The tem­per­a­ture in Leh is 44 degrees in the after­noon and −6-6 degrees at night. How much did it drop? A shop­keeper owes ₹7 and the debt is can­celled. How much bet­ter off is he? Both ques­tions are sub­trac­tions with inte­gers. This les­son shows you how to do them on a num­ber line, and why tak­ing away a neg­a­tive num­ber turns into adding.

Sub­tract­ing means tak­ing one num­ber away from another.

Inte­gers are the whole num­bers. They go up from zero. They go down past zero too. Half num­bers are not inte­gers.

Num­bers below zero are called neg­a­tive num­bers. You write a minus sign in front of them, like −3-3. Num­bers above zero are called pos­i­tive.

With inte­gers the num­ber you take away can be neg­a­tive. That is the only new part here.

Tak­ing away moves you left

Pic­ture a num­ber line. It is a straight line with the num­bers writ­ten along it in order. Small ones sit on the left. Big ones sit on the right.

Adding moves you to the right. Tak­ing away moves you to the left.

The inte­gers carry on both ways: …,−3,−2,−1,0,1,2,3,4,…\dots, -3, -2, -1, 0, 1, 2, 3, 4, \dots

Start at 88 and take away 33. You move three steps left. You land on 55. So 8−3=58 - 3 = 5.

You can start below zero too. Start at −2-2 and take away 33. You move three steps left again. You land on −5-5. So −2−3=−5-2 - 3 = -5.

You can also cross zero on the way. Start at 22 and take away 66. Two steps left take you to 00. Four more take you to −4-4. So 2−6=−42 - 6 = -4.

Read the words in the right order

A ques­tion might be writ­ten like this: Sub­tract 77 from 1212.

The num­ber after the word from is the one you start with. So you start at 1212. You write 12−712 - 7. The answer is 55.

Remem­ber. Sub­tract 77 from 1212 means 12−712 - 7. The num­ber after from comes first.

It is easy to write this the wrong way round. 12−712 - 7 is 55. But 7−127 - 12 is −5-5. The two are not the same. Check the order every time.

Why tak­ing away a neg­a­tive adds

Some­times you take away a neg­a­tive num­ber. Then two minus signs sit side by side. Take −7-7 away from 1212 and you write 12−(−7)12 - (-7).

Here is why they turn into a plus. Face along the num­ber line. Adding walks you to the right. Tak­ing away turns you round first. So you walk to the left.

A minus sign in front of a num­ber means the other way. Walk­ing 77 means seven steps right. Walk­ing −7-7 means the same seven steps, but the other way. So you turn round.

Tak­ing away −7-7 turns you twice. Two turns leave you fac­ing right again. So you walk seven steps right.

This is not the same as −3−5-3 - 5. There the two minus signs are apart. Each one is a walk to the left, and the answer is −8-8. Here the minus sign sits straight in front of a neg­a­tive num­ber. That is what turns you round twice.

Money gives the same answer. Think of −7-7 as a debt: seven rupees you owe some­one. Take that debt away and you no longer owe it. So you are seven rupees bet­ter off. You end up with more.

A pat­tern shows it too. Watch the num­ber you take away go down by one each line, while the answer goes up by one:

12−2=1012−1=1112−0=1212−(−1)=1312−(−2)=14\begin{aligned}12 - 2 &= 10 \\ 12 - 1 &= 11 \\ 12 - 0 &= 12 \\ 12 - (-1) &= 13 \\ 12 - (-2) &= 14\end{aligned}

The pat­tern only keeps going if tak­ing away −1-1 gives one more, just like adding 11.

Remem­ber. Tak­ing away a neg­a­tive is the same as adding. 12−(−7)=12+7=1912 - (-7) = 12 + 7 = 19.

Worked exam­ples

Three number lines from -20 to 20: an arrow from 12 right to 19, an arrow from -12 right to -5, and an arrow from -12 left to -19, each labelled with its subtraction
The three worked exam­ples below. Tak­ing away -7 walks right; tak­ing away 7 walks left.

Sub­tract −7-7 from 1212

Start at 1212. You take away −7-7, so you add 77. Move seven steps right.

Sub­tract −7-7 from 1212

12−(−7)=12+7=19\begin{aligned}&12 - (-7) \\ &= 12 + 7 \\ &= 19\end{aligned}

Sub­tract −7-7 from −12-12

Start at −12-12 this time. You still add 77. Move seven steps right from −12-12. Seven is less than twelve, so you do not reach zero. The answer stays neg­a­tive.

Sub­tract −7-7 from −12-12

−12−(−7)=−12+7=−5\begin{aligned}&-12 - (-7) \\ &= -12 + 7 \\ &= -5\end{aligned}

Sub­tract 77 from −12-12

Here you take away a plain 77. The two minus signs are not together here. The first one belongs to the −12-12. Only the sec­ond one means take away. So start at −12-12 and move seven steps left.

Sub­tract 77 from −12-12

−12−7=−19\begin{aligned}&-12 - 7 \\ &= -19\end{aligned}

Sub­tract −15-15 from −6-6

Start at −6-6. Tak­ing away −15-15 means adding 1515, so walk right. Six steps bring you to 00. Nine more bring you to 99.

−6−(−15)=−6+15=9\begin{aligned}&-6 - (-15) \\ &= -6 + 15 \\ &= 9\end{aligned}

The tem­per­a­ture in Leh

The drop is the after­noon read­ing take away the night read­ing. Sub­tract −6-6 from 44.

4−(−6)=4+6=10\begin{aligned}&4 - (-6) \\ &= 4 + 6 \\ &= 10\end{aligned}

The tem­per­a­ture dropped by 1010 degrees. Count it on a ther­mome­ter: four degrees down to zero, then six more.

What you are tak­ing awayWhat to doExam­ple
The num­ber you take away is neg­a­tiveTurn it into adding12−(−7)=1912 - (-7) = 19
The num­ber you take away is pos­i­tiveTake it away as usual, and move left−12−7=−19-12 - 7 = -19

The sign on the num­ber you start at does not change what you do. It only changes where you start.

Look at the order once more. Sub­tract −7-7 from 1212 gives 1919. The wrong way round is −7−12-7 - 12. That gives −19-19. Swap­ping the two num­bers gives the same dis­tance. But you walk it the other way. So you get the same num­ber with the other sign.

Your turn

Work these out on a num­ber line. Say the start num­ber out loud first. That is the one that catches peo­ple out. Say­ing it fixes it in your head before you begin.

When the num­bers get big, you do not have to count every step. Work out where you start. Work out which way you are going. Then do the take-away or the adding in your head. The num­ber line is there to tell you which way to go.

  1. Sub­tract −4-4 from 99.
  2. Sub­tract −6-6 from −10-10.
  3. Sub­tract 55 from −8-8.
  4. Sub­tract −9-9 from 22.
  5. Sub­tract −3-3 from −3-3.
  6. Sub­tract 1111 from −4-4.

Each answer below shows the sum writ­ten the right way round. It also shows the num­ber you land on.

Ques­tionAnswerQues­tionAnswer
9−(−4)9 - (-4)13132−(−9)2 - (-9)1111
−10−(−6)-10 - (-6)−4-4−3−(−3)-3 - (-3)00
−8−5-8 - 5−13-13−4−11-4 - 11−15-15

Now try these. Some have two minus signs together. Some do not. Look at each one before you start.

1) 7−(−2)7 - (-2)4) 0−(−8)0 - (-8)7) 3−83 - 8
2) −5−(−9)-5 - (-9)5) 20−(−15)20 - (-15)8) −1−(−1)-1 - (-1)
3) −6−4-6 - 46) −7−(−12)-7 - (-12)9) 10−(−10)10 - (-10)
Ques­tionAnswerQues­tionAnswerQues­tionAnswer
7−(−2)7 - (-2)990−(−8)0 - (-8)883−83 - 8−5-5
−5−(−9)-5 - (-9)4420−(−15)20 - (-15)3535−1−(−1)-1 - (-1)00
−6−4-6 - 4−10-10−7−(−12)-7 - (-12)5510−(−10)10 - (-10)2020

Remem­ber. Tak­ing away a pos­i­tive num­ber moves you left. Tak­ing away a neg­a­tive num­ber moves you right.

Com­mon mis­takes

  • Writ­ing "sub­tract aa from bb" as a−ba - b. The num­ber after "from" always comes first.
  • Turn­ing every pair of minus signs into a plus. Only a minus sign straight in front of a neg­a­tive num­ber does that: −3−5=−8-3 - 5 = -8, not 22.
  • Walk­ing the wrong way. Tak­ing away a neg­a­tive walks right; tak­ing away a pos­i­tive walks left.
  • Drop­ping the sign of the answer when you cross zero, as in 2−6=−42 - 6 = -4.
  • Think­ing a num­ber take away itself is any­thing but zero, even when it is neg­a­tive: −3−(−3)=0-3 - (-3) = 0.

Key terms

Inte­ger
A whole num­ber, zero, or a whole num­ber below zero, such as −4-4, 00 or 99.
Neg­a­tive num­ber
A num­ber below zero, writ­ten with a minus sign in front.
Pos­i­tive num­ber
A num­ber above zero.
Num­ber line
A straight line with the inte­gers marked in order, small on the left and big on the right.
Sub­trac­tion
Tak­ing one num­ber away from another; on a num­ber line it is a walk to the left.
Debt
Money you owe, which can be thought of as a neg­a­tive amount.

Answers

First set:

  1. 9−(−4)=9+4=139 - (-4) = 9 + 4 = 13
  2. −10−(−6)=−10+6=−4-10 - (-6) = -10 + 6 = -4
  3. −8−5=−13-8 - 5 = -13
  4. 2−(−9)=2+9=112 - (-9) = 2 + 9 = 11
  5. −3−(−3)=−3+3=0-3 - (-3) = -3 + 3 = 0
  6. −4−11=−15-4 - 11 = -15

Sec­ond set:

  1. 7−(−2)=7+2=97 - (-2) = 7 + 2 = 9
  2. −5−(−9)=−5+9=4-5 - (-9) = -5 + 9 = 4
  3. −6−4=−10-6 - 4 = -10
  4. 0−(−8)=0+8=80 - (-8) = 0 + 8 = 8
  5. 20−(−15)=20+15=3520 - (-15) = 20 + 15 = 35
  6. −7−(−12)=−7+12=5-7 - (-12) = -7 + 12 = 5
  7. 3−8=−53 - 8 = -5
  8. −1−(−1)=−1+1=0-1 - (-1) = -1 + 1 = 0
  9. 10−(−10)=10+10=2010 - (-10) = 10 + 10 = 20