Square Root
ArithmeticThe square root of a number is the value that, when multiplied by itself, gives that number.
Formula
\sqrt{n} = x \text{ means } x^2 = n \ (x \ge 0)
Definition
The square root of a number is what you multiply by itself to get that number. The square root symbol is the radical sign: sqrt.
Example
$\sqrt{25} = 5$ because $5 \times 5 = 25$. $\sqrt{9} = 3$ because $3 \times 3 = 9$. $\sqrt{100} = 10$.
Key Insight
Finding the square root is the opposite (inverse) of squaring. If squaring a number means building a square, the square root tells you the side length of that square.
Definition
The principal square root of $n \ge 0$ is the non-negative real number $x$ such that $x^2 = n$, written $\sqrt{n}$ or $n^{1/2}$. The square root of a non-perfect-square positive integer is irrational. $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ and $\sqrt{a/b} = \sqrt{a}/\sqrt{b}$ for $a,b \ge 0$.
Example
Simplify $\sqrt{72}$: $72 = 36 \times 2 = 6^2 \times 2$. So $\sqrt{72} = 6\sqrt{2} \approx 8.485$. $\sqrt{(-3)^2} = \sqrt{9} = 3 = |-3|$.
Key Insight
$\sqrt{x^2} = |x|$, not $x$. The absolute value appears because the square root is always non-negative, but $x$ itself could be negative.
Definition
The square root function $\sqrt{\cdot}: [0,\infty) \to [0,\infty)$ is continuous, strictly increasing, and concave. As a special case of $b^{1/n}$, $\sqrt{x} = x^{1/2}$ and satisfies $\frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}}$. In complex numbers, every non-zero $z$ has two square roots: $\sqrt{|z|}\, e^{i\arg(z)/2}$ and its negative. The principal square root takes $\arg(z)$ in $(-\pi, \pi]$.
Example
Newton's method for $\sqrt{a}$: $x_{n+1} = (x_n + a/x_n)/2$. Starting with $x_0=1$ for $\sqrt{2}$: $x_1=1.5$, $x_2=1.4167$, $x_3=1.41422$. Converges quadratically.
Key Insight
The irrationality of $\sqrt{2}$ (proved by the Pythagoreans) was one of the first mathematical crises in history, shattering the belief that all quantities could be expressed as ratios of integers. It motivated the development of the real number system.