What Is a Square Root?
The square root of a number n is the value x such that x² = n. In other words, it's the side length of a square whose area is n. Written as √n or n^(1/2).
Perfect squares have integer square roots: √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.
Non-perfect squares have irrational square roots — decimals that never terminate or repeat. √2 = 1.41421356... √3 = 1.73205080... √5 = 2.23606797... These are some of the most famous irrational numbers, and proving that √2 is irrational is one of the oldest proofs in mathematics (attributed to the ancient Greeks, possibly Hippasus, around 500 BC).
How to Estimate Square Roots Without a Calculator
Estimating square roots mentally is a useful skill. The trick: find the nearest perfect squares above and below your number, then interpolate.
For √50: √49 = 7, √64 = 8. So √50 is between 7 and 8, closer to 7. Better estimate: 7 + (50−49)/(64−49) = 7 + 1/15 ≈ 7.07. (True answer: 7.071.)
For √200: √196 = 14, √225 = 15. So √200 is between 14 and 15. 14 + (200−196)/(225−196) = 14 + 4/29 ≈ 14.14. (True answer: 14.142.)
The Babylonian method (also called Heron's method) is an ancient iterative algorithm for refining estimates: start with any guess x₀, then repeatedly apply x_{n+1} = (x_n + N/x_n) / 2. For √10, starting at 3: x₁ = (3 + 10/3)/2 = 3.167. x₂ = (3.167 + 10/3.167)/2 = 3.1623. Already very close to the true answer of 3.16228.