Letters that stand for numbers

Have you ever seen a puz­zle like "I am think­ing of a num­ber"? The num­ber is real, but nobody has told you what it is yet. Math­e­mat­ics has a neat way to write such a num­ber: it uses a let­ter. You will meet let­ters like this in for­mu­las for area, in rules for pat­terns, and in every alge­bra les­son that fol­lows. This les­son is your first step.

Every sum so far was made of num­bers you could see. Look at $7 + 5 = 12$. It tells you both num­bers. It asks you for the answer. But you are not always told each num­ber. This les­son shows you what to do then. You put a let­ter in place of that num­ber.

A let­ter is not a new kind of thing. It is a num­ber. You have just not been told which one. So all you know still works. Adding is still adding. Mul­ti­ply­ing is still a short way to write a long addi­tion. Most of the work below is count­ing.

Vari­able

Start with two words you will need. A sym­bol is a mark you write on the page. The let­ter $a$ is a sym­bol. So is the num­ber $5$. The value of a sym­bol is the num­ber it stands for.

Some sym­bols can stand for var­i­ous num­bers, not just one. A sym­bol like that is called a vari­able. It has a sec­ond name as well: a lit­eral. That word is Latin for a let­ter. Both names mean the same thing. Where a book says lit­eral, read vari­able.

The let­ters $x$, $y$, $z$, $a$, $b$ and $c$ are used most. The word var­i­ous does the real work there. It means the value is not fixed. $a$ may stand for $3$ in one ques­tion. It may stand for $50$ in the next. Nei­ther one is more right. The let­ter is a place kept open for a num­ber.

Con­stant

Other sym­bols can stand for one num­ber only. Their value is fixed. A sym­bol like that is called a con­stant. The num­bers $5$, $2$, $3$ and $7$ are con­stants.

A con­stant has nowhere to move. The sym­bol $5$ means five in this ques­tion. It means five in the next one. It will mean five in every sum ever writ­ten. That is the whole dif­fer­ence between the two. A vari­able is free to change. A con­stant is not.

Remem­ber. A vari­able takes var­i­ous val­ues. A con­stant keeps one.

Adding a let­ter to itself

You know that mul­ti­ply­ing is a short way to write a long addi­tion. Take four threes added up: $3 + 3 + 3 + 3 = 12$. We write that as $3 \times 4 = 12$. Turn­ing a mul­ti­pli­ca­tion round does not change it. So we can write $4 \times 3 = 12$ too. The same short way works when you add a let­ter. With a let­ter we use the sec­ond order. The count goes in front. The thing being counted comes after it.

$a + a$ is $a$ taken twice. That is $2 \times a$. Between a num­ber and a let­ter we leave the times sign out. So $2 \times a$ is writ­ten sim­ply as $2a$.

$a + a$

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

$a + a + a$

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

That did not depend on the count being two or three. Add $a$ on $n$ times and you get $n$ lots of $a$.

$a$ added $n$ times

$$\begin{aligned}a + a + a + \cdots \; n \text{ times} &= n \times a \\ &= na\end{aligned}$$

Try it with a num­ber. Then you can see it is not a trick. Let $a$ stand for $4$. Now $a + a$ is $4 + 4 = 8$. And $2a$ is $2 \times 4 = 8$. The two agree. They will agree what­ever $a$ stands for. Both of them mean the same thing: two lots of $a$.

Count­ing the a's

Once $a$ can be counted, adding and tak­ing away become count­ing. Three $a$'s and two more $a$'s make five $a$'s. It works just like sweets. Three sweets and two sweets make five sweets. The let­ter is car­ried along untouched. Only the num­ber in front of it changes.

$3a + 2a$

$$\begin{aligned}3a + 2a &= (3 + 2)a \\ &= 5a\end{aligned}$$

$7a - 3a$

$$\begin{aligned}7a - 3a &= (7 - 3)a \\ &= 4a\end{aligned}$$

The bracket is the whole rea­son this works. $3a$ is three lots of $a$. $2a$ is two lots of $a$. So together there are $3 + 2$ lots of $a$. The count­ing hap­pens inside the bracket. The $a$ waits out­side. It comes back unchanged.

Put a num­ber in and check. Let $a$ stand for $10$. Then $3a$ is $3 \times 10 = 30$. And $2a$ is $2 \times 10 = 20$. Their total is $30 + 20 = 50$. And $5a$ is $5 \times 10 = 50$. The rule and the num­bers say the same thing.

Three counts in one line

Work out $4b + 3b - 2b$. Go from left to right and keep the let­ter.

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

Check with $b = 2$: $8 + 6 - 4 = 10$, and $5 \times 2 = 10$. They agree.

Dif­fer­ent let­ters do not mix

What about $3a + 2b$? Here the counts are of dif­fer­ent things, like three apples and two bananas. You can­not call that five apples or five bananas. So $3a + 2b$ stays as it is. Only counts of the same let­ter join together. If you are told later that $a = 1$ and $b = 4$, then you can find the value: $3 + 8 = 11$. Until then, the two parts are kept side by side.

Remem­ber. Add the counts and keep the let­ter. $3a + 2a = 5a$. That holds what­ever num­ber $a$ turns out to be.

Mul­ti­ply­ing a let­ter by itself

Adding $a$ to itself gave $2a$. Mul­ti­ply­ing $a$ by itself gives some­thing quite dif­fer­ent. We write it in a new way too.

$a \times a$

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

$a^{2}$ is read as a squared, or as a power 2.

$a \times a \times a$

$$a \times a \times a = a^{3}$$

$a^{3}$ is read as a cubed, or as a power 3.

The pic­ture below shows why $2a$ and $a^{2}$ are so dif­fer­ent. With $a = 5$, two rows of five make $10$ squares, but five rows of five make a whole square of $25$. That is where the name "squared" comes from: $a \times a$ is the area of a square whose side is $a$.

The small raised num­ber is a count, not an answer. It tells you how many $a$'s are being mul­ti­plied together. It tells you noth­ing else. Writ­ten out in full, $a^{5}$ is $a \times a \times a \times a \times a$. That is five of them, and slow to read. The small $5$ does the count­ing for you.

Take care not to read $a^{2}$ as $a \times 2$. The two are not the same. Let $a$ be $5$. Then $2a = 10$ but $a^{2} = 25$. Let $a$ be $3$. Then $2a = 6$ but $a^{2} = 9$. They do agree when $a$ is $2$. Both ways give $2 \times 2 = 4$. But among the count­ing num­bers $1, 2, 3, \dots$ it is the only value they agree on. Try any other one. The two answers will not match.

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

Expo­nen­tial form

Three let­ters work together here. Each one has its own job. The let­ter $a$ is the num­ber you mul­ti­ply. The let­ter $n$ says how many of them to use. So mul­ti­ply that many $a$'s together. The answer you get is called $N$. Then $N = a^{n}$. You read that as a power n. And $a^{n}$ is called the expo­nen­tial form of $N$.

Take $a = 2$ and $n = 3$. Then $N$ is $2 \times 2 \times 2 = 8$. So the expo­nen­tial form of $8$ is $2^{3}$. The num­ber $8$ is what you get. $2^{3}$ says what it is made of.

$a$ is the base. It is the num­ber being mul­ti­plied. $n$ says how many times it appears. Give both of those. Then the num­ber has been described in full.

Expo­nen­tial is a long name for a short idea. The small raised num­ber has a name of its own. It is called the expo­nent. The long name is built from that word. So the name is telling you where to look. Look at the small num­ber.

This way of writ­ing is worth hav­ing. It says what the num­ber is made of. Ten 2s mul­ti­plied together is long to write. It is easy to mis­count. And one look at it tells you noth­ing. $2^{10}$ is short. You can­not mis­count it. And one look tells you the num­ber is made of 2s and noth­ing else.

Express 1024 in expo­nen­tial form, whose base is 2

The base is given to you. So only one thing is left to find. How many times must $2$ be mul­ti­plied? Take the 2s out one at a time. You do that by halv­ing.

$1024$ can be halved exactly. You can share it into two equal parts with remain­der $0$. That is what divis­i­ble by $2$ means. You have met this idea before. The last les­son split the num­bers into even ones and odd ones. The even ones are the ones you can halve exactly. So one 2 comes out of $1024$. That leaves $512$. $512$ is divis­i­ble by $2$ as well. So another 2 comes out. Keep going until $1$ is all that is left.

Each divi­sion took one 2 out of the num­ber. You reached $1$ with noth­ing left over. So put every one of those 2s back. They must give $1024$ again.

$1024$ as $2$s mul­ti­plied together

$$1024 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$$

Count the 2s on that line. There are ten. The base is $2$ and it appears ten times. So the expo­nen­tial form of $1024$ is $2^{10}$. You read that as 2 power 10.

You can check it from the other end. This time you dou­ble. Start at $2$. Dou­ble one row at a time until you reach $1024$.

Each row after the first is the row above it dou­bled. The small num­ber climbs by one as you go down. There is a rea­son for that. Dou­bling means mul­ti­ply­ing by $2$ once more. So each row puts one more 2 into the mul­ti­pli­ca­tion. And the tenth row lands on $1024$.

Remem­ber. In $a^{n}$ the base $a$ is what you mul­ti­ply. $n$ is how many times you mul­ti­ply it. The small num­ber counts. It does not answer.

Another exam­ple: 243 with base 3

The same method works for any base. Take the 3s out one at a time by divid­ing by $3$.

$$\begin{aligned}243 \div 3 &= 81 \\ 81 \div 3 &= 27 \\ 27 \div 3 &= 9 \\ 9 \div 3 &= 3 \\ 3 \div 3 &= 1\end{aligned}$$

Five divi­sions reached $1$. So $243 = 3 \times 3 \times 3 \times 3 \times 3 = 3^{5}$. Check by mul­ti­ply­ing up: $3, 9, 27, 81, 243$.

Your turn

Vari­able or con­stant?

Count the let­ters

Write these with a power

Express in expo­nen­tial form

1. Express $8$ in expo­nen­tial form, whose base is $2$
2. Express $64$ in expo­nen­tial form, whose base is $2$
3. Express $256$ in expo­nen­tial form, whose base is $2$
4. Express $27$ in expo­nen­tial form, whose base is $3$
5. Express $81$ in expo­nen­tial form, whose base is $3$
6. Express $125$ in expo­nen­tial form, whose base is $5$
7. Express $49$ in expo­nen­tial form, whose base is $7$
8. Express $1000$ in expo­nen­tial form, whose base is $10$

Say these aloud

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 $5a^{2}$ or $5$.
• Join­ing dif­fer­ent let­ters, as in writ­ing $3a + 2b = 5ab$.
• Think­ing a let­ter always has the same value. A vari­able can take var­i­ous val­ues from ques­tion to ques­tion.
• Mis­count­ing the base in expo­nen­tial form. Count the divi­sions that reach $1$, not the num­bers in the chain.
• Writ­ing $2^{10}$ as $20$. The expo­nent is a count of 2s, not a num­ber to mul­ti­ply by.

Key terms

Sym­bol

A mark writ­ten on the page, such as $a$ or $5$.

Vari­able (lit­eral)

A sym­bol that can stand for var­i­ous num­bers.

Con­stant

A sym­bol whose value is fixed, such as $7$.

Base

The num­ber being mul­ti­plied in $a^{n}$; here it is $a$.

Expo­nent

The small raised num­ber that counts how many times the base is mul­ti­plied.

Expo­nen­tial form

A num­ber writ­ten as a base with an expo­nent, such as $1024 = 2^{10}$.

Squared and cubed

Names for a power 2 and a power 3.

Answers

Vari­able or con­stant?

1) $x$: vari­able. 2) $9$: con­stant. 3) $b$: vari­able. 4) $12$: con­stant. 5) $z$: vari­able. 6) $100$: con­stant.

Count the let­ters

1) $4a$ 2) $8a$ 3) $13a$ 4) $9a$ 5) $12a$ 6) $3b$ 7) $3a$ 8) $5a$ 9) $7a$ 10) $9a$ 11) $7y$ 12) $2x$

Write these with a power

1) $a^{2}$ 2) $b^{3}$ 3) $x^{2}$ 4) $y^{4}$ 5) $z^{2}$ 6) $a^{5}$

Express in expo­nen­tial form

1. $8 = 2^{3}$
2. $64 = 2^{6}$
3. $256 = 2^{8}$
4. $27 = 3^{3}$
5. $81 = 3^{4}$
6. $125 = 5^{3}$
7. $49 = 7^{2}$
8. $1000 = 10^{3}$

Say these aloud

Model answers: 1) "a squared", or "a power two". 2) "a cubed", or "a power three". 3) "two a". 4) "three a". 5) "a power n". 6) "n a".