What is a perfect square?

A number that is the square of an integer. 16 is a perfect square (4²), but 20 is not, since √20 ≈ 4.47.

How many perfect squares should I memorize?

Aim for 1² through 25² = 625. This covers most competition problems and makes algebra much faster.

What is the square root of a negative number?

In the real number system, it is undefined. In complex numbers, √(−1) = i, and √(−n) = i√n.

Squares & Roots

Instant recall for perfect squares

Squaring a number means multiplying it by itself. Square roots reverse the operation. Knowing perfect squares up to 25² by heart is a massive advantage in competitive math, algebra, and mental calculation.

Key Concepts

Perfect Squares

Numbers that are squares of integers: 1, 4, 9, 16, 25, 36, ... These should be memorized up to at least 15² = 225.

Square Roots

The square root of a perfect square is an integer. √49 = 7 because 7 × 7 = 49.

Squaring Two-Digit Numbers

Use (a+b)² = a² + 2ab + b². For 13²: (10+3)² = 100 + 60 + 9 = 169.

Squaring Numbers Ending in 5

Take the tens digit n, compute n × (n+1), then append 25. 35² = 3×4 = 12 → 1225.

Tips & Tricks

Memorize squares up to 25: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625.
Square a number ending in 5: n5² = n × (n+1) | 25. 75² = 7 × 8 = 56 → 5625.
To square a number near 50: use (50 + d)² = 2500 + 100d + d². 52² = 2500 + 200 + 4 = 2704.
Cube a single digit: just memorize 1-10: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.