You mathematicians may think my question silly, but I'm stuck on it and I need your help. I know that it's common to say that, for example, 0.8 = 0.80. I cannot accept this. I know that 8/10 = 80/100. So, I should have no problem just adding a zero in the hundredths place to make 0.8 into 0.80. But, I also consider 0.8 to be any number between 0.75 and 0.84! I don't know what digit should be put in the hundredths place of 0.8. Why should it be zero?!!

I'm doing Maths with Physics at university at the moment, and I think there's a bit of both subjects in this question. What we really need to think of is what we are talking about when we say "0.8".

There are two ways which we can look at the number "0.8": From a mathematical point of view, and from a practical (physics) point of view -

Firstly, if we look at "0.8" from a mathematical point of view, then you have already argued why we should say "0.8" = "0.80". The key to how mathematicians represent numbers in our number system (known as "decimal"), is the fact that 1=1, 1/10=0.1, 1/100=0.01, and so on. Therefore, you correctly say that 8/10 = 80/100 and so "0.8"="0.80".

The other way of looking at "0.8" is to look at it as a physical measurement, or an approximation of something. What we have to realise is that this is not exactly the number we are talking about. Well, by what we said above, if "0.8" is not exactly the number we are talking about, then "0.80" is not exactly the number we are talking about(as 0.8 and 0.80 are exactly the same), so it may well be in the range 0.75 to 0.84.

(Strictly speaking we should say 0.75 £ x < 0.85, and the range we define here is simply our way of saying how we are going to say what x "rounded" to 1 dedcimal place is. Notice also that the range we have picked is such that for every number we can find one and only one number such that 0.75 £ x < 0.85, or 0.65 £ x < 0.75, or 0.85 £ x < 0.95, etc.)

I think some of what I have said here might not be 100% crystal clear, but I hope you can understand it.

Hope this helps,

Andrew R