What is Square Root – Definition, Formula & Examples

The square root of a number $n$ is the value that, when multiplied by itself, gives $n$ — written $\sqrt{n}$.

Quick Reference:

Definition: $\sqrt{n} = x$ means $x^2 = n$

Symbol: $\sqrt{\phantom{n}}$ (radical sign), introduced by Christoph Rudolff (c. 1525)

Principal root: The non-negative root (both $+3$ and $-3$ square to 9, but $\sqrt{9} = 3$ by convention)

Perfect squares: $\sqrt{1}=1$, $\sqrt{4}=2$, $\sqrt{9}=3$, $\sqrt{16}=4$, $\sqrt{25}=5$, ...

Irrational roots: $\sqrt{2} \approx 1.414$, $\sqrt{3} \approx 1.732$ — non-terminating, non-repeating decimals

Formula (Pythagorean use): $c = \sqrt{a^2 + b^2}$

Type: Arithmetic operation — inverse of squaring

Used in: Algebra, geometry, physics, statistics (standard deviation), engineering

Full Definition

The square root of $n$ is one of the two numbers whose square equals $n$. For any positive number $n$, there are two square roots: $+\sqrt{n}$ and $-\sqrt{n}$. By convention, the radical symbol $\sqrt{n}$ refers to the principal (positive) square root only.

Square roots are defined for all non-negative real numbers. The square root of a negative number is not a real number — it introduces imaginary numbers ($\sqrt{-1} = i$), studied in complex number theory.

Origin and History

The concept of square roots appears in ancient Babylonian mathematics (c. 1800 BCE) on clay tablets such as YBC 7289, which gives $\sqrt{2} \approx 1.41421$ to five decimal places — an approximation accurate to within 0.000006 of the true value. Christoph Rudolff (c. 1499–1543, Germany) introduced the modern radical symbol $\sqrt{}$ in his 1525 work Coss. The symbol was extended to $\sqrt[3]{}$, $\sqrt[4]{}$ etc. for cube and fourth roots by later mathematicians.

How To Find What is Square Root of Any Number

For perfect squares, square roots are exact whole numbers. For non-perfect-square integers, the square root is irrational — it never terminates or repeats as a decimal.

Three methods for finding square roots:

1. Memorise perfect squares up to $\sqrt{225} = 15$ for quick calculation.

2. Prime factorisation — factor the number and take one factor from each pair: $\sqrt{36} = \sqrt{2^2 \times 3^2} = 2 \times 3 = 6$.

3. Long division method — a systematic algorithm for finding square roots to any decimal precision, taught in many curricula.

Worked Examples of Square Root

Example 1: Perfect square root

Find $\sqrt{144}$.

$12^2 = 144$, so $\sqrt{144} = 12$.

Final answer: $12$

Example 2: Using the Pythagorean theorem

A right triangle has legs $a = 3$ cm and $b = 4$ cm. Find the hypotenuse.

$$c = \sqrt{a^2 + b^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ cm}$$

Final answer: $c = 5$ cm

Example 3: Simplifying a surd

Simplify $\sqrt{72}$.

$$\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$$

Final answer: $6\sqrt{2}$

Common Confusions About Square Root

$\sqrt{n^2} = |n|$, not necessarily $n$. The square root always returns the non-negative value: $\sqrt{(-3)^2} = \sqrt{9} = 3$, not $-3$.

$\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$. This is one of the most common algebraic errors. $\sqrt{9 + 16} = \sqrt{25} = 5$, but $\sqrt{9} + \sqrt{16} = 3 + 4 = 7 \neq 5$.

$\sqrt{n}$ is not the same as $n^2$. Squaring and square-rooting are inverse operations. $\sqrt{4} = 2$; $4^2 = 16$ — these are entirely different.

Where Square Roots Appear Beyond Arithmetic

Square roots appear in the distance formula between two points: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$, in the standard deviation formula in statistics ($\sigma = \sqrt{\frac{\sum(x-\mu)^2}{n}}$), and in the quadratic formula ($x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$). In physics, the period of a pendulum is $T = 2\pi\sqrt{L/g}$ — the square root of the length divided by gravitational acceleration. Every time a distance or spread is computed from squared values, a square root appears.