Adding integers

The tem­per­a­ture in Shimla on a win­ter night can be $-4$ degrees and rise by 9 degrees by noon. A shop­keeper can gain ₹146 on one day and lose ₹238 the next. When­ever amounts like these are put together, you are adding inte­gers. In this les­son you add inte­gers on the num­ber line first, then learn two short rules that work for any inte­gers at all, even very big ones.

Adding is a walk on the num­ber line#

Think of the num­ber line. A plus sign tells you to walk right. A minus sign tells you to walk left. The num­ber tells you how many steps. Where you stop is the answer.

The mag­ni­tude of a num­ber is its size when you ignore the sign. The mag­ni­tude of $-7$ is 7, and the mag­ni­tude of $7$ is also 7. The mag­ni­tude tells you how far to walk. The sign tells you which way.

The four cases#

Every adding sum of two inte­gers is one of these four.

Both signs the same#

In $5 + 8$, both walks go right: 5 steps and then 8 more. So you add: $5 + 8 = 13$.

In $-5 - 8$, both walks go left: 5 steps left and then 8 more steps left. So you add the mag­ni­tudes and keep the minus sign:

$$-5 - 8 = -(5 + 8) = -13$$

The two minus signs do not can­cel here. Each one is another walk to the left.

Signs dif­fer­ent#

In $-5 + 8$, you walk 5 steps left and 8 steps right. The walks pull against each other. The shorter walk undoes part of the longer one. What is left is the dif­fer­ence, $8 - 5 = 3$. The walk right was longer, so the answer is pos­i­tive:

$$-5 + 8 = +(8 - 5) = 3$$

In $5 - 8$, you walk 5 right and 8 left. The dif­fer­ence is again 3, but now the walk left was longer, so the answer is neg­a­tive:

$$5 - 8 = -(8 - 5) = -3$$

The two rules#

Remem­ber. If both signs are the same, add the mag­ni­tudes and keep that sign.

Remem­ber. If the signs are dif­fer­ent, find the dif­fer­ence of the mag­ni­tudes (the big­ger take away the smaller), and give the answer the sign of the num­ber with the big­ger mag­ni­tude.

To show "the dif­fer­ence", always the big­ger num­ber take away the smaller, we can write a small $(-)$ between the two num­bers. So $18 \,(-)\, 17$ means the dif­fer­ence between 18 and 17, which is 1. This way you never take a big­ger num­ber from a smaller one.

Exam­ple 1#

Work out $17 + 18$, $-17 - 18$, $-17 + 18$ and $17 - 18$.

• $17 + 18 = 35$. Both pos­i­tive, so add.
• $-17 - 18 = -35$. Both neg­a­tive, so add $18 + 17 = 35$ and keep the minus.
• $-17 + 18 = 1$. Signs dif­fer: $18 \,(-)\, 17 = 1$. The 18 is big­ger and pos­i­tive, so the answer is $+1$.
• $17 - 18 = -1$. Signs dif­fer: $18 \,(-)\, 17 = 1$. The 18 is big­ger and neg­a­tive, so the answer is $-1$.

Notice the last two. The dig­its of the answer are the same. Only the sign changes, and it comes from the big­ger num­ber.

Big­ger num­bers#

The rules do not change for big num­bers. Do the adding or the dif­fer­ence at the side, then put the sign in front.

Exam­ple 2#

Work out $234 + 79$.

Both pos­i­tive, so add: $234 + 79 = 313$.

Exam­ple 3#

Work out $-146 - 238$.

Both neg­a­tive, so add the mag­ni­tudes: $238 + 146 = 384$. Keep the minus sign.

$$-146 - 238 = -384$$

Exam­ple 4#

Work out $46 - 128$.

Signs dif­fer. Find the dif­fer­ence: $128 \,(-)\, 46 = 82$. The big­ger num­ber, 128, has a minus sign, so the answer is neg­a­tive.

$$46 - 128 = -82$$

Exam­ple 5#

Work out $-79 + 115$.

Signs dif­fer: $115 \,(-)\, 79 = 36$. The big­ger num­ber, 115, is pos­i­tive.

$$-79 + 115 = 36$$

Exam­ple 6#

Work out $3196 - 528$ and $31532 - 4369$.

In $3196 - 528$ the big­ger num­ber, 3196, is pos­i­tive, so the answer is pos­i­tive: $3196 \,(-)\, 528 = 2668$.

In $31532 - 4369$ the big­ger num­ber is again pos­i­tive. Work the dif­fer­ence down the page with care:

$$\begin{array}{r} 31532 \\ -\,4369 \\ \hline 27163 \end{array}$$

So $31532 - 4369 = 27163$. Check by adding back: $27163 + 4369 = 31532$.

Exam­ple 7#

Work out $-347 + 168$.

Signs dif­fer, and the big­ger num­ber, 347, is neg­a­tive. So the answer is neg­a­tive. To find $347 \,(-)\, 168$, you can count up from 168 in two jumps: from 168 to 200 is 32, and from 200 to 347 is 147.

$$32 + 147 = 179$$

So $-347 + 168 = -179$.

Exam­ple 8#

Work out $-789 - 1369$.

Both neg­a­tive: $1369 + 789 = 2158$. So $-789 - 1369 = -2158$.

Longer sums#

When a sum has more than two num­bers, you may take the steps in any order. So gather all the pos­i­tive num­bers together and all the neg­a­tive num­bers together. Then you have just one sum of two num­bers.

Exam­ple 9#

Work out $23 + 59 - 32 - 94$.

The pos­i­tive num­bers: $59 + 23 = 82$. The neg­a­tive num­bers: $94 + 32 = 126$.

$$\begin{aligned}23 + 59 - 32 - 94 &= 82 - 126 \\ &= -44\end{aligned}$$

The dif­fer­ence is $126 \,(-)\, 82 = 44$, and the big­ger group was neg­a­tive, so the answer is $-44$.

Exam­ple 10#

Work out $57 - 123 + 248 - 39$.

The pos­i­tive num­bers: $248 + 57 = 305$. The neg­a­tive num­bers: $123 + 39 = 162$.

$$\begin{aligned}57 - 123 + 248 - 39 &= 305 - 162 \\ &= 143\end{aligned}$$

The big­ger group was pos­i­tive, so the answer is $+143$.

Exam­ple 11#

Work out $12 + 38 - 11 - 32$ in two ways.

Group­ing the signs: $50 - 43 = 7$.

Tak­ing pairs: $12 - 11 = 1$ and $38 - 32 = 6$, and $1 + 6 = 7$. Both ways agree.

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?

$$-6 + 11 = +(11 - 6) = 5$$

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

Com­mon mis­takes#

• Think­ing two minus signs always make a plus. In $-5 - 8$ both walks go left, so the answer is $-13$, not 13.
• Tak­ing the sign of the first num­ber. In $-79 + 115$ the answer takes the sign of 115, the big­ger num­ber, so it is pos­i­tive.
• Adding when the signs are dif­fer­ent. In $46 - 128$ you find the dif­fer­ence, 82, not the total, 174.
• Drop­ping a num­ber in a long sum. Tick each num­ber as you move it into the pos­i­tive or neg­a­tive group.

Key terms#

• Mag­ni­tude: the size of a num­ber when its sign is ignored.
• Dif­fer­ence: the big­ger mag­ni­tude take away the smaller, writ­ten with a small $(-)$.
• Oppo­site: the inte­ger the same dis­tance from zero on the other side, such as 7 and $-7$. They add to zero.

Prac­tice#

1. $5 + 8$
2. $-7 - 12$
3. $-13 + 18$
4. $-18 + 7$
5. $-153 + 392$
6. $2379 + 467$
7. $-7301 - 58012$
8. $-34201 + 5832$
9. $247 + 72 - 138 - 59$
10. $168 - 79 + 234 - 349$
11. $1238 - 1784 + 342$
12. A shop­keeper gains ₹146 on Mon­day and loses ₹238 on Tues­day. Write this as an adding sum of inte­gers and find his total gain or loss.

Answers#