[S2 Expository] Negation and Swapping a difference

Introduction

     In this article, I introduce a neat and very useful trick for algebra and arithmetic.  Let us say you want to buy something that costs  $5,  but you only have  $2.  You need to borrow from your friend to help pay for it.  How much do you owe?  $3  Obviously.  Using negative numbers to denote debts, we can write out the story as  2 – 5 = -(5 – 2) = -3.  It is a ‘-’ because you know you are owing something.  To calculate the amount you owe, you just swap the  2  and the  5.

     In other words, you are instinctively using the following identity without realising it.                                             

This algebraic identity (something that is always true, not just sometimes true) actually works for all types numbers that the student will encounter, including positive and negative numbers.  It basically says that swapping a difference is the same as negating it. 

     How do we know it is always true?  This is easily shown:-

                      RHS = -(b – a) = -b + a = a + (-b) = a – b = LHS

The equality relationship is symmetrical.  Since  RHS = LHS,  we have  LHS = RHS.  Equations work both ways.

How to use it?

     (1) You can use it for calculations involving positive and negative numbers, as above.

     (2) Whenever you see difference e.g.  pet – cat,  you can swap the quantities, put brackets around it and attach a negative sign in front to get  -(cat – pet).  Remember that equations work both ways.  You can use it in the other direction.  Hence

     (3) Whenever you see a negated difference e.g.  -(train – van),  you can imagine that the negative sign causes the two quanties to swap.  After swapping, the ‘-’  is used up and you end up with  van – train.
     May the swapping power be with you!

Suitable Levels

* Lower Secondary Mathematics

* other syllabuses that involve algebra and negative numbers