Square Root Calculator

What is a Square Root?

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 144 is 12, because 12 × 12 = 144. It is denoted by the symbol √ (radical).

Perfect Squares

A number is a perfect square if its square root is a whole number.

• √1 = 1, √4 = 2, √9 = 3, √16 = 4, √25 = 5
• √36 = 6, √49 = 7, √64 = 8, √81 = 9, √100 = 10
• √121 = 11, √144 = 12, √169 = 13, √196 = 14, √225 = 15

Calculation Methods

1. Trial method — find a number whose square equals the given number
2. Prime factorization — decompose the number into prime factors
3. Newton's method — iterative algorithm for approximate calculation

Properties of Square Roots

• √(a × b) = √a × √b
• √(a / b) = √a / √b
• (√a)² = a
• √a² = |a|

Practical Applications

• Geometry — Pythagorean theorem, diagonal calculations
• Physics — velocity, acceleration, electrical quantities
• Statistics — standard deviation and variance
• Finance — volatility calculation, Black-Scholes model

Interesting fact: the square root of 2 (approximately 1.41421356...) was the first irrational number discovered by ancient Greeks. It cannot be expressed as a fraction.