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If you divide 2222 by 77, the decimals repeat in a predictable cycle forever, but if you measure the true circumference of a circle, the decimals of Ο€\pi never repeat and never end.
Every real number on the number line belongs to one of two strictly separate teams: rational or irrational.

The Two Number Families

A rational number is any value that can be written as a fraction ab\frac{a}{b}, where aa and bb are integers and b≠0b \neq 0.
An irrational number cannot be written as a fraction of two integers because its decimal expansion goes on forever without any repeating pattern.
What tells you immediately whether a square root belongs in the rational or irrational box?

Radicals and Roots

A radicand is the number sitting inside a square root symbol, like the 2525 inside 25\sqrt{25}.
If the radicand is a perfect square (like 0,1,4,9,16,250, 1, 4, 9, 16, 25), the square root evaluates to an exact integer, making it rational.
If the radicand is any positive whole number that is NOT a perfect square, its square root is always irrational.
Around 500 BCE, the Greek philosopher Hippasus proved that the diagonal of a 1Γ—11 \times 1 square, 2\sqrt{2}, could never be expressed as a ratio of whole numbers β€” shocking the ancient mathematical world.