A long time ago I was determined to watch every interview with David Tennant so I listened to Christian O'Connell Breakfast Show few times. Once there was a conversation among the hosts (David wasn’t there yet) about common fractions and why are they still being taught at schools. It started with Christian talking about how he was helping, or trying to help, his daughter with her homework the day before and soon we learned that neither of the hosts was able to express 6.85 as a common fraction. Then David came and was asked the question. He easily answered it's six and eighty-five hundredths, and started to divide 85 and 100 by five but had a little problem with it without a piece of paper. Before the topic was finally abandoned, one of the hosts asked: what do we even need common fractions for, in a decimal, computerised world?
(as usual, click on pictures for sources)
And this is what I’d like to talk about. About the audition, let me just clarify that I 1) don’t believe you should be able to do calculations without a piece of paper at six in the morning and 2) don’t believe you should remember everything you learnt on Maths classes. But shouldn’t you rather read a page from your child’s textbook, think a little and get it, instead of talking in the studio the following day how difficult and useless it is?
Why do we learn about common fractions
Our intuition of fractions is common fractions. A fraction is a part of a whole. We intiutively understand fractions as results of division. A half of an apple is when you cut the apple into two equal parts and take one. When you get a part of a cake that has been cut into four equal pieces, you get a quarter. One of six equal slices of a pizza is more than one of eight equal slices of the pizza. Compare: what’s 0.4 of a pizza? Why cutting something in half gives you 0.5?
Decimal fractions are often better for calculations – and that’s why we learn about them and use them in banks and other places - but they work as a way of writing, not a definition. To understand them, you must know the decimal system, while you can use common fractions even with not knowing how to write.
One can argue that we should tell children about common fractions so they understand fractions as the idea and create correct intuition in their minds, then tell them how to converse common fraction into decimals and let them forget about common fractions complicated more than a third. Despite the fact that the writing with numerators and denominators is convenient in Mathemetics: rational numbers have many special properties, decimals can be infinitely long, there are rational functions etc., common fractions have some huge real life advantages. They can give more information than decimals: 4/7 of our cars are Toyota tells you the company has seven cars, while 0.7 (or most likely: 70%) of our employees are men doesn’t tell you how many employees are there or even if there's ten of them or a hundred. Besides, you can easily turn a common fraction into a ratio. We often use proportion: our TV screens are 16:9 and there are two parts butter for three parts flour notes in the cookbooks. In that context, 1.5 doesn't give us such rich information as 3/2 (or 3:2).
Decimal fractions have their traps. Let me show you something interesting. Are you sure you really understand the decimal system? Answer me then: how much 0.(9) is?
It's easy, right? It's 0.999999999..., with the infinite number of nines. Quite right. Tell me then, how much is 1-0.(9)? In other words, how much smaller 0.(9) is than one?
Many people says it's a very small number, but they can't tell exactly what it is. The corrent answer is - and it might be surprising when you hear it for the first time - that 0.999999999... equals one. I'm not kidding. There are many ways to prove it and the one with infinite series is the most 'convincing' for me but I think some people might be satisfied with a simplier explanation. We know that 0.(3) = 1/3. Now multiply both sides of the equation by 3. We get 0.(9) = 1.
This is what decimal system plus infinity (an idea that often goes against intuition) does to you. Some numbers have two representations in the decimal system and number one is one of them.
Decimal fractions as a part of decimal system are extremely useful. When you see 6.85 as a price you probably think of the number as a bit less than seven and it’s enough for making a decision. You don’t have to convert all decimals into common fractions. But that doesn’t mean common fractions aren’t useful. They come handy when you’re in a kitchen or trying to get 3/5 of council’s votes. They give you a better perspective. So maybe, in the era of information, instead of bragging about our gaps, let’s fill them (or just be quiet). If not four our's sake, so at least we can help our kids with their homework.
If you want to listen to the audition, here's the link. It's from 2015. The topic starts about 5 minute.