Square Root

Square Root

 

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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.

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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

 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  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  (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

 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.

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