The principal square root function f(x)=$\sqrt{x}$ (usually just referred to as the "square root function") is a function which maps the set of non-negative real numbers onto itself, and, like all functions, always returns a unique value. In geometrical terms, the square root function maps the area of a square to its side length.
The principal square root function f(x)=$\sqrtx$ (usually just referred to as the "square root function") is a function which maps the set of non-negative real numbers onto itself, and, like all functions, always returns a unique value. In geometrical terms, the square root function maps the area of a square to its side length.
The square root of x is rational if and only if x is a rational number which can be represented as a ratio of two perfect squares. See square root of 2 for proofs that this is an irrational number, and quadratic irrational for a proof for all non-square natural numbers. The square root function maps rational numbers into algebraic numbers (a superset of the rational numbers).
For every real numbers x,
Square Roots Of Numbers" title="Properties Of Square Roots Of Numbers" width="203" height="45" />
For all non-negative real numbers x and y,
Properties of Square Roots of Complex and Negative 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 with a more inclusive set of numbers, called the complex numbers, that does contain solutions to the square root of a negative number. This is done by introducing a new number, denoted by i (sometimes j, especially in the context of electricity where "i" traditionally represents electric current) and called the imaginary unit, which is defined such that i2 = ?1. Using this notation, we can think of i as the square root of ?1, but notice that we also have (?i)2 = i2 = ?1 and so ?i is also a square root of ?1. By convention, the principal square root of ?1 is i, or more generally, if x is any positive number, then the principal square root of ?x is
$\sqrt{-x}$ = i $\sqrt{x}$
because
(i $\sqrt{x}$ )2 = i2($\sqrt{x}$ )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 ?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.
Properties of Imaginary Square Root
The square root of i is given by
$\sqrt{i}$ = $\frac{1}{\sqrt{2}}{(1+i)}$
This result can be obtained algebraically by finding a and b such that
i = (a+bi)2
or equivalently
i = a2 + 2abi - b2
This gives the two equations
2ab = 1
a2 - b2 = 0,
which are easily solved to
a = b =± $\frac{1}{\sqrt{2}}$
The choice of the principal root then gives
a = b = $\frac{1}{\sqrt{2}}$
The result can also be obtained by using De Moivre's theorem and setting
i = cos ($\frac{\pi}{2}$ ) + isin ($\frac{\pi}{2}$)
which produces
$\sqrt{i}$ = (cos ($\frac{\pi}{2}$)) + isin($\frac{\pi}{2}$))$\frac{1}{2}$
= cos ($\frac{\pi}{4}$)) + isin($\frac{\pi}{4}$)
= $\frac{1}{\sqrt{2}}$ + i $\frac{1}{\sqrt{2}}$
= $\frac{1}{\sqrt{2}}$ (1+ i)
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