Squares and Square Roots

Squaring a number

To square a number, multiply it by itself. 5 squared, written $5^{2}$, is 5 × 5 = 25. Lay 25 tiles in 5 rows of 5 and they make a square with side 5, which is where the name comes from.

The results 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100, from $1^{2}$ up to $10^{2}$, are the square numbers. Each one is the number of tiles in a square.

5 rows of 5 tiles make a square with side 5: $5^{2} = 5 \times 5 = 25$.

The square root

The square root goes the other way. It starts with the tiles and asks for the side. $\sqrt{25}$, the square root of 25, is the number that makes 25 when it is multiplied by itself. 25 tiles make a square with side 5, so $\sqrt{25}$ is 5.

Squaring and taking the square root undo each other: $7^{2} = 49$, and $\sqrt{49}$ is 7.

A negative number squared is positive too: (−5) × (−5) = 25. The sign $\surd$ always means the positive square root, so $\sqrt{25}$ is 5, the length of the side.

Grow the square from its corner until it holds 25 tiles. Side 4 holds 16, too few, and side 6 holds 36, too many. Side 5 holds exactly 25, so $\sqrt{25}$ is 5.

Square numbers have whole roots

Only the square numbers have whole-number square roots, because only they make a complete square of tiles. 36 tiles make a 6 by 6 square, so $\sqrt{36}$ is 6. 20 tiles cannot make a square: a 4 by 4 square uses 16 of them and leaves 4 over, and a 5 by 5 square needs 25.

36 tiles make a square with side 6, so $\sqrt{36}$ is 6.

Between two square numbers

$\sqrt{20}$ is not a whole number, but you can still say where it lies. 20 is between the square numbers 16 and 25, so $\sqrt{20}$ lies between $\sqrt{16}$ and $\sqrt{25}$, which are 4 and 5.

20 is nearer to 16 than to 25, so $\sqrt{20}$ is nearer to 4. It is about 4.47, and 4.47 × 4.47 = 19.9809, very close to 20. To estimate any square root, find the square numbers on either side of it.

$\sqrt{20}$ lies between $\sqrt{16}$, which is 4, and $\sqrt{25}$, which is 5, a little nearer to 4.

The usual mistakes

Squaring is not doubling. $6^{2}$ is 6 × 6 = 36, not 6 × 2 = 12.

The square root is not half. $\sqrt{36}$ is 6, not 18. Check any square root by multiplying it by itself: 6 × 6 = 36 gives back the number you started with, and 18 × 18 = 324 does not.