Square numbers and square roots

What it is

A square number is a whole number multiplied by itself: 7 · 7 = 7² = 49. The name comes from geometry: a square with side 7 has area 49. The first square numbers are

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …

The square root of a number a, written √a, is the non-negative number whose square is a. √49 = 7, because 7² = 49. The square root is the opposite of squaring, and can be written as a power: √a = a^(1/2).

• √81 = 9, because 9 · 9 = 81.
• √0 = 0 and √1 = 1.
• √50 is about 7.07: 50 lies between the squares 49 and 64, so the root lies between 7 and 8, close to 7. It is not a whole number, and not a fraction either.
• √0.25 = 0.5, because 0.5 · 0.5 = 0.25.

Non-examples:

• 50 is not a square number, and neither are 2, 3, 5 or 18. A square number never ends in 2, 3, 7 or 8.
• √(−4) does not exist as a real number. A real number squared is never negative.

Why it matters

The square root appears whenever you go from an area or a squared relationship back to a length: the side of a square with area 49 is √49 = 7 units. Pythagoras’ theorem, the distance formula and the solution of quadratic equations rest on it.

• Root of a product: √(a · b) = √a · √b, for example √36 = √4 · √9 = 2 · 3.
• Root of a quotient: √(a/b) = √a / √b.
• Fractional exponent: √a is written a^(1/2), and then the laws of powers apply.
• Radical equations: equations with the unknown under the root sign, where the solutions have to be checked.

All squares up to 20² are in the table of squares and cubes.

Common misconceptions

• “√a + √b = √(a + b).” No: √9 + √16 = 7 but √25 = 5.
• “The square root halves the number.” √16 = 4, not 8: 8 · 8 = 64.
• “Squaring means multiplying by 2.” 7² = 49, not 14. The mix-up is common because 2 · 2 = 4 = 2².
• “√49 = ±7.” The root sign gives only 7. It is the equation x² = 49 that has the solutions 7 and −7.
• “(−3)² = −9.” (−3)² = (−3) · (−3) = 9. It is −3² (without brackets) that is −9.

Common questions