**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.

(3) Whenever you see a
negated 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*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

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