p
that are less than or equal to pk
, ..., p
, and there are p
of them. Therefore the other pk
numbers are all relatively prime to pk
Proof of the formula:
The fundamental theorem of arithmetic states that if n
> 1 there is a unique expression for n
< ... < pr
are prime numbers and each ki
≥ 1. (The case n
= 1 corresponds to the empty product.)
Repeatedly using the multiplicative property of φ and the formula for φ(pk
This is Euler's product formula.
In words, this says that the distinct prime factors of 36 are 2 and 3; half of the thirty-six integers from 1 to 36 are divisible by 2, leaving eighteen; a third of those are divisible by 3, leaving twelve that are coprime to 36. And indeed there are twelve: 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, and 35.
The totient is the discrete Fourier transform of the gcd, evaluated at 1: (Schramm (2008))
The real part of this formula is
Note that unlike the other two formulae (the Euler product and the divisor sum) this one does not require knowing the factors of n
. However, it does involve the calculation of the greatest common divisor of n
and every positive integer less than n
, which would suffice to provide the factorization anyway.
Euler's classical formula
where the sum is over all positive divisors d
, can be proven in several ways. (see Arithmetical function for notational conventions.)
One way is to note that φ(d
) is also equal to the number of possible generators of the cyclic group Cd
; specifically, if Cd
>, then gk
is a generator for every k
coprime to d
. Since every element of Cn
generates a cyclic subgroup, and all φ(d
) subgroups of Cd
are generated by some element of Cn
, the formula follows. In the article Root of unity Euler's formula is derived by using this argument in the special case of the multiplicative group of the n
th roots of unity.
This formula can also be derived in a more concrete manner. Let n
= 20 and consider the fractions between 0 and 1 with denominator 20:
Put them into lowest terms:
First note that all the divisors of 20 are denominators. And second, note that there are 20 fractions.
Which fractions have 20 as denominator? The ones whose numerators are relatively prime to 20
By definition this is φ(20) fractions.
Similarly, there are φ(10) = 4 fractions with denominator 10
φ(5) = 4 fractions with denominator 5
and so on. Since the same argument works for any number, not just 20, the formula is established.
Möbius inversion gives
where μ is the Möbius function.
This formula may also be derived from the product formula by multiplying out
The first 99 values (sequence in OEIS) are shown in the table and graph below:
The top line in the graph, y
− 1, is a true upper bound. It is attained whenever n
The lower line, y
which connects the points for n
= 30, 60, and 90 is misleading. If the plot were continued, there would be points below it.
(Examples: for n
= 210 = 7×30, φ(n
) ≈ 0.229 n
; for n
= 2310 = 11×210 φ(n
) ≈ 0.208 n
; and for n
= 30030 = 13×2310 φ(n
) ≈ 0.192 n
In fact, there is no lower bound that is a straight line of positive slope; no matter how gentle the slope of a line is, there will eventually be points of the plot below the line.
This states that if a
are relatively prime then
This follows from Lagrange's theorem and the fact that φ(n
) is the order of the n
multiplicative group of integers modulo .
is the Euler constant).
> 1 is a positive integer and ω(m
) is the number of distinct prime factors of m
) is a standard abbreviation for gcd(a
In 1965 P. Kesava Menon proved
) = σn
is the number of divisors of n
Schneider found a pair of identities connecting the totient function, the golden ratio and the Möbius function
. In this section
is the totient function, and
is the golden ratio.
Subtracting them gives
Applying the exponential function to both sides of the preceding identity yields an infinite product formula for Euler's constant
The proof is based on the formulae
The Dirichlet series for φ(n
) may be written in terms of the Riemann zeta function as:
The Lambert series generating function is
which converges for |q
| < 1.
Both of these are proved by elementary series manipulations and the formulae for φ(n
In the words of Hardy & Wright, φ(n
) is "always ‘nearly n
but as n
goes to infinity, for all δ > 0
These two formulae can be proved by using little more than the formulae for φ(n
) and the divisor sum function σ(n
In fact, during the proof of the second formula, the inequality
true for n
> 1, is proven.
We also have
Here γ is Euler's constant, γ = 0.577215665..., eγ = 1.7810724..., e−γ = 0.56145948... .
Proving this doesn't quite require the prime number theorem. Since log log (n
) goes to infinity, this formula shows that
In fact, more is true.
Concerning the second inequality, Ribenboim says "The method of proof is interesting, in that the inequality is shown first under the assumption that the Riemann hypothesis is true, secondly under the contrary assumption."
For the average order, we have
due to Arnold Walfisz, its proof exploiting estimates on exponential sums due to I. M. Vinogradov and N.M. Korobov (this is currently the best known estimate of this type). The "Big O" stands for a quantity that is bounded by a constant times the function of "n" inside the parentheses (which is small compared to n
This result can be used to prove that the probability of two randomly chosen numbers being relatively prime is
In 1950 Somayajulu proved
In 1954 Schinzel and Sierpiński strengthened this, proving that the set
is dense in the positive real numbers. They also proved that the set
is dense in the interval (0, 1).
A totient number
is a value of Euler's totient function: that is, an m
for which there is at least one x
for which φ(x
) = m
. The valency
of a totient number m
is the number of solutions to this equation. A nontotient
is a natural number which is not a totient number: there are infinitely many nontotients, and indeed every odd number has a multiple which is a nontotient.
The number of totient numbers up to a given linit x
for a constant C
= 0.8178146... .
If counted accordingly to multiplicity, the number of totient numbers up to a given limit x
where the error term R
is of order at most
for any positive k
It is known that the multiplicity of m
δ infinitely often for any δ < 0.55655.
Ford (1999) proved that for every integer k
≥ 2 there is a totient number m
of multiplicity k
: that is, for which the equation φ(x
) = m
has exactly k
solutions; this result had previously been conjectured by Wacław Sierpiński, and it had been obtained as a consequence of Schinzel's hypothesis H. Indeed, each multiplicity that occurs, does so infinitely often.
However, no number m
is known with multiplicity k
= 1. Carmichael's totient function conjecture is the statement that there is no such m
In the last section of the Disquisitiones
Gauss proves that a regular n
-gon can be constructed with straightedge and compass if φ(n
) is a power of 2. If n
is a power of an odd prime number the formula for the totient says its totient can be a power of two only if a) n
is a first power and b) n
− 1 is a power of 2. The primes that are one more than a power of 2 are called Fermat primes, and only five are known: 3, 5, 17, 257, and 65537. Fermat and Gauss knew of these. Nobody has been able to prove whether there are any more.
Thus, a regular n
-gon has a straightedge-and-compass construction if n
is a product of distinct Fermat primes and any power of 2.
The first few such n
are 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, ... . (sequence in OEIS)
Setting up an RSA system involves choosing large prime numbers p
, computing n
), and finding two numbers e
such that ed
≡ 1 (mod k
). The numbers n
(the "encryption key") are released to the public, and d
(the "decryption key") is kept secure.
A message, represented by an integer m
, where 0 < m
, is encrypted by computing S
It is decrypted by computing t
). Euler's Theorem can be used to show that if 0 < t
, then t
The security of an RSA system would be compromised if the number n
could be factored or if φ(n
) could be computed without factoring n
is prime, then φ(p
) = p
− 1. In 1932 D. H. Lehmer asked if there are any composite numbers n
such that φ(n
) | n
− 1. None are known.
In 1933 he proved that if any such n
exists, it must be odd, square-free, and divisible by at least seven primes (i.e. ω(n
) ≥ 7). In 1980 Cohen and Hagis proved that n
> 1020 and that ω(n
) ≥ 14. Further, Hagis showed that if 3 divides n
> 101937042 and ω(n
) ≥ 298848.
This states that there is no number n
with the property that for all other numbers m
) ≠ φ(n
). See Ford's theorem above.
As stated in the main article, if there is a single counterexample to this conjecture, there must be infinitely many counterexamples, and the smallest one has at least ten billion digits in base 10.
The Disquisitiones Arithmeticae
has been translated from Latin into English and German. The German edition includes all of Gauss' papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.
References to the Disquisitiones
are of the form Gauss, DA, art. nnn
For more information, see the page on Japanese Wikipedia.
Time Shock , is a game show airing in Japan. Its original run, titled Quiz Time Shock , was from 1969 to 1986; a revival aired from 1989 to 1990. A second revival, Time Shock 21, ran from 2000 to 2002. Afterwards, a series of periodic specials titled Time Shock aired from 2002 through 2007, plus one in 2007 titled New Time Shock . The most recent revival, Super Time Shock , debuted in 2008.
First, the contestant was raised to the top of the on-set clock set piece in an elevator chair. After a brief interview, the contestant selected one of 10 packets of questions; each packet contained 12 questions. The contestant had 60 seconds to answer all 12 questions. The questions were read by the announcer at five-second intervals; players could interrupt the reading of the question to give an answer. Each correct answer earned an increasing amount of money, as follows:
At the end of the round, the contestant was lowered back to floor level; if the contestant failed to answer at least four questions correctly, the chair would be rapidly spun as the contestant was lowered. In the 1989-1990 run, jets of dry ice would also be activated, and the prize money would be lost; in this instance, the contestant would win a plush doll. Also in the 1989-90 run, certain packets could contain "shock" questions; failing to answer a shock question correctly would end the round and forfeit the prize money.
In Time Shock 21, teams of five contestants played for the right to face "Final Time Shock" (played the same way as the original Time Shock). In earlier episodes, five teams competed over three rounds, or "stages," with low-scoring teams being eliminated along the way.
Each team faced a round of 10 questions, to be answered in 60 seconds. Each player was directed a question in turn, and had 6 seconds to answer (including the time used to read the question). No conferring or passing was allowed. The three teams with the highest scores advanced (tiebreaker questions were asked, if necessary). In addition, a team had to answer at least five questions correctly to qualify.
A series of visual puzzles were played. These puzzles often made extensive use of CGI. Typical puzzles might be identifying a picture as a Rubik's Cube solves itself (with the picture replacing one of the colors), identifying a famous person based on paintings and clues by grade school kids, or identifying an object as a karate expert kicks a block of stone into the shape of the object. Each puzzle had a time limit of either 20 or 60 seconds; when a player buzzed in, the timer stopped. If the player was right, their team scored a point; if not, that player was eliminated for the rest of the puzzle, and the time resumed. In addition, an incorrect answer in the last 10 seconds of a 20-second puzzle, or the last 20 seconds of a 60-second puzzle eliminated a second player from the puzzle. 60-second puzzles sometimes had three parts, such as spotting three differences between two otherwise-identical pictures or video clips, or spotting three things that are wrong with the scene; in these instances, each part counted for one point. The first two teams to score three points advanced to "Data Time Shock."
Each team was given a question in which the top 10 answers were ranked in some way. Each player, in turn, tried to identify one of the top 10 answers. No passing was allowed; a player had to give an answer (whether one of the top 10 or not) to allow the next player to offer an answer. Each team had 60 seconds to give as many answers as possible, and the team that scored higher won the game and played Final Time Shock. In addition, a score of 5 correct answers or better was required to qualify.
The team elected one of its players to face Final Time Shock, which was played in the same way as the original series (60 seconds to answer 12 questions; 5 seconds, including time used to read it, to answer each question). The prize money earned was divided among the team, and was as follows:
If the contestant failed to accumulate at least 6 correct answers, they were subjected to a "Tornado Spin," in which the chair would flip the contestant upside down in addition to rotating as the player descended.
In later episodes, only two teams would compete against each other in "Visual Time Shock," played the same way as in episodes with five teams. The first team to correctly solve three puzzles would play "Final Time Shock."
Under two-team rules, the team's representative had to answer 6 questions correctly in "Final Time Shock" to win any money. The payouts were as follows:
As before, a player failing to answer 6 questions correctly would be subjected to a "Tornado Spin." In addition, the team would have two or three players attempt "Final Time Shock," with the team winning the prize for the best score achieved (for example, if the first player got 8 questions right and the second player got 10, the team would win ¥4,000,000 for the 10 correct answers).
World Maths Day (World Math Day in American English) is an online international mathematics competition, powered by educational resource provider 3P Learning (the same organisation behind the school resources Mathletics, Spellodrome and IntoScience). Smaller elements of the wider Mathletics program effectively power the World Maths Day event.
The first World Maths Day was held on March 14 2007 (Pi Day), and has been held on the 1st Wednesday in March in subsequent years. Despite these origins, the phrases "World Maths Day" and "World Math Day" are trademarks and not to be confused with other competitions such as the International Mathematical Olympiad or days such as Pi Day. In 2010, World Maths Day created a Guinness World Record for the Largest Online Maths Competition.
In 2011, the team behind the competition added a second event World Spelling Day - and officially rebranded as the World Education Games. In 2012, a third event was added - World Science Day. The World Maths Day and World Education Games games are now sponsored by Samsung, and supported by UNICEF as global charity partner.
World Maths Day involves participants playing 60 second games,with the platform heavily based on "Live Mathletics" found in Mathletics. The contests involve mental math problems appropriate for each age group which test the accuracy and speed of the students as they compete against other students across the Globe. The simple but innovative idea of combining the aspects of multi-player online gaming with math problems has contributed to its popularity around the world.
The inaugural World Maths Day was held on March 13 2007. 287 000 students from 98 countries answered 38 904 275 questions.The student numbers and the participating countries have steadily increased in the following years as evidenced by the Guinness Record created in 2010.In 2012 as the World Education Games, over 5.9 Million students from 240 Countries and Territories around the World registered to take part in the games with the World Maths Day being the biggest attraction. In 2013, it was held between 5-7th March and the awards were presented at the Sydney Opera House to the Champions.
A number of awards are offered to the students who take part and for those who do well in the event. Additionally the Champions and the Top Ten students in the world are awarded Gold medals every year.
There are also a number of National lead-up events in different regions around the world which are also based on the Mathletics format.The winners become the Ambassadors of their Country for the World Education Games the following year and help to promote the event.
The complete list of individual Champions,School,Class and other winners over the years can be found in the World Education Games website.
The Individual Gold Medal Winners through the years are listed below
Sha Tin College, Hong Kong
Throughout the year Mathletics host several National Mathletics challenges in the lead up to World Maths Day. These challenges and the winners list are as follows:
The American Math Challenge :Winner- Alek K, Haddonfields schools, Null.
The Australian Maths Challenge :Winner- Parker C, Home Education, NSW.
The Canadian Math Challenge :Winner- Shekar S, North Kipling Junior Middle School, ON.
The European Schools Maths Challenge:Winner- Anna S, British International School of Ljubljana, Slovenia.
The Middle East Schools Maths Challenge:Winner- Zakria Y, Australian International School,UAE.
The NZ Maths Challenge :Winner- Vlad B, St Mary's School ,Christchurch.
The South African Maths Challenge :Winner- Jaden D, Wilton House, GT.
The UK Four Nations Maths Challenge :Winner- Sharan Maiya, Glasgow Academy, Scotland.
The American Math Challenge :Winner- Sayan Das, Team USA, Minnesota.
The Australian Maths Challenge :Winner- Tatiana Devendranath, Team Australia, VIC.
The Canadian Math Challenge :Winner- Tom.L, MPS, Etobicoke.
The European Schools Maths Challenge:Winner-
The Middle East Schools Maths Challenge:Winner- .
The NZ Maths Challenge :Winner- Jake Dawson,Springbank School.
The Pakistan Maths Challenge: Winner- Dilsher A, The International School of Choueifat.
The South African Maths Challenge :Winner- Jaden D, Team ZAF.
The UK Four Nations Maths Challenge :Winner- Sharan Maiya, Glasgow Academy, Scotland, United Kingdom.
The American Math Challenge :Winner- Zhe W, Team USA, Massachusetts
The Australian Maths Challenge :Winner- Aaron Herrmann,, Seaford 6-12 School, South Australia
The Canadian Math Challenge :Winner- Hanting C, Maywood Community School, Canada
The European Schools Maths Challenge:Winner- Filip Szary, Team Poland
The Middle East Schools Maths Challenge:Winner- Pushp raj P, MES Indian School ,Qatar
The NZ Maths Challenge :Winner- Willem Ebbinge, Remarkables Primary School, Otago
The Pakistan Maths Challenge: Winner- Husnain Ali Abid, Team Pakistan, Punjab
The South African Maths Challenge :Winner- Bradley P, Merrifield College, Eastern Cape
The UK Four Nations Maths Challenge :Winner- Ryan Conlan, Team GBR ,Scotland
The Little Professor is a backwards calculator designed for children ages 5 to 9. Instead of providing the answer to a mathematical expression entered by the user, it generates unsolved expressions and prompts the user for the answer.
When the user turns the Little Professor on and selects a difficulty level, an incomplete equation such as "3 x 6 =" appears on the LED screen. The user has three chances to enter the correct number. If the answer is incorrect, the screen displays "EEE". After the third wrong answer, the correct answer is displayed. If the answer supplied is correct, the Little Professor goes to the next equation. The Little Professor displays the number of correct first answers after each set of 10 problems.
The Little Professor was first released by Texas Instruments on June 13, 1976. As the first electronic educational toy, the Little Professor is a common item on calculator collectors' lists.
In 1976, the Little Professor cost less than $20. More than 1 million units sold in 1977.
An emulator of the Little Professor for Android was published in 2012.
An anomalous cancellation or accidental cancellation is a particular kind of arithmetic procedural error that gives a numerically correct answer. An attempt is made to reduce a fraction by canceling individual digits in the numerator and denominator. This is not a legitimate operation, and does not in general give a correct answer, but in some rare cases the result is numerically the same as if a correct procedure had been applied.
Examples of anomalous cancellations which still produce the correct result include:
The article by Boas analyzes two-digit cases in bases other than base 10, e.g., 32/13 = 2/1 is the only solution in base 4.