On Fractions

I think these are great: more adaptable and more accurate than decimals.

But it's the division thing which always bugs me. You know: "turn the divisor upside down and multiply" - what the hell? Well, ok, I'll do it....

Well, doodling during a lesson the other day (yes I know I am supposed to be the teacher), I worked out a kind of rationalisation.

Take 3/6 divided by 1/2.

Flip over the 1/2 to make 2.

3/6 x 2 = 6/6 = 1.

So a half goes into 3/6 once. Makes sense, as 3/6 is a half.


But think of it without fractions.

6 / 3 = 2, right?

but 6 x 1/3 also = 2.

Because division is the inverse of multiplication, you are basically just doing the inverse. Also because multiplication is a scaling process, when you times by a number less than 1, the answer will be less than the number you were trying to scale. It scales downwards. 4x2 = 8 but 4x1/2 = 2.


And reciprocals are essential here. A reciprocal is any number you multiply any other number by to make 1. In practice this means that the reciprocal of a whole number is 1/that number.

1/6 x 6 = 1

But the reciprocal of any fraction is that fraction turned upside down.


3/4 - 4/3.

This is because you can imagine any whole number as n/1.

I guess you could say the reciprocal of a number is the inverse of that number...

So, 10/2 = 5. Ie 2 goes into 10 five times.

And 10 x 1/2 = 5. ie 10 lots of 1/2 make 5.

And 10 / 1/2 = 10 x2 = 20. ie a half goes into 10 20 times, which is the same as multiplying ten by two.

The fact that this holds makes it a lot easier to divide fractions, so it is a trick, but a useful one. There are proper proofs of it, but this isn't supposed to be a proof, only a sort of meditation on the subject.