TOPICS
Search

Square Root


SqrtReal
Min Max
Powered by webMathematica

A square root of x is a number r such that r^2=x. When written in the form x^(1/2) or especially sqrt(x), the square root of x may also be called the radical or surd. The square root is therefore an nth root with n=2.

Note that any positive real number has two square roots, one positive and one negative. For example, the square roots of 9 are -3 and +3, since (-3)^2=(+3)^2=9. Any nonnegative real number x has a unique nonnegative square root r; this is called the principal square root and is written r=x^(1/2) or r=sqrt(x). For example, the principal square root of 9 is sqrt(9)=+3, while the other square root of 9 is -sqrt(9)=-3. In common usage, unless otherwise specified, "the" square root is generally taken to mean the principal square root. The principal square root function sqrt(x) is the inverse function of f(x)=x^2 for x>=0.

SqrtReImAbs
Min Max
Re
Im Powered by webMathematica

Any nonzero complex number z also has two square roots. For example, using the imaginary unit i, the two square roots of -9 are +/-sqrt(-9)=+/-3i. The principal square root of a number z is denoted sqrt(z) (as in the positive real case) and is returned by the Wolfram Language function Sqrt[z].

When considering a positive real number x, the Wolfram Language function Surd[x, 2] may be used to return the real square root.

The square roots of a complex number z=x+iy are given by

 sqrt(x+iy)=+/-(x^2+y^2)^(1/4){cos[1/2tan^(-1)(x,y)]+isin[1/2tan^(-1)(x,y)]}.
(1)

In addition,

 sqrt(x+iy)=1/2sqrt(2)[sqrt(sqrt(x^2+y^2)+x)+isgn(y)sqrt(sqrt(x^2+y^2)-x)].
(2)

As can be seen in the above figure, the imaginary part of the complex square root function has a branch cut along the negative real axis.

There are a number of square root algorithms that can be used to approximate the square root of a given (positive real) number. These include the Bhaskara-Brouncker algorithm and Wolfram's iteration. The simplest algorithm for sqrt(n) is Newton's iteration:

 x_(k+1)=1/2(x_k+n/(x_k))
(3)

with x_0=1.

The square root of 2 is the irrational number sqrt(2) approx 1.41421356 (OEIS A002193) sometimes known as Pythagoras's constant, which has the simple periodic continued fraction [1, 2, 2, 2, 2, 2, ...] (OEIS A040000). The square root of 3 is the irrational number sqrt(3) approx 1.73205081 (OEIS A002194), sometimes known as Theodorus's constant, which has the simple periodic continued fraction [1, 1, 2, 1, 2, 1, 2, ...] (OEIS A040001). In general, the continued fractions of the square roots of all positive integers are periodic.

A nested radical of the form sqrt(a+/-bsqrt(c)) can sometimes be simplified into a simple square root by equating

 sqrt(a+/-bsqrt(c))=sqrt(d)+/-sqrt(e).
(4)

Squaring gives

 a+/-bsqrt(c)=d+e+/-2sqrt(de),
(5)

so

a=d+e
(6)
b^2c=4de.
(7)

Solving for d and e gives

 d,e=(a+/-sqrt(a^2-b^2c))/2.
(8)

For example,

 sqrt(5+2sqrt(6))=sqrt(2)+sqrt(3)
(9)
 sqrt(3-2sqrt(2))=sqrt(2)-1.
(10)

The Simplify command of the Wolfram Language does not apply such simplifications, but FullSimplify does. In general, radical denesting is a difficult problem (Landau 1992ab, 1994, 1998).

A counterintuitive property of inverse functions is that

 sqrt(z)sqrt(1/z)={-1   for I[z]=0 and R[z]<0; undefined   for z=0; 1   otherwise,
(11)

so the expected identity (i.e., canceling of the sqrt(z)s) does not hold along the negative real axis.


See also

Cube Root, nth Root, Nested Radical, Newton's Iteration, Principal Square Root, Pythagoras's Constant, Quadratic Surd, Radical, Root, Root of Unity, Square Number, Square Root Algorithms, Square Root Inequality, Square Triangular Number, Surd, Theodorus's Constant Explore this topic in the MathWorld classroom

Related Wolfram sites

http://functions.wolfram.com/ElementaryFunctions/Sqrt/

Explore with Wolfram|Alpha

References

Landau, S. "A Note on 'Zippel Denesting.' " J. Symb. Comput. 13, 31-45, 1992a.Landau, S. "Simplification of Nested Radicals." SIAM J. Comput. 21, 85-110, 1992b.Landau, S. "How to Tangle with a Nested Radical." Math. Intell. 16, 49-55, 1994.Landau, S. "sqrt(2)+sqrt(3): Four Different Views." Math. Intell. 20, 55-60, 1998.Sloane, N. J. A. Sequences A002193/M3195, A002194/M4326, A040000, and A040001 in "The On-Line Encyclopedia of Integer Sequences."Spanier, J. and Oldham, K. B. "The Square-Root Function sqrt(bx+c) and Its Reciprocal," "The bsqrt(a^2-x^2) Function and Its Reciprocal," and "The bsqrt(x^2+a) Function." Chs. 12, 14, and 15 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 91-99, 107-115, and 115-122, 1987.Williams, H. C. "A Numerical Investigation into the Length of the Period of the Continued Fraction Expansion of sqrt(D)." Math. Comput. 36, 593-601, 1981.

Referenced on Wolfram|Alpha

Square Root

Cite this as:

Weisstein, Eric W. "Square Root." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SquareRoot.html

Subject classifications