Hi,

If h/f and h'/f' are two successive terms of a Farey series (i.e. 0 < = h/f < h'/f' < = 1), then how can I show that h/f < (h+h')/(f+f') < h'/f'? There might be something really obvious that I have overlooked, but I just can't see it.

Thanks,
Olof.

I'm not clear what a Farey Series is. Could you explain?

Google gives http://www.cut-the-knot.com/blue/Farey.html which seems to explain what one is.

You're not be any chance reading Hardy and Wright are you? That's the only place where I've ever heard any mention of Farey Series.

James.

This is interesting - I've never heard of a Farey series before. So a Farey series simply means that the terms are strictly increasing and bounded between 0 and 1?

Also I think we need to assume h,f,h',f' are positive. Otherwise setting h = 1, f = 2, h' = -1, f' = -1 we get:

1/2 < 0 < 1

which isn't altogether accurate.


So basically the question I think is:

If a/x < b/y then show a/x < (a + b)/(x + y) < b/y, given a,b,x,y > 0. (I'll write it like this to save writing 's everywhere.)

Firstly the inequality is totally obvious, intuitively. Imagine you sit an examination which has two sections, A and B. Section A is marked out of x and section B is marked out of y. In section A you score a total of "a" marks and in section B you get "b" marks.

So in section A you get a percentage of a/x and in section B your percentage is b/y. And you do better in section A as a/x < b/y.

Your overall percentage is (a + b)/(x + y). Now clearly your overall percentage is between the percentages in the individual sections (can you imagine getting 80% in one section and 70% in the other section, but overall getting 65%???) Therefore (a + b)/(x + y) is trapped between a/x and b/y, as required.

This of course is extremely wishy-washy, but if you want to make it rigorous you can see that the overall mark must be a weighted mean of the section A and section B marks, i.e.

(a + b)/(x + y) = p x a/x + q x b/y

where p + q = 1, and are both between 0 and 1. (In fact p = x / (x + y) and q = y / (x + y).) From here the result quickly follows.

To quote Hardy and Wright (chapter 3):

'The Farey series F_n of order n is the ascending series of irreducible fractions between 0 and 1 whose denominators do not exceed n. Thus h/k belongs to F_n if

0 < = h < = k < = n, (h,k) = 1.

The numbers 0 and 1 are included in the forms 0/1 and 1/1. For example, F₅ is

0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1.'

James.

I am indeed reading Hardy and Wright. I'm surprised that it's not mentioned anywhere else, it's quite an interesting series in it's own right.

Michael - that was what was so annoying: it was intuitively obvious, but I couldn't show it algebraically.

How about this...

h/f < h'/f'
hf' < h'f
hf' - h'f < 0
h(f + f') - f(h + h') < 0
(h(f + f') - f(h + h'))/(f(f + f')) < 0
h/f - (h + h')/(f + f') < 0
h/f < (h + h')/(f + f')

James.

Brilliant! Thanks again. The result (h + h')/(f + f') < h'/f' follows in the same way.
I've made the same mistake with this so many times, so it's not just university Maths students.

In actual fact I just realised that among all my jottings I had done virtually the same thing as you did above, but I hadn't realised it. Whoops!

Regards,
Olof.

Hi Olof,

This is off-topic, but can you recommend any good math text books that that help the jump from a-level to uni maths, in particular books you liked. You seem to have an interesting collection of books from what you have talked about in this forum.

Thanks
Hal

I haven't read all that many books, but there are a couple that I've found good.
I'm quite interested in Number Theory, and I'm finding "An Introduction to The Theory of Numbers" by Hardy and Wright very useful. It doesn't go through everything with lots of explanation, so quite often I have to sit there for quite a while and just try to figure out what they've done. The thing I originally asked in this thread was an example of this: they just said "The 'mediant' (h+h')/(f+f') of h/f and h'/f' falls in the interval (h/f, h'/f')", and didn't actually show it. This is a good thing, however, as it makes you think for yourself, rather than them laying everything out nice and neatly for you. I think this is one of the standard Number Theory text books; they even had it in my school library.

Another book, which I just received, is called "A pathway into number theory", by R. P. Burn. This focuses more on letting you prove things for yourself, and just points you in the right direction. I'm still on Section 1, but so far it's been a really good book.

I'm not sure if either of those help making the transition from A-Level to Uni. (seeing as I'm still doing A-Levels, but they're good books nevertheless.

"Fermat's Last Theorem", by Simon Singh, is also a good book, even though it's more of a popular mathematical book, and provides you with an historical background to quite a few Mathematicians and other things, as well as Andrew Wiles' story.

One book that I'm reading that is definitely aimed at people going into further mathematical study, is "Alice in Numberland", by Haggarty and someone (I can't find it at the moment), which, although getting a bit old I believe, is fairly good.

Oh and there's one book every Mathematician should have: "The Penguin Dictionary of Curious and Interesting Numbers", by David Wells. Just for fun really.

This turned into a rather long post. Hope I've been of some help anyway!

Regards,
Olof.

Thanks Olof- I'll be sure to check those books out!