Oh no! Not Fractions!

This is part one in a series of posts on fraction arithmetic. The other posts can be found here:

Fractions freak me out. Am I weird? Am I just too dumb to get it?

No.

Out of every student I’ve ever tutored, at any level of maths study and ability, probably the single most common issue is fraction arithmetic. So you can at least be safe in the knowledge that you are far from being alone in being freaked out by fractions.

I don’t know why it is. Maybe there’s simply not enough reinforcement of the ideas at primary school. Maybe once kids start using calculators it’s just easier to handle everything in decimals. I don’t know. That’s above my pay grade. I’ll leave the answering of that question to the Ministry of Education.

Now, I’m not going to try to cover everything in one post. Today I’ll just cover the basic anatomy of a fraction. In future posts I’ll look closer at how we perform the various arithmetic operations with fractions.

So, here’s a fraction:

$\large{\boldsymbol{\frac{1}{3}}}$

Breathe. It’s ok, it won’t bite.

It’s just one integer (a whole number) on top of another. The number on the top is called the numerator (the 1 in the fraction above) and the number on the bottom is called the denominator (the 3).

If the denominator is larger than the numerator, we call it a proper fraction:

$\large{\boldsymbol{\frac{1}{3}\quad\frac{7}{8}\quad\frac{2}{37}\quad\frac{389}{411}}}$

If not, then it is an improper fraction:

$\large{\boldsymbol{\frac{4}{3}\quad\frac{8}{8}\quad\frac{286}{37}\quad\frac{412}{411}}}$

A fraction can be read as numerator divided by denominator:

$\large{\boldsymbol{\frac{1}{3}=1\div 3}}$

If we write the answer to that division as a decimal we get:

$\large{\boldsymbol{\frac{1}{3}=1\div 3=0.33\dot{3}}}$

That dot above the last 3 means “recurring”. In other words, the 3 is repeated forever… and ever. This leads to one of the reasons why it’s handy to be able to do fraction arithmetic. Some fractions (like $\large{\frac{1}{3}}$ ) cannot be precisely written as a decimal.

If I convert $\large{\frac{1}{3}}$ to a decimal, no matter how many 3’s I include after the decimal point, I am going to have to round the number slightly which will introduce a small error into my calculation.

Often this won’t matter, but if that rounding error occurs in the middle of a larger calculation then it can get amplified and result in a significant inaccuracy.

That’s all for now.

Oh No! Not Fractions! Part 2

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