What Is a Square Root?

What is a square root? If n² = m, then n is a square root of m. That is the square root definition from OpenStax Prealgebra 2e, section 9.1. The simplest way to see it is with a square whose area you already know. Imagine a square that covers 16 unit squares. Each side of that square measures 4 units because 4 × 4 = 16. The side length is the square root of the area. This area diagram works for any perfect square: a 25‑unit square has sides of 5 units; a 36‑unit square has sides of 6 units. The square root is the inverse of squaring. Squaring 4 gives 16; taking the square root of 16 brings you back to 4.

The square root symbol is √. It always means the principal square root, the non‑negative one. The number written under the symbol is the radicand. In √25, the radicand is 25 and the result is 5. If you see, √25, that means the negative square root,, 5. The symbol alone never gives both; you have to write ±√ to show both roots. Every positive number has two square roots, but the symbol picks one.

Solve x² = 9 and you get x = 3 and x =, 3 because 3 × 3 = 9 and (, 3) × (, 3) = 9. Both are square roots of 9. But √9 equals 3 only. The principal square root is the non‑negative one. Mistaking √9 for ±3 is the most common error students make. The ± symbol appears when you are solving an equation, not when you evaluate a radical. OpenStax Prealgebra 2e section 9.1 defines the principal square root of m as √m and the negative square root as, √m.

Why Two Roots Exist

Squaring a negative number gives a positive product. So, 4 and 4 both square to 16. Every positive real number has exactly two real square roots: one positive, one negative. Zero is the exception: it has one square root, zero itself.

A perfect square is a number whose square root is an integer. The perfect squares from 0 to 144 are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Their square roots are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. Most numbers are not perfect squares. The square root of 2 is about 1.4142, and the decimal never ends or repeats. That is an irrational square root. It cannot be written as a simple fraction. The same is true for √3, √5, √7, and every non‑perfect‑square integer. An irrational square root is still an exact value; the decimal is only an approximation. A calculator shows 1.414213562, but that is not √2.

Four errors show up repeatedly in classrooms and homework. Avoid them by knowing what the square root actually does.

Misconception: The square root of a fraction is smaller than the original number

For a fraction between 0 and 1, the square root is larger, not smaller. √0.25 equals 0.5. The radicand 0.25 is less than 1, and its square root 0.5 is greater than 0.25. This holds for every positive fraction less than 1.

Misconception: √(a + b) equals √a + √b

√(9 + 16) is √25, which is 5. √9 + √16 is 3 + 4 = 7. Five does not equal seven. The square root of a sum is not the sum of the square roots. There are no shortcuts for addition under a radical.

Misconception: Negative numbers have no square root at all

In real numbers, √(, 4) is not defined. But in the complex number system, √(, 4) = 2i, where i = √(, 1) is the imaginary unit. OpenStax Intermediate Algebra 2e section 8.8 covers the complex number system. For middle‑school and high‑school work, the radicand must be greater than or equal to 0 to get a real result.

The product rule √(a × b) = √a × √b holds only when a and b are non‑negative. Applying it to negatives gives contradictions: √(, 4 ×, 9) = √36 = 6, but √(, 4) × √(, 9) = 2i × 3i =, 6.

The square root function f(x) = √x is defined only for x ≥ 0 in the real numbers. If you enter a negative radicand into a real‑number calculator, it returns an error or NaN. That is not a bug. It is the domain restriction. To get a result for a negative radicand, you need a calculator that supports complex numbers or you work with the imaginary unit i manually.