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In mathematics, a square root of a number x is a number whose square (the result of multiplying the number by itself) is x. Every non-negative real number x has a unique non-negative square root, called the principal square root and denoted

sqrt x

. For example, the principal square root of 9 is 3 (denoted

sqrt 9 = 3

) because

3^2 = 3	imes3 = 9

. The other square root of 9 (not the principal square root) is −3.
Square roots often arise when solving quadratic equations, or equations of the form

ax^2+bx+c=0

, due to the variable

x

being squared.

Per the fundamental theorem of algebra, there are two solutions to the equation defining the square roots of any number (although these roots may not be distinct, as in the square root of zero). For a positive real number, the two square roots are the principal square root and the negative square root (denoted

-sqrt x

). Together, the principal and negative square roots of a number are denoted

pmsqrt x

. For negative real numbers, the concept of imaginary and complex numbers has been developed to provide a mathematical framework to deal with the results. Square roots of objects other than numbers can also be defined.

Square roots of integers that are not perfect squares are always irrational numbers, i.e., numbers not expressible as a ratio of two integers. For example,

sqrt 2

cannot be written exactly as m/n, where n and m are integers. Nonetheless, it is exactly the length of the diagonal of a square with side length 1. This has been known since ancient times, with the discovery that

sqrt 2

is irrational attributed to Hippasus, a disciple of Pythagoras. (See square root of 2 for proofs)

The square root symbol (

sqrt

) was first used during the 16th century. It has been suggested that it originated as an altered form of lowercase r, representing the Latin radix (meaning "root").

    Square root
        Properties
        Computation
        Square roots of negative and complex numbers
        Square roots of matrices and operators
        Infinitely nested square roots
        Square roots of the first 20 positive integers
        Geometric construction of the square root
        History
        Notes
        See also

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Properties

The principal square root function

f(x) = sqrt

is a function which maps the set of non-negative real numbers

mathbb^+ cup

onto itself.

The principal square root function

f(x) = sqrt

always returns a unique value.

To obtain both roots of a positive number, take the value given by the principal square root function as the first root (root₁) and obtain the second root (root₂) by subtracting the first root from zero (ie root₂ = 0 − root₁).

The following important properties of the square root functions are valid for all positive real numbers

x

and

y

$sqrt = sqrt x sqrt y qquad Rightarrow qquad sqrt , = , 10 cdot sqrt y$

$sqrt = rac qquad Rightarrow qquad sqrt = rac$

$sqrt x = x^$

The square root function maps rational numbers to algebraic numbers; also,

sqrt x

is rational if and only if

x

is a rational number which can be represented as a ratio of two perfect squares. In particular,

sqrt 2

is irrational.

In geometrical terms, the square root function maps the area of a square to its side length.

Contrary to popular belief,

sqrt

does not necessarily equal

x

. The equality holds for non-negative

x

, but when

x < 0

sqrt

is positive by definition, and thus

sqrt = -x

. Therefore,

sqrt = left|x

ight| for real

x

(see absolute value).

Suppose that

x

and

a

are real numbers, and that

x^2 = a

, and we want to find

x

. A common mistake is to "take the square root" and deduce that

x = sqrt a

. This is incorrect, because the principal square root of

x^2

is not

x

, but the absolute value

left| x

ight|, one of our above rules. Thus, all we can conclude is that

left| x 
ight| = sqrt a

, or equivalently

x = pmsqrt a

In calculus, for instance when proving that the square root function is continuous or differentiable, or when computing certain limits, the following identity often comes handy:

$sqrt x - sqrt y = rac,$

valid for all non-negative numbers $x$ and $y$ which are not both zero.

The function

f(x) = sqrt x

has the following graph, made up of half a parabola lying on its side:

The function is continuous for all non-negative

x,

and differentiable for all positive

x

(it is not differentiable for

x=0

since the slope of the tangent there is ∞). Its derivative is given by

$f'(x) = rac.$

The Taylor series of

sqrt

about

x=0

can be found using the binomial theorem:

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Computation

Many methods of calculating square roots exist today, some meant to be done by hand and some meant to be done by machine.

Many, but not all pocket calculators have a square root key. Computer spreadsheets and other software are also frequently used to calculate square roots. Computer software programs typically implement good routines to compute the exponential function and the natural logarithm, and then compute the square root of x using the identity

$sqrt = e^$

The same identity is exploited when computing square roots with logarithm tables or slide rules.

The most common method of square root calculation by hand is known as the "Babylonian method". It involves a simple algorithm, which will bring you closer and closer to the actual square root each time it is repeated. To find r, the square root of a real number x:

Start with an arbitrary positive start value r (the closer to the square root of x, the better).

Replace r by the average between r and x / r. (It is sufficient to take an approximate value of the average, not too close to the previous value of r and x / r in order to ensure convergence.)

Repeat steps 2 and 3.

The best known time complexity for computing a square root with n digits of precision is the same as that for multiplying two n-digit numbers.

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Square roots of negative and complex numbers

The square of any positive or negative number is positive, and the square of 0 is 0. Therefore, no negative number can have a real square root. However, it is possible to work in a larger number system, called complex numbers, in which negative numbers have square roots. This is done by introducing a new number, called the imaginary unit, which is defined to be a square root of -1. This number is usually denoted by

i

(sometimes j , especially in the context of electricity). Using this notation, the square root of any negative number

-x

$sqrt = isqrt x$

because

$(isqrt x)^2 = i^2(sqrt x)^2 = (-1)x = -x$ .

By the argument given above, i can be neither positive nor negative. This creates a problem: for the complex number z, we cannot define

sqrt z

to be the "positive" square root of

z

.

For every non-zero complex number z there exist precisely two numbers w such that w² = z. The usual definition of √z is by introducing the following branch cut: if

z = r^

is represented in polar coordinates with

-pi < phi 
 leq pi

, then we set

sqrt = r^

. Thus defined, the square root function is holomorphic everywhere except on the non-positive real numbers (where it isn't even continuous). The above Taylor series for

sqrt

remains valid for complex numbers x with |x| < 1.

When the number is in rectangular form the following formula can be used:

$sqrt = sqrt pm i sqrt$

where

left|x+iy
ight| = sqrt

(the absolute value or modulus of the complex number),
and the sign of the imaginary part of the root is the same as the sign of the imaginary part of the original number.

Note that because of the discontinuous nature of the square root function in the complex plane, the law

sqrt = sqrt z cdot sqrt w

is in general not true. Wrongly assuming this law underlies several faulty "proofs", for instance the following one showing that -1 = 1:

$-1 = i cdot i = sqrt cdot sqrt = sqrt = sqrt = 1$

The third equality cannot be justified. (See invalid proof.)

This problem can arise as a misuse of the principal square root notation √ defined in the beginning of the article, or neglecting to account for the branch cut in the definition of the complex square root function. With the general (two-valued) square root concept, it is indeed true that one of the two square roots of 1 is -1.

However the law can only be wrong by a factor -1 (it is right up to a factor -1), √(zw) = ±√(z)√(w), is true for either ± as + or as -. Note that √(c²) = ±c, therefore √(a²b²) = ±ab and therefore √(zw) = ±√(z)√(w), using a = √(z) and b = √(w).

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Square roots of matrices and operators

If A is a positive-definite matrix or operator, then there exists precisely one positive definite matrix or operator B with B² = A; we then define √A = B.

More generally, to every normal matrix or operator A there exist normal operators B such that B² = A. In general, there are several such operators B for every A and the square root function cannot be defined for normal operators in a satisfactory manner. Positive definite operators are akin to positive real numbers, and normal operators are akin to complex numbers.

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Infinitely nested square roots

Under certain conditions infinitely nested radicals such as

$x = sqrt$

represent rational numbers. This rational number can be found by realizing that x also appears under the radical sign, which gives the equation

$x = sqrt.$

If we solve this equation, we find that x = 2. This approach can also be used to show that generally, if n > 0, then:

$sqrt = rac.$

The same procedure also works to get

$sqrt = rac.$

This method will give a rational

x

value for all values of

n

such that

$= + . ,$

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Square roots of the first 20 positive integers

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Geometric construction of the square root
A square root can be constructed with a compass and straightedge.In his Elements, Euclid (fl. 300 BC) gave the construction of the geometric mean of two quantities in two different places: Proposition II.14 and Proposition VI.13. Since the geometric mean of

a

and

b

sqrt

, one can construct

sqrt

simply by taking

b=1

.

The construction is also given by Descartes in his La Géométrie, see figure 2 on page 2. However, Descartes made no claim to originality and his audience would have been quite familiar with Euclid.

Another method of geometric consruction uses right triangles and induction:

sqrt = 1

can, of course, be constructed, and once

sqrt

has been constructed, the right triangle with 1 and

sqrt

for its legs has a hypotenuse of

sqrt

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History

The Rhind Mathematical Papyrus is a copy from 1650 BCE of an even earlier work and shows us how the Egyptians extracted square roots.

In Ancient India, the knowledge of theoretical and applied aspects of square and square root was at least as old as the Sulba Sutras, dated around 800-500 B.C. (possibly much earlier). A method for finding very good approximations to the square roots of 2 and 3 are given in the Baudhayana Sulba Sutra. Aryabhata in the Aryabhatiya (section 2.4), has given a method for finding the square root of numbers having many digits.

D.E. Smith in History of Mathematics, says, about the existing situation in Europe: "In Europe these methods (for finding out the square and square root) did not appear before Cataneo (1546). He gave the method of Aryabhata for determining the square root".

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Notes

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