Adding integers

What this les­son is about

Inte­gers are every­where once you look for them. The tem­per­a­ture in Shimla on a win­ter night can be $-4$ degrees. A lift can go to base­ment level $-2$. A shop­keeper who spends more than he earns in a day has a loss, which is a neg­a­tive amount.

When­ever two of these amounts are put together, you are adding inte­gers. This les­son shows you how to do it on a num­ber line first, and then with two short rules that work for any inte­gers at all.

What inte­gers are

Inte­gers are the num­bers you count with, like $1$, $2$ and $3$.

Zero is an inte­ger too.

So are the same num­bers below zero.

A num­ber below zero has a minus sign in front of it.

There are no halves or bits of num­bers among the inte­gers.

A set is a col­lec­tion of num­bers gath­ered together.

You write the num­bers of a set inside curly brack­ets.

Math­e­mati­cians use the let­ter $Z$ as a short name for the set of inte­gers.

The set looks like this: $Z = \{ \dots, -3, -2, -1, 0, 1, 2, 3, 4, \dots \}$.

The dots at each end mean the list car­ries on for ever.

The num­ber line

A num­ber line puts the inte­gers in order from left to right.

The order runs $-3$, $-2$, $-1$, $0$, $1$, $2$, $3$, $4$.

Zero sits between the two sides.

Num­bers to the right of zero are pos­i­tive.

Num­bers to the left of zero are neg­a­tive.

Adding is a walk along this line.

A plus sign tells you to walk right.

A minus sign tells you to walk left.

The num­ber tells you how many steps to take.

The place where you stop is the answer.

Mag­ni­tude

Mag­ni­tude is a word for the size of a num­ber.

You look at the num­ber and ignore its sign.

The mag­ni­tude of $-7$ is $7$.

The mag­ni­tude of $7$ is $7$ as well.

So $-7$ and $7$ have the same mag­ni­tude.

They sit the same dis­tance from zero.

One is on the left and one is on the right.

The mag­ni­tude tells you how far you walk.

The sign tells you which way.

The four rules

There are four rules.

Each one is a short walk on the line.

Both signs pos­i­tive: $3 + 5 = 8$

Start at $3$.

Walk $5$ steps to the right.

You stop at $8$.

Both walks go the same way, so they add up.

Both signs neg­a­tive: $-3 - 5 = -8$

Start at zero and walk $3$ steps left.

Now walk $5$ more steps left.

Both walks go the same way again.

So you add $3$ and $5$ to get $8$ steps.

Every step went left, so the answer is $-8$.

In this sum the two minus signs do not undo each other.

Each one is one more walk to the left.

Tak­ing away a neg­a­tive num­ber, as in $12 - (-7)$, is a dif­fer­ent thing.

That case comes in the sub­trac­tion les­son.

$-3 - 5$

$$\begin{aligned}&-3 - 5 \\ &= -(3 + 5) \\ &= -8\end{aligned}$$

The bracket holds the two step counts.

You walked $3$ left and then $5$ left, so the bracket is $3 + 5$.

The minus sign in front of the bracket is the way you walked.

Dif­fer­ent signs: $-3 + 5 = 2$

Start at $-3$.

Walk $5$ steps to the right.

The first $3$ steps bring you back to zero.

You still have $2$ steps to take.

Those $2$ steps take you past zero, so you stop at $2$.

$-3 + 5$

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

You walked $5$ right and $3$ left, so the bracket is $5 - 3$.

The walk to the right was the longer one.

So the sign in front of the bracket is a plus.

Dif­fer­ent signs: $3 - 5 = -2$

Start at $3$.

Walk $5$ steps to the left.

The first $3$ steps bring you to zero.

Two steps are left over.

They carry you past zero to $-2$.

$3 - 5$

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

You walked $3$ right and $5$ left, so the bracket is $5 - 3$ again.

This time the walk to the left was the longer one.

So the sign in front of the bracket is a minus.

The two notes

Remem­ber. When both signs are the same, add the two mag­ni­tudes. Then keep that sign.

Remem­ber. When the signs are dif­fer­ent, take the smaller mag­ni­tude from the larger one. Then keep the sign of the larger one.

Your book words the first note in another way.

It says to keep the sign of the num­ber with the greater mag­ni­tude.

When both signs are the same, that is just the sign they share.

The first note works because both walks go the same way.

Nei­ther walk undoes the other, so the two lengths join into one.

The sec­ond note works because the two walks pull against each other.

The shorter walk undoes part of the longer one.

What is left over is how much longer the one walk was.

The sign comes from whichever walk was longer.

That is the way you were still going when you stopped.

Worked exam­ples

Here are the same four rules with big­ger num­bers.

Look at the last two rows.

The dig­its in the answer are the same each time.

Only the sign is dif­fer­ent.

The sign comes from whichever walk was longer.

More worked exam­ples

Here are a few more, each with every step shown.

$-8 + 3$

$$\begin{aligned}&-8 + 3 \\ &= -(8 - 3) \\ &= -5\end{aligned}$$

The signs are dif­fer­ent, so you take the smaller mag­ni­tude from the larger: $8 - 3 = 5$. The walk left was longer, so the answer is neg­a­tive.

$-4 - 9$

$$\begin{aligned}&-4 - 9 \\ &= -(4 + 9) \\ &= -13\end{aligned}$$

Both walks go left, so the lengths join: $4 + 9 = 13$ steps left.

$-45 + 45$

The two walks are the same length and go oppo­site ways. They undo each other exactly, so the answer is $0$. A num­ber and its oppo­site always add to zero.

A story sum. Early in the morn­ing the tem­per­a­ture in Leh is $-6$ degrees. By noon it has risen by $11$ degrees. What is the noon tem­per­a­ture?

$$\begin{aligned}&-6 + 11 \\ &= +(11 - 6) \\ &= 5\end{aligned}$$

The rise was the longer walk, so the noon tem­per­a­ture is $5$ degrees, above zero.

A longer sum

Some sums have more than two num­bers.

You can take the steps in any order you like.

You still fin­ish in the same place.

So gather the pos­i­tive num­bers into one group.

Gather the neg­a­tive num­bers into another group.

$12 + 13 - 38 - 56$

$$\begin{aligned}&12 + 13 - 38 - 56 \\ &= 25 - 94 \\ &= -69\end{aligned}$$

The pos­i­tive num­bers make $25$.

The neg­a­tive num­bers make $94$.

You walk $25$ steps right and $94$ steps left.

The left walk is longer, so you fin­ish left of zero.

$94 - 25 = 69$, so the answer is $-69$.

One sum, two ways

Here is the same sum done two ways.

Both ways give the same answer.

Way one: group the signs

$$\begin{aligned}&12 + 38 - 11 - 32 \\ &= 50 - 43 \\ &= 7\end{aligned}$$

The pos­i­tive num­bers make $50$.

The neg­a­tive num­bers make $43$.

Take $43$ from $50$ to get $7$.

The sec­ond way pairs each pos­i­tive num­ber with a neg­a­tive one.

Way two: take them in pairs

$$\begin{aligned}12 - 11 &= 1 \\ 38 - 32 &= 6 \\ 1 + 6 &= 7\end{aligned}$$

Each pair is one walk right and one walk left, taken together.

The pairs keep the num­bers small, which is eas­ier to do in your head.

Pick the way you find eas­ier.

The walk on the line is the same either way.

One more long sum: $25 - 60 + 17 - 2$

$$\begin{aligned}&25 - 60 + 17 - 2 \\ &= (25 + 17) - (60 + 2) \\ &= 42 - 62 \\ &= -20\end{aligned}$$

The pos­i­tive num­bers make $42$ and the neg­a­tive num­bers make $62$. The walk left is longer by $20$, so the answer is $-20$.

Prac­tice

Work these out, then check your answers below.

In ques­tion $8$ you land back on zero.

You walk $40$ steps right and $40$ steps left.

The two walks were the same length, so they undo each other.

Com­mon mis­takes

• Think­ing two minus signs always make a plus. In $-3 - 5$ both walks go left, so the answer is $-8$, not $8$.
• Tak­ing the sign of the first num­ber instead of the num­ber with the larger mag­ni­tude: $-3 + 5$ is $2$, not $-2$.
• Adding the mag­ni­tudes when the signs are dif­fer­ent. In $12 - 13$ you take away, which gives $-1$, not $-25$.
• Drop­ping a num­ber when group­ing a long sum. Tick each num­ber as you move it into the pos­i­tive or neg­a­tive group.
• Believ­ing that $-7$ is big­ger than $-2$ because 7 is big­ger than 2. On the num­ber line $-7$ is fur­ther left, so it is smaller.

Key terms

Inte­ger

A whole num­ber, zero, or the neg­a­tive of a whole num­ber. The set is writ­ten $Z$.

Pos­i­tive inte­ger

An inte­ger to the right of zero on the num­ber line.

Neg­a­tive inte­ger

An inte­ger to the left of zero, writ­ten with a minus sign.

Num­ber line

A line on which the inte­gers are marked in order from left to right.

Mag­ni­tude

The size of a num­ber when its sign is ignored; how far it is from zero.

Oppo­site

The inte­ger the same dis­tance from zero on the other side, such as $7$ and $-7$. The two add to zero.

Answers

1. $7 + 6 = 13$ (same signs: add)
2. $-7 - 6 = -(7 + 6) = -13$
3. $-7 + 6 = -(7 - 6) = -1$
4. $7 - 6 = +(7 - 6) = 1$
5. $-15 + 40 = +(40 - 15) = 25$
6. $18 - 25 = -(25 - 18) = -7$
7. $-11 - 19 = -(11 + 19) = -30$
8. $14 + 26 - 9 - 31 = 40 - 40 = 0$
9. $-20 + 8 + 5 = 13 - 20 = -7$
10. $23 - 40 + 12 = 35 - 40 = -5$