Perfect square

The first twelve

$n$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$
$n^2$	$1$	$4$	$9$	$16$	$25$	$36$	$49$	$64$	$81$	$100$	$121$	$144$

Recognising these turns a square root into recall rather than calculation: $\sqrt{81}$ is $9$ because $81$ is in the table.

Why they are called square

A perfect square is exactly the area of a square with whole-number sides. $36$ unit tiles form a $6 \times 6$ square with none left over, and $35$ tiles do not form a square at all.

Trapping a root between two of them

Most numbers are not perfect squares, and the two nearest ones fix the answer between whole numbers:

$$49 < 50 < 64 \quad\Rightarrow\quad 7 < \sqrt{50} < 8$$

$50$ sits just above $49$, so $\sqrt{50}$ sits just above $7$ — about $7.07$, not halfway to $8$. See square roots for the estimating method.

Where they show up in algebra

A quadratic whose discriminant is a perfect square factors over the whole numbers, so solving by factoring will work on it. When it is not a perfect square, the roots are irrational and the quadratic formula is the reliable route.