Square Root

Square Root


Min Max

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 .

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  or r=sqrt(x). For example, the principal square root of 9 is , 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  for x>=0.

Min Max

Any nonzero complex number  also has two square roots. For example, using the imaginary unit i, the two square roots of  are +/-sqrt(-9)=+/-3i. The principal square root of a number  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  are given by


In addition,


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  is Newton’s iteration:


with x_0=1.

The square root of 2 is the irrational number  (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  (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.

nested radical of the form  can sometimes be simplified into a simple square root by equating


Squaring gives



a = d+e
b^2c = 4de.

Solving for d and e gives


For example,


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,

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

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