How to Calculate Square Root: Methods and Examples
What Is a Square Root?
The square root of a number is the value that, when multiplied by itself, gives the original number. Written as √n, the square root of 25 is 5 because 5 × 5 = 25. Every positive number has two square roots: a positive root (+5) and a negative root (-5). By convention, √ refers to the positive root.
Perfect Squares Reference
| Number (n) | Square Root (√n) | n² |
|---|---|---|
| 1 | 1 | 1 |
| 4 | 2 | 4 |
| 9 | 3 | 9 |
| 16 | 4 | 16 |
| 25 | 5 | 25 |
| 36 | 6 | 36 |
| 49 | 7 | 49 |
| 64 | 8 | 64 |
| 81 | 9 | 81 |
| 100 | 10 | 100 |
| 144 | 12 | 144 |
| 169 | 13 | 169 |
| 196 | 14 | 196 |
| 225 | 15 | 225 |
Method 1: Newton's Approximation (Babylonian Method)
For non-perfect squares, this iterative method converges quickly:
- Make an initial guess g₀ (a nearby perfect square's root works well).
- Apply the formula: g₁ = (g₀ + n/g₀) ÷ 2
- Repeat with g₁ as the new guess until the answer is accurate enough.
Example: √50
g₀ = 7 (since 7² = 49 is close)
g₁ = (7 + 50/7) ÷ 2 = (7 + 7.143) ÷ 2 = 7.071
g₂ = (7.071 + 50/7.071) ÷ 2 = 7.0711 ✓
Actual: √50 ≈ 7.0711
Method 2: Estimation by Bracketing
Find the two perfect squares your number falls between:
Example: √75
8² = 64 and 9² = 81. So √75 is between 8 and 9.
75 is 11 out of 17 (81-64) of the way from 64 to 81 → approximately 8 + (11/17) ≈ 8.65.
Actual: √75 ≈ 8.660 — very close!
Square Roots in Real Life
- Pythagorean theorem: a² + b² = c². Finding the hypotenuse of a right triangle: √(a² + b²).
- Distance formula: Distance between two points = √((x₂-x₁)² + (y₂-y₁)²)
- Physics: RMS (root mean square) voltage in AC circuits = V_peak ÷ √2 ≈ V_peak × 0.707
- Statistics: Standard deviation = √(variance)
- Finance: Sharpe ratio and volatility calculations use square roots
Irrational Square Roots
Most square roots are irrational — they cannot be expressed as a simple fraction and their decimal expansion never repeats or terminates. √2 ≈ 1.41421356..., √3 ≈ 1.73205080..., √5 ≈ 2.23606797... These go on forever with no pattern. This was proven by the ancient Greeks and was considered a mathematical scandal — "irrational" numbers were thought impossible.
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