Square root

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 $3^{2}=3\times 3=9$ . 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 $\pm {\sqrt {x}}$ . 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").

Properties

The principal square root function $f(x)={\sqrt {x}}$ is a function which maps the set of non-negative real numbers $\mathbb {R} ^{+}\cup \{0\}$ onto itself.

The principal square root function $f(x)={\sqrt {x}}$ 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 {xy}}={\sqrt {x}}{\sqrt {y}}\qquad \Rightarrow \qquad {\sqrt {100\cdot y}}\,=\,10\cdot {\sqrt {y}}

{\sqrt {\frac {x}{y}}}={\frac {\sqrt {x}}{\sqrt {y}}}\qquad \Rightarrow \qquad {\sqrt {\frac {x}{100}}}={\frac {\sqrt {x}}{10}}

{\sqrt {x}}=x^{1/2}

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, ${\sqrt {x^{2}}}$ does not necessarily equal $x$ . The equality holds for non-negative $x$ , but when $x<0$ , ${\sqrt {x^{2}}}$ is positive by definition, and thus ${\sqrt {x^{2}}}=-x$ . Therefore, ${\sqrt {x^{2}}}=\left|x\right|$ 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 $\left|x\right|$ , one of our above rules. Thus, all we can conclude is that $\left|x\right|={\sqrt {a}}$ , or equivalently $x=\pm {\sqrt {a}}$ .

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}}={\frac {x-y}{{\sqrt {x}}+{\sqrt {y}}}},

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)={\frac {1}{2{\sqrt {x}}}}.

The Taylor series of ${\sqrt {x+1}}$ about $x=0$ can be found using the binomial theorem:

${\sqrt {x+1}}\,\!$	$=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{(1-2n)n!^{2}4^{n}}}x^{n}$
	$=1+{\frac {1}{2}}x-{\frac {1}{8}}x^{2}+{\frac {1}{16}}x^{3}-{\frac {5}{128}}x^{4}+\dots$
for $\left\|x\right\|<1$ .

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 {x}}=e^{{\frac {1}{2}}\ln x}

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 of r and x / r.
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.

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 the 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 $-x$ is

{\sqrt {-x}}=i{\sqrt {x}}

because

(i{\sqrt {x}})^{2}=i^{2}({\sqrt {x}})^{2}=(-1)x=-x

.

By the argument given above, i can be neither positive nor negative. Thus one drawback of working with complex numbers is that the terms "positive" and "negative" lose their meaning. This creates another problem: 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 as follows: if $z=r^{i\phi }$ is represented in polar coordinates with $-\pi <\phi \leq \pi$ , then we set ${\sqrt {z}}=r^{i\phi \over 2}$ . 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 ${\sqrt {1+x}}$ remains valid for complex numbers x with |x| < 1.

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

{\sqrt {x+iy}}={\sqrt {\frac {\left|x+iy\right|+x}{2}}}\pm i{\sqrt {\frac {\left|x+iy\right|-x}{2}}}

where 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 ${\sqrt {zw}}={\sqrt {z}}\cdot {\sqrt {w}}$ 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 {-1}}\cdot {\sqrt {-1}}={\sqrt {-1\cdot -1}}={\sqrt {1}}=1

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

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).

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.

Infinitely nested square roots

Under certain conditions infinitely nested radicals such as

x={\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+\cdots }}}}}}}}

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 {2+x}}.

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 {n+{\sqrt {n+{\sqrt {n+{\sqrt {n+\cdots }}}}}}}}={\frac {1+{\sqrt {1+4n}}}{2}}.

The same procedure also works to get

{\sqrt {n-{\sqrt {n-{\sqrt {n-{\sqrt {n-\cdots }}}}}}}}={\frac {-1+{\sqrt {1+4n}}}{2}}.

This method will give a rational $x$ value for all values of $n$ such that

{n}={x^{2}}+{x}.\,

Square roots of the first 20 positive integers

${\sqrt {1}}$	$=\,$	1
${\sqrt {2}}$	$\approx$	1.4142135623 7309504880 1688724209 6980785696 7187537694 8073176679 7379907324 78462
${\sqrt {3}}$	$\approx$	1.7320508075 6887729352 7446341505 8723669428 0525381038 0628055806 9794519330 16909
${\sqrt {4}}$	$=\,$	2
${\sqrt {5}}$	$\approx$	2.2360679774 9978969640 9173668731 2762354406 1835961152 5724270897 2454105209 25638
${\sqrt {6}}$	$\approx$	2.4494897427 8317809819 7284074705 8913919659 4748065667 0128432692 5672509603 77457
${\sqrt {7}}$	$\approx$	2.6457513110 6459059050 1615753639 2604257102 5918308245 0180368334 4592010688 23230
${\sqrt {8}}$	$\approx$	2.8284271247 4619009760 3377448419 3961571393 4375075389 6146353359 4759814649 56924
${\sqrt {9}}$	$=\,$	3
${\sqrt {10}}$	$\approx$	3.1622776601 6837933199 8893544432 7185337195 5513932521 6826857504 8527925944 38639
${\sqrt {11}}$	$\approx$	3.3166247903 5539984911 4932736670 6866839270 8854558935 3597058682 1461164846 42609
${\sqrt {12}}$	$\approx$	3.4641016151 3775458705 4892683011 7447338856 1050762076 1256111613 9589038660 33818
${\sqrt {13}}$	$\approx$	3.6055512754 6398929311 9221267470 4959462512 9657384524 6212710453 0562271669 48293
${\sqrt {14}}$	$\approx$	3.7416573867 7394138558 3748732316 5493017560 1980777872 6946303745 4673200351 56307
${\sqrt {15}}$	$\approx$	3.8729833462 0741688517 9265399782 3996108329 2170529159 0826587573 7661134830 91937
${\sqrt {16}}$	$=\,$	4
${\sqrt {17}}$	$\approx$	4.1231056256 1766054982 1409855974 0770251471 9922537362 0434398633 5730949543 46338
${\sqrt {18}}$	$\approx$	4.2426406871 1928514640 5066172629 0942357090 1562613084 4219530039 2139721974 35386
${\sqrt {19}}$	$\approx$	4.3588989435 4067355223 6981983859 6156591370 0392523244 4936890344 1381595573 28203
${\sqrt {20}}$	$\approx$	4.4721359549 9957939281 8347337462 5524708812 3671922305 1448541794 4908210418 51276

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 $b$ is ${\sqrt {ab}}$ , one can construct ${\sqrt {a}}$ 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: ${\sqrt {1}}=1$ can, of course, be constructed, and once ${\sqrt {x}}$ has been constructed, the right triangle with 1 and ${\sqrt {x}}$ for its legs has a hypotenuse of ${\sqrt {x+1}}$ .

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.^[1]

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.^[2] 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".^[3]

Notes

^ Anglin, W.S. (1994). Mathematics: A Concise History and Philosophy. New York: Springer-Verlag.
^ Joseph, ch.8.
^ Smith, p. 148.

References

Smith D.E., History of Mathematics (book 2)
Joseph, George G., The Crest of the Peacock: Non-European Roots of Mathematics, 2nd ed. Penguin Books, London. (2000). ISBN 0-691-00659-8.

External links

Rational Mean: High-Order Square-Root methods
MATHPATH: Square roots via Mediants
Japanese soroban techniques - Professor Fukutaro Kato's method
Japanese soroban techniques - Takashi Kojima's method
Algorithms, implementations, and more - Paul Hsieh's square roots webpage
Square root of positive real numbers with implementation in Rexx.

[1] Anglin, W.S. (1994). Mathematics: A Concise History and Philosophy. New York: Springer-Verlag.

[2] Joseph, ch.8.

[3] Smith, p. 148.

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