Square Root
Definition
For a number $x$, a number $r$ satisfying $r^{2} = x$ is called a square root of $x$.
$$ \text{$r$ is a square root of $x$} \iff r^{2} = x $$
Explanation
For a positive $x \gt 0$, there exist two numbers $r \gt 0$ and $-r$ that are square roots of $x$ whose absolute values are equal and whose signs are opposite. The positive one $r$ is called the positive square root of $x$, and is denoted as follows.
$$ r = \sqrt{x} $$
$-r = -\sqrt{x}$ is called the negative square root of $x$.
The symbol $\sqrt{\ }$ itself is called the radical symbol, and $\sqrt{x}$ is read as [square root x]. Perhaps the most difficult point when first learning square roots is distinguishing between 「the square roots of $x$」 and 「the square root $x$」. The square roots of $x$, by the above definition, refer to all numbers which, when squared, equal $x$. That is, all numbers satisfying $r^{2} = x$. By contrast, the square root $x$ is the reading of $\sqrt{x}$, so it denotes the positive square root of $x$. Therefore, for positive $x \gt 0$, the square root $x$ ($=\sqrt{x}$) is always positive.
$$ \text{$x$의 제곱근} = \left\{ r : r^{2} = x \right\} = \left\{ \sqrt{x}, -\sqrt{x} \right\} $$
$$ \text{제곱근 $x$} = \sqrt{x} ( \gt 0 ) $$
Properties
For $a, b \gt 0$ the following properties hold.
(a) $\sqrt{a \pm b} \ne \sqrt{a} \pm \sqrt{b}$ $(a \gt b)$
(b) $\sqrt{a b} = \sqrt{a} \sqrt{b}$
(b’) $p\sqrt{a} \times q\sqrt{b} = pq \sqrt{a b}$
(c) $\dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\dfrac{a}{b}}$
Proof
(b)
Squaring $\sqrt{a} \sqrt{b}$,
$$ ( \sqrt{a} \sqrt{b} )^{2} = ( \sqrt{a} )^{2} ( \sqrt{b} )^{2} = a b $$
This means $\sqrt{a}\sqrt{b}$ is the positive square root of $ab$. Hence $\sqrt{a b} = \sqrt{a} \sqrt{b}$.
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$0$ and square roots of negative numbers
$\sqrt{0}$ is defined as $0$.
There is no real number that when multiplied by itself yields a negative number. To talk about square roots of negative numbers one must extend the number system to the complex numbers. We define the square root $-1$, i.e. $\sqrt{-1}$, to be the complex number $i$.
$$ \sqrt{-1} := i $$
Following the above caveat, 「the square roots of $-1$」 are two numbers $\left\{ i, -i \right\}$, whereas 「the square root $-1$」 $(=\sqrt{-1})$ refers only to $i$. For a positive $x \gt 0$, we define the square root of $-x$ as follows.
$$ \sqrt{-x} := i \sqrt{x} $$
One can verify without difficulty that the above properties also hold for square roots of negative numbers.