square root

A square root of a number is a value that, when squared, gives that number back. It undoes squaring. Since 5 × 5 = 25, the number 5 is a square root of 25; and since (−5) × (−5) = 25 as well, −5 is also a square root of 25.

Every positive number has two square roots, one positive and one negative, but the symbol sqrt( ) by convention means only the nonnegative one, called the principal square root. So sqrt(25) = 5, not ±5; you write −sqrt(25) when you want the negative one. The square root of 0 is just 0.

Within the real numbers you cannot take the square root of a negative number, because no real value squares to a negative result (a square is always 0 or positive). Such roots require imaginary numbers. Most square roots are irrational — sqrt(2) ≈ 1.41421… runs on forever without repeating — and are usually left in radical form for exactness.