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
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