What is a Square Root?
The square root of a number n, written √n, is the value that, when multiplied by itself, gives n. For example, √9 = 3 because 3 × 3 = 9, and √144 = 12 because 12 × 12 = 144. Every positive number has two square roots: a positive one and a negative one. By convention, √n refers to the principal (positive) square root.
When the square root of a number is an integer, the number is called a perfect square. The perfect squares within our table are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196. All other numbers have irrational square roots — they cannot be expressed as a fraction and their decimal expansion never terminates or repeats.
Computing Square Roots
Several methods exist for computing square roots. Newton's method (also called the Babylonian method) starts with an initial guess x₀ and iteratively refines it: xn+1 = (xn + n/xn) / 2. This converges quadratically — each iteration roughly doubles the number of correct digits. The ancient Babylonians used this method over 3,500 years ago.
The long division method for square roots pairs digits from the decimal point outward and extracts the root digit by digit, similar to long division. Modern computers use optimized algorithms based on Newton's method or lookup tables with interpolation.
Famous Irrational Square Roots
√2 ≈ 1.41421356... is the most famous irrational square root. Its irrationality was proven by the ancient Greeks, reportedly causing a crisis in Pythagorean philosophy (which held that all numbers were rational). √2 is the length of the diagonal of a unit square. √3 ≈ 1.73205080... appears frequently in geometry (height of equilateral triangles). √5 ≈ 2.23606797... is connected to the golden ratio: φ = (1 + √5) / 2.
Properties of Square Roots
Square roots obey several important rules: √(ab) = √a × √b (product rule), √(a/b) = √a / √b (quotient rule), (√a)² = a (inverse relationship). However, √(a + b) ≠ √a + √b — this is a common error in algebra.
Applications
Square roots are essential in geometry (Pythagorean theorem, distance formula), physics (RMS values, wave equations), statistics (standard deviation = square root of variance), finance (volatility calculations), and engineering (signal processing, impedance calculations). The distance formula d = √((x₂−x₁)² + (y₂−y₁)²) is one of the most widely used formulas involving square roots.