Squares of numbers

Arrange 16 chairs in a hall so that there are as many rows as there are chairs in each row. You get 4 rows of 4. Num­bers like 16 have a spe­cial name: they are squares. In this les­son you will learn what the square of a num­ber is, learn the squares from 1 to 30 by heart, and spot some pat­terns that let you square many num­bers in your head.

The square of a num­ber#

When a num­ber is mul­ti­plied by itself, the answer is called the square of that num­ber.

$$a \times a = a^2$$

We read $a^2$ as "a squared" or "a power 2". The small raised $2$ tells you that two copies of $a$ are mul­ti­plied.

$$\begin{aligned} 1^2 &= 1 \times 1 = 1 \\ 2^2 &= 2 \times 2 = 4 \\ 3^2 &= 3 \times 3 = 9 \end{aligned}$$

The name comes from shapes. A square with sides of $3$ units can be filled with $3$ rows of $3$ small unit squares, which is $9$ in all.

A num­ber that is the square of a whole num­ber is called a per­fect square. So $1$, $4$, $9$, $16$ and $25$ are per­fect squares. The num­ber $10$ is not, because no whole num­ber times itself gives $10$.

Squares from 1 to 30#

Learn this table well. You will use it again and again in this chap­ter.

If you for­get one, mul­ti­ply it out. For $22^2$:

$$\begin{array}{r} 22 \\ \times\ 22 \\ \hline 44 \\ 440 \\ \hline 484 \end{array}$$

Pat­tern 1: around 25#

Look at pairs of num­bers that are the same dis­tance from $25$.

In each pair the squares end in the same two dig­its. Only the hun­dreds change. So if you know $24^2 = 576$, then $26^2$ is $100$ more: $676$.

Pat­tern 2: mul­ti­ples of 10#

To square a mul­ti­ple of $10$, square the digit in front and write two zeros after it.

$$\begin{aligned} 10^2 &= 100 \\ 20^2 &= 400 \\ 30^2 &= 900 \\ 40^2 &= 1600 \\ 50^2 &= 2500 \\ 60^2 &= 3600 \\ 70^2 &= 4900 \\ 80^2 &= 6400 \\ 90^2 &= 8100 \\ 100^2 &= 10000 \end{aligned}$$

For exam­ple, $70^2 = 70 \times 70$. That is $7 \times 7 = 49$, with the two zeros put back: $4900$.

Pat­tern 3: num­bers end­ing in 5#

Here is a lovely short cut. To square a num­ber that ends in $5$:

1. Take the dig­its in front of the $5$. Mul­ti­ply that num­ber by the next num­ber up.
2. Write $25$ after the answer.

$$\begin{aligned} 15^2 &= 225 & \quad 1 \times 2 &= 2 \\ 25^2 &= 625 & \quad 2 \times 3 &= 6 \\ 35^2 &= 1225 & \quad 3 \times 4 &= 12 \\ 45^2 &= 2025 & \quad 4 \times 5 &= 20 \\ 55^2 &= 3025 & \quad 5 \times 6 &= 30 \\ 65^2 &= 4225 & \quad 6 \times 7 &= 42 \\ 75^2 &= 5625 & \quad 7 \times 8 &= 56 \\ 85^2 &= 7225 & \quad 8 \times 9 &= 72 \\ 95^2 &= 9025 & \quad 9 \times 10 &= 90 \end{aligned}$$

Exam­ple 1#

Find $105^2$.

The dig­its in front of the $5$ make $10$. The next num­ber up is $11$, and $10 \times 11 = 110$. Write $25$ after it.

$$105^2 = 11025$$

Pat­tern 4: adding odd num­bers#

Add the odd num­bers in order, start­ing from $1$.

$$\begin{aligned} 1 &= 1 = 1^2 \\ 1 + 3 &= 4 = 2^2 \\ 1 + 3 + 5 &= 9 = 3^2 \\ 1 + 3 + 5 + 7 &= 16 = 4^2 \end{aligned}$$

The sum of the first $n$ odd num­bers is always $n^2$.

Exam­ple 2#

Find $1 + 3 + 5 + 7 + 9 + 11 + 13 + 15$ with­out adding.

There are $8$ odd num­bers, start­ing from $1$. So the sum is $8^2 = 64$.

Pat­tern 5: the last digit#

Look at the last digit of every square in the table. It is always $0$, $1$, $4$, $5$, $6$ or $9$. It is never $2$, $3$, $7$ or $8$.

So a num­ber end­ing in $2$, $3$, $7$ or $8$ can never be a per­fect square. For exam­ple, $2768$ ends in $8$, so it is not a per­fect square. But be care­ful: a num­ber end­ing in $1$, such as $11$, need not be a per­fect square either. The last digit can only rule a num­ber out.

Remem­ber. $a^2 = a \times a$. A per­fect square is the square of a whole num­ber. Per­fect squares never end in $2$, $3$, $7$ or $8$.

Com­mon mis­takes#

• Writ­ing $5^2 = 10$. The $2$ means mul­ti­ply $5$ by itself, not by $2$: $5^2 = 25$.
• For­get­ting a zero when squar­ing a mul­ti­ple of $10$. $40^2$ is $1600$, not $160$.
• In the end­ing-in-5 short cut, mul­ti­ply­ing by the same num­ber instead of the next one. For $35^2$ use $3 \times 4$, not $3 \times 3$.

Prac­tice#

1. Find $13^2$.
2. Find $19^2$.
3. Find $40^2$.
4. Find $65^2$ using the short cut.
5. Find $85^2$ using the short cut.
6. Given $23^2 = 529$, find $27^2$ with­out mul­ti­ply­ing.
7. Is $10$ a per­fect square? Give a rea­son.
8. Can $5437$ be a per­fect square? Give a rea­son.
9. Find $1 + 3 + 5 + \ldots + 19$, the sum of the first $10$ odd num­bers.
10. Find $115^2$ using the short cut.

Answers#