Negatives and Subtraction


One of those irritating confusions from earlier teaching lies in the use of the minus sign. It would help if you could view  2 - 3 [say “two minus three”] as 2 + (⁻3) [“two plus negative three”]. It would help to distinguish the two and the negative, so I will try to write this as 2 – 3 = 2 + 3


This idea would allow you to view any list of terms, such as 2 + 5 – 7 + 9 – 13 as both a mixture of additions and subtractions and as a list of numbers, some of which are negative. This is another of my many versions of doublethink, but as ever the intention is to help, not confuse.

When you treat the ‘minus’ sign as being ‘plus a negative” you separate the
doing part (adding, subtracting) from the descriptor (the positive or negative part). In grammar terms, you’re separating a verb and an adjective – the noun is the (signless) number.

Now it is possible to see all addition and subtraction as the same operation, so that the idea of an expression being a string (list) of terms separated by plus signs (or minus signs) makes much more sense. If it is a list, then you can change the order without changing the sense.

Try changing your thinking: instead of doing ‘taking away’, do ‘adding a negative’. Some of the problems below are very much easier if you rearrange the terms. Be careful of the brackets.


Exercise: simplify these:

2 + 8 + 12 – 11 – 3 – 223 – 21 + 53 - 32

12 + 18 – 21 – 31 +22 -13 -12x + 5 – x + 7 - 12

x + y – x + y – (x+y) + (x-y)(x+3) + (x-2) – (x-5)


Typical test questions ask you how far it is between marks on a thermometer (e.g. the temperature drops from 5º C  to ⁻12ºC; what change is this?). If you are liable to confusion, it may help to visualize the number line, like this:

Hold a finger in front of you; this is zero. Is 5 above or below? Is -12 (sorry, 12) above or below? So how far is it from 12 to +5 ?

Some people prefer to work from left to right, so  …. -3 -2 -1 0  1  2  3 …..

But I find that working vertically avoids all the dyslexic confusions – we don’t seem to have any problem with ‘up’ meaning ‘more positive’.


DJS 20060821

Rewriting the whole in 2017 I wondered why this page was among the sixth form work. Obviously it belongs in with the Lower School.

© David Scoins 2017