What is a ‘basic idea’ in maths?

Liping Ma (2010) states that inverse operations are one of the most basic ideas of maths.  A basic idea is really just a basic attitude a person has or a basic mathematical principle. A person who holds a profound understanding of fundamental mathematics (PUFM) is able to build these basic ideas as strong foundations for more advanced mathematics they may come across. In a person with PUFM these basic ideas become so intrinsic in their mathematical thinking, that they no longer have to think about them at all! This leaves all their brain power free to work on the tricky stuff.

What are inverse operations?

Luckily not one of these!

“The word ‘inverse’ means reverse in direction or position. It comes from the Latin word ‘inversus,’ which means to turn upside down or inside out. In mathematics, an inverse operation is an operation that undoes what was done by the previous operation.” (Study.com, 2015)

The four operations of adding, subtracting, multiplying and dividing are the basic foundations for all mathematics.  The inverse of addition is subtraction, and the same is true in reverse, and the inverse of multiplication is division, and the same is also true in reverse here too.  

 

Let’s look at some examples…

If….5 + 12 = 17, then the inverse of this would be 17 – 12 = 5

As you can see this takes us right back to the start.

And, if….7×3 = 21, then the inverse of this would be 21 ÷ 3 = 7

And lastly… If 10² = 100, then the inverse of this would be √100 = 10

You can find a short clip to explain this in more detail here.

Lets see if we can find inverse operations in the big wide world…

A man, let’s call him Frank,  goes out to a bar for the night. On his way there he stops off at the cash machine and takes out  3 X £10 = £30. When he gets to the bar he orders a drink while he waits for his friends. It costs £3. His friends are late. Frank orders another drink and some peanuts, it costs £3.95. He’s still waiting… He orders another drink now, a double whisky, that costs £7.50. Then he decides he’s had enough and he’s going to go home where drinks are cheaper!

Frank gets outside and realises he can’t drive home because he’s been drinking (sensible Frank!). So he has a choice of walking or getting a taxi but he’s not sure if he has enough. Frank uses the inverse operation of addition to work out if he has enough:

So, when he took money out addition was used:

£10 + £10 + £10 = £30

Now he wants to work out how much is left…

£30 – £3 – £3.75 – £7.50 = £15.75

Which means, using the inverse operation of addition (which is what he did when he started) Frank has worked out that, not only does he have enough money for a taxi, he also has enough to buy a kebab too!

But, but do inverse operations only occur in the pub?! 

No, inverse operations are everywhere!

We can take this basic principle of inverse operations to an even more basic level.

Black – is the inverse of – White

 

Heavy – is the inverse of – Light

Hot – is the inverse of – Cold

The list of inverse operations that can be found in the big wide world could go on and on.

I wonder how many you can identify next time you have some spare time.