Fitting in with Fractions
The rational numbers are a brilliantly simple solution to all the problems caused by division in the integers and yet fractions are so little loved by learners of mathematics in school. Take this opportunity to get better acquainted with the rationals in their different forms and see how (& why) they make our lives so much better. Fractions are friends.
As well as having different kinds of numbers we also have different ways of expressing the same numerical values and from a young age you will have learnt how to 'translate' between the different representations. Since that one night in 1971 when the UK went to sleep with 12 pennies to the shilling and 20 shillings to the pound and then woke up with 100 pence in every pound, the number translations that occupy most of our school time are those between fractions and decimals.
We learn that there are numbers which fit nicely: a half is 0.5, and numbers which have more difficulty fitting in: a third is 0.33333... But why does this happen? And whose fault is it? Is there something wrong with the value of a third or is there something wrong with decimals? Clearly some investigation is necessary. Through each of these next exercises it is expected that you will know 'how' to do what is necessary, the aim is to think about why that 'how' is working. What are the underlying truths which make the 'how' work?
Let's start with the straight forward ones. Terminating decimals are decimal numbers whose digits end at some point, like 54.281. The question is, can you find an equivalent fraction representation for them? Any of them? All of them? Go on then.
Back to Bus Stops
Translating from terminating decimals into fractions is easy enough then, what about fractions into decimals? You might have more than one method here, explore them & consider what works best where, this will help you to think about the reasoning behind your categorisation.
A Most Hateful Truth
We can explain now why all fractions must terminate or recur in decimal form. Before we investigate what else decimals might do it is sensible to spend a little more time getting to know the recurring decimals and enjoying some of the face-melting quandaries they throw up.
We'll begin with a very simple question: Is 0.99999... the same value as 1?
Still, stubbornly, I long for no. They should be different. To me, 0.99999... is the mathematical expression of nearly-ness, it is nearly 1, so nearly 1, as nearly as nearly gets, but still not there. Unfortunately though, hatefully, we can prove that they are the same.
Here are a couple of reasons and a painfully bullet-proof proof.
Sorry to have put you through that pain but we must obey the mathematics.
As much as we might want 0.99999... and 1 to be different values, they are provably equal and all that is provable is true.
To wriggle out of this hateful truth in my head I reason that they are different somewhere, it's just not somewhere we can go.
I find this thought soothes the face-melting adequately so I can go about my day, feel free to think it too.
Forever To Play
As troublesome as that particular example might be, there is something a little thrilling about it. 0.99999... and all other recurring decimals go on forever, and yet we can manipulate them so easily. For all the problems it causes us with 'fitting in' the decimal system here gives us the opportunity to play with forever using skills as simple as multiplying by 10 and subtracting. The predictability of recurring decimals gives us power over their infinitely repeating patterns, all we have to do is play a match up game. Try these treats, they will help you forgive the recurring decimals for the above atrocity.