Letters that stand for numbers

Have you ever played "I am think­ing of a num­ber"? The num­ber is real, but nobody has told you what it is yet. Math­e­mat­ics writes such a num­ber with a let­ter. In this les­son you meet let­ters that stand for num­bers, learn the short way to write them, and learn to add and take away counts of the same let­ter.

Vari­ables#

A sym­bol is a mark you write on the page. The let­ter $a$ is a sym­bol, and so is the num­ber 5.

A sym­bol which takes var­i­ous numer­i­cal val­ues is called a vari­able. Its other name is a lit­eral. The let­ters used most are $x, y, z, a, b, c$.

"Var­i­ous" means the value is not fixed. In one ques­tion $a$ may stand for 3. In the next ques­tion it may stand for 50. The let­ter is a place kept open for a num­ber.

Con­stants#

A sym­bol which has a fixed value is called a con­stant. The num­bers $1, 2, 3, 4, \dots$ are con­stants.

The sym­bol 5 means five in this ques­tion, in the next ques­tion, and in every sum ever writ­ten. That is the dif­fer­ence. A vari­able can change. A con­stant can­not.

Writ­ing a num­ber times a let­ter#

Between a num­ber and a let­ter we leave out the times sign, and the num­ber goes in front.

Look at $1 \times a$. One lot of $a$ is just $a$, so we never write the 1. In the same way, a minus sign in front of the num­ber stays in front, and $-1 \times a$ is writ­ten sim­ply as $-a$.

Adding a let­ter to itself#

Mul­ti­ply­ing is a short way to write a long addi­tion. You know that $3 + 3 + 3 + 3 = 3 \times 4$. Let­ters work the same way.

$$\begin{aligned}a + a &= 2a \\ a + a + a &= 3a \\ a + a + a + a &= 4a\end{aligned}$$

Try it with a num­ber to see it is not a trick. Let $a$ stand for 4. Then $a + a = 4 + 4 = 8$, and $2a = 2 \times 4 = 8$. They agree, what­ever num­ber $a$ stands for.

Adding and tak­ing away counts of a let­ter#

Three $a$'s and two more $a$'s make five $a$'s, just as three sweets and two sweets make five sweets. Add or take away the num­bers in front. The let­ter stays as it is.

Exam­ple 1#

Work out $3a + 2a$.

$$3a + 2a = (3 + 2)a = 5a$$

Exam­ple 2#

Work out $9a + 4a$.

$$9a + 4a = (9 + 4)a = 13a$$

Exam­ple 3#

Work out $5a - 3a$ and $9a - 5a$.

$$\begin{aligned}5a - 3a &= (5 - 3)a = 2a \\ 9a - 5a &= (9 - 5)a = 4a\end{aligned}$$

Exam­ple 4#

Work out $3a + 7a + 8a$.

$$3a + 7a + 8a = (3 + 7 + 8)a = 18a$$

Check with $a = 2$: $6 + 14 + 16 = 36$, and $18 \times 2 = 36$. They agree.

Exam­ple 5#

Work out $3b - 2b$.

$$3b - 2b = (3 - 2)b = 1b = b$$

One lot of $b$ is writ­ten sim­ply as $b$.

Dif­fer­ent let­ters do not mix#

What about $3a + 2b$? These are counts of dif­fer­ent things, like three apples and two bananas. You can­not call them five apples. So $3a + 2b$ stays as it is. Only counts of the same let­ter join together.

Remem­ber. Add or take away the num­bers in front, and keep the let­ter.

Adding a let­ter, or mul­ti­ply­ing it#

Adding $a$ to itself gives $2a$. Mul­ti­ply­ing $a$ by itself is quite dif­fer­ent, and it is writ­ten in a new way:

$$a \times a = a^{2}$$

$a^{2}$ is read as "$a$ squared" or "$a$ power 2". The small raised 2 counts how many $a$'s are mul­ti­plied.

The pic­ture shows how dif­fer­ent they are. With $a = 5$, two rows of five make 10 squares, but five rows of five make a whole square of 25.

So $2a$ and $a^{2}$ are not the same. When $a = 5$, $2a$ is 10 but $a^{2}$ is 25. When $a = 3$, $2a$ is 6 but $a^{2}$ is 9. You will learn much more about pow­ers in the next les­son.

Remem­ber. $2a$ counts the $a$'s that are added. $a^{2}$ counts the $a$'s that are mul­ti­plied.

Com­mon mis­takes#

• Read­ing $a^{2}$ as $2a$. The small 2 means mul­ti­ply two $a$'s, not add them.
• Chang­ing the let­ter when adding. $3a + 2a$ is $5a$, not $5a^{2}$ and not 5.
• Join­ing dif­fer­ent let­ters. $3a + 2b$ is not $5ab$. It stays as it is.
• Writ­ing $1a$. One lot of $a$ is writ­ten $a$.

Key terms#

• Vari­able (lit­eral): a sym­bol that can take var­i­ous val­ues, such as $x$ or $a$.
• Con­stant: a sym­bol with a fixed value, such as 7.
• $a$ squared: $a \times a$, writ­ten $a^{2}$.

Prac­tice#

1. Is $z$ a vari­able or a con­stant?
2. Is 12 a vari­able or a con­stant?
3. Write $5 \times b$ the short way.
4. Write $b + b + b$ the short way.
5. Write $b + b + b + b$ the short way.
6. Work out $3a + 2a$.
7. Work out $8a - 3a$.
8. Work out $6x + 6x$.
9. Work out $12y - 7y$.
10. Work out $4b + 3b - 2b$.
11. Write $b \times b$ with a power.
12. If $a = 6$, find the value of $2a$ and of $a^{2}$.

Answers#