At its simplest, squaring a number means multiplying that number by itself.
If we have a number $a$, the product $a \times a$ is written as $a^2$.
How to read it: You can say “$a$ square” or “$a$ to the power of 2.”
The Formula: $\therefore a^2 = a \times a$
Memorizing squares up to 30 is a game-changer for speed. Here is a quick reference list to get you started:
| Number | Square | Number | Square | Number | Square | ||
| $1^2$ | 1 | $11^2$ | 121 | $21^2$ | 441 | ||
| $2^2$ | 4 | $12^2$ | 144 | $22^2$ | 484 | ||
| $3^2$ | 9 | $13^2$ | 169 | $23^2$ | 529 | ||
| $4^2$ | 16 | $14^2$ | 196 | $24^2$ | 576 | ||
| $5^2$ | 25 | $15^2$ | 225 | $25^2$ | 625 | ||
| $6^2$ | 36 | $16^2$ | 256 | $26^2$ | 676 | ||
| $7^2$ | 49 | $17^2$ | 289 | $27^2$ | 729 | ||
| $8^2$ | 64 | $18^2$ | 324 | $28^2$ | 784 | ||
| $9^2$ | 81 | $19^2$ | 361 | $29^2$ | 841 | ||
| $10^2$ | 100 | $20^2$ | 400 | $30^2$ | 900 |
Check out the relationship between squares in the 20s!
$21^2$ (441) and $29^2$ (841) differ by 400.
$22^2$ (484) and $28^2$ (784) differ by 300.
$23^2$ (529) and $27^2$ (729) differ by 200.
$24^2$ (576) and $26^2$ (676) differ by 100.
Patterns like these make memorization much easier!
Squaring multiples of 10 is very straightforward. You simply square the first digit and double the number of zeros.
$(10)^2 = 100$
$(20)^2 = 400$
$(50)^2 = 2500$
$(100)^2 = 10,000$
There is a brilliant trick for squaring any number that ends in 5 (like 15, 25, 35, etc.).
The last two digits will always be 25.
To find the first digits, take the first number and multiply it by the next consecutive integer (e.g., for 35, multiply 3 by 4 to get 12).
Let’s see it in action:
$(15)^2 = 225$
$(25)^2 = 625$
$(35)^2 = 1225$
$(45)^2 = 2025$
$(75)^2 = 5625$
$(95)^2 = 9025$