Letters that stand for numbers

This les­son is the first step into alge­bra. Alge­bra lets you write about num­bers even when you do not know them yet, by using let­ters. You meet it in rules like "the perime­ter of a square is four times its side", in the fare chart of a bus, and in almost every chap­ter of math­e­mat­ics that comes after this one. It all begins with two sim­ple ideas: the vari­able and the con­stant.

Some­one gives you a closed box of sweets. Nobody has counted them. You can­not say how many are inside.

There is a num­ber in that box. It is a def­i­nite num­ber. You just do not know it yet.

You still want to talk about it. So you give that num­ber a short name. You can call it $x$.

Remem­ber. $x$ is not the box. $x$ is the num­ber of sweets in the box.

Why a name helps

A name lets you work before you count. If you dropped $6$ more sweets in, the box would hold $x + 6$ sweets.

You did not open the box. You still wrote down the new amount. That is what let­ters are for.

You can take sweets away in the same way. If you eat $2$ of them, the box holds $x - 2$ sweets. Later, when you open the box and find $7$, you can work out the num­bers.

$$\begin{aligned}x + 6 &= 7 + 6 = 13 \\ x - 2 &= 7 - 2 = 5\end{aligned}$$

Putting a num­ber in place of the let­ter like this is called sub­sti­tut­ing.

A vari­able

One day you open the box. You count $7$ sweets. So $x$ stands for $7$ in that box.

Your friend has a box too. Hers holds $70$ sweets. In her box $x$ stands for $70$.

The same let­ter fits both boxes. The let­ter did not change. The num­ber it stands for changed.

The num­ber a sym­bol stands for is called its value. So $x$ had the value $7$, then the value $70$.

A sym­bol like this is called a vari­able. This chap­ter also calls it a lit­eral. It is one idea with two names.

Remem­ber. A vari­able is a sym­bol which can take var­i­ous numer­i­cal val­ues. It can stand for dif­fer­ent num­bers at dif­fer­ent times. The chap­ter writes vari­able (or) lit­eral. Exam­ples: $x$, $y$, $z$, $a$, $b$, $c$.

A con­stant

Your box has $5$ printed on the lid. Some­one printed it there. You can­not rub a bit off it.

That $5$ does not tell you how many sweets are inside. It is $5$ today. It is $5$ tomor­row.

Nobody can make it $6$. Its value is fixed. A sym­bol like this is called a con­stant.

A sym­bol is any mark you write to stand for a num­ber. A let­ter is one kind. A writ­ten num­ber like $5$ is another kind.

Remem­ber. A con­stant is a sym­bol which has a fixed value. It always stands for the same num­ber. Exam­ples: $5$, $-3$, $2$.

$-3$ is a num­ber too. It sits below $0$. So does $-8$. Each one is locked to one value, so both are con­stants.

The two side by side

A let­ter can hide a num­ber you know

A let­ter is not always a mys­tery. You may know the num­ber and still use a let­ter.

Say your school bag weighs $4$ kilo­grams. You can write $w$ for its weight.

Now one short let­ter does the work. Two bags like yours weigh $2w$. You wrote that with­out say­ing $4$ once.

Tomor­row you may take a book out. The bag will weigh some­thing else. Only the num­ber $w$ stands for changes.

So $w$ is a vari­able. It just hap­pens to be $4$ today. If the weight could never change, $w$ would be a con­stant.

And a num­ber nobody knows

Some num­bers nobody has found yet. Nobody has weighed the bag. Nobody has counted the box.

A let­ter still lets you write about that num­ber. You can add to it and take from it.

Alge­bra is the part of maths where you work with num­bers you have not been told yet. Let­ters are what let you do it.

Remem­ber. A vari­able can take many val­ues. A con­stant keeps one fixed value.

Vari­ables and con­stants work­ing together

Most expres­sions in alge­bra mix the two. Look at the cost of pens that sell for ₹$5$ each. If you buy $n$ pens, you pay $5n$ rupees. Here $n$ is a vari­able, because you might buy any num­ber of pens. The $5$ is a con­stant, because the price does not change.

Now look at $3x + 4$. The let­ter $x$ is the vari­able. The num­bers $3$ and $4$ are con­stants. When $x = 2$, the expres­sion is worth $3 \times 2 + 4 = 10$. When $x$ changes, the value changes, but the $3$ and the $4$ stay just as they are.

A square with side $s$ has perime­ter $4s$. The side can be any length, so $s$ is a vari­able. A square always has four sides, so $4$ is a con­stant.

Your turn

Vari­able or con­stant?

Write vari­able or con­stant for each one. Ask your­self one ques­tion. Can this sym­bol stand for a dif­fer­ent num­ber another day?

Com­mon mis­takes

• Think­ing a let­ter stands for an object. $x$ is the num­ber of sweets, not the sweets or the box.
• Think­ing a let­ter is always a mys­tery. A vari­able may stand for a num­ber you already know today.
• Call­ing a neg­a­tive num­ber a vari­able. $-3$ is fixed, so it is a con­stant.
• For­get­ting that $0$ is a con­stant. It always has the value zero.
• Read­ing $5n$ as fifty-some­thing. It means $5 \times n$.

Key terms

Vari­able

A sym­bol which can take var­i­ous numer­i­cal val­ues, such as $x$, $y$ or $a$. Also called a lit­eral.

Con­stant

A sym­bol which has a fixed value, such as $5$, $-3$ or $2$.

Value

The num­ber a sym­bol stands for.

Sym­bol

Any mark writ­ten to stand for a num­ber, such as a let­ter or a numeral.

Sub­sti­tute

Put a num­ber in place of a let­ter, as in $x + 6 = 7 + 6$ when $x = 7$.

Alge­bra

The part of math­e­mat­ics that works with num­bers through let­ters.

Answers

1. $x$: vari­able
2. $9$: con­stant
3. $a$: vari­able
4. $-3$: con­stant
5. $m$: vari­able
6. $0$: con­stant
7. $p$: vari­able
8. $12$: con­stant
9. $t$: vari­able
10. $-8$: con­stant
11. $b$: vari­able
12. $100$: con­stant

Every let­ter here can stand for a dif­fer­ent num­ber on another day, so each let­ter is a vari­able. Every writ­ten num­ber always means the same amount, so each num­ber is a con­stant.