Square Roots and Irrational Numbers

The square root of 49 is 7, because 7 × 7 = 49. The cube root of 27 is 3.

Most numbers so far have been of one kind.

Rational number is one that can be written as a fraction of two integers.

And some cannot

√2 cannot be written as any fraction. Its decimal never ends and never repeats.

Estimating one

√30 sits between 5 and 6, because 25 is below 30 and 36 is above it.

What a square root is

√25 = 5 because 5 × 5 = 25. The square root of a number is the side length of a square with that area. That geometric reading is where the name comes from and it makes the operation concrete.

Two numbers square to give 25

Both 5 and −5 square to 25. The symbol √ means the positive one by convention, so √25 = 5. When solving x² = 25 you must give both answers: x = 5 or x = −5. The distinction matters and is easy to miss.

Most square roots are irrational

√2 cannot be written as any fraction. Its decimal never terminates and never repeats. Only perfect squares have rational roots, and perfect squares are rare — so irrational numbers are the normal case, not the exception.

Leave it exact when you can

√2 is exact; 1.414 is rounded. Carrying the root symbol through a calculation and only converting at the end avoids accumulating rounding errors, and it is usually what an answer is expected to look like.