The square root of a number is the value that, multiplied by itself, gives that number. Geometrically it is the side length of a square of that area, which is where the name comes from and why it shows up in every distance and area calculation you will ever do.
Every positive number has two square roots, one positive and one negative, since (−4)² and 4² both give 16. The radical symbol √ conventionally refers to the positive one, called the principal root.
Perfect squares worth memorising
| n | n² | n | n² |
|---|---|---|---|
| 1 | 1 | 9 | 81 |
| 2 | 4 | 10 | 100 |
| 3 | 9 | 11 | 121 |
| 4 | 16 | 12 | 144 |
| 5 | 25 | 13 | 169 |
| 6 | 36 | 15 | 225 |
| 7 | 49 | 20 | 400 |
| 8 | 64 | 25 | 625 |
Estimating without a calculator
Find the perfect squares either side and interpolate. For √50: 49 is 7² and 64 is 8², and 50 is barely above 49, so the answer is a little over 7 — 7.07 in fact. For a better estimate, use the Babylonian method: guess, divide the number by your guess, and average the two. For √50 starting at 7: 50 ÷ 7 = 7.14, average 7.07. One iteration is usually accurate to two decimal places.
Where square roots appear
- Pythagoras: the hypotenuse of a right triangle is √(a² + b²) — the basis of every distance calculation in geometry and graphics.
- Standard deviation: the square root of the variance, which is what converts it back into the original units.
- Quadratic formula: the discriminant sits under a radical, and its sign decides how many real solutions exist.
- Physics: pendulum period, escape velocity and RMS voltage are all square-root relationships.
- Screen and paper sizes: the A-series paper ratio is 1:√2, chosen so halving a sheet preserves its proportions.
Irrational roots
The square root of any number that is not a perfect square is irrational — it cannot be written as a fraction and its decimal expansion never repeats. √2 ≈ 1.41421356 was reportedly the discovery that most disturbed the Pythagoreans, who had assumed all quantities were ratios of whole numbers. Negative numbers have no real square root at all, which is the gap imaginary numbers were invented to fill.