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## Homework Statement

Fractions ##\frac{a}{b}## and ##\frac{c}{d}## are called neighbor fractions if their difference ##\frac{ad - bc}{bd}## has numerator ##\pm 1##, that is, ##ad - bc = \pm 1##.

Prove that:

(a) in this case neither fraction can be simplified (that is, neither has any common factors in numerator and denominator);

(b) if ##\frac{a}{b}## and ##\frac{c}{d}## are neighbor fractions, then ##\frac{a + c}{b + d}## is between them and is a neighbor fraction for both ##\frac{a}{b}## and ##\frac{c}{d}##; moreover,

(c) no fraction ##\frac{e}{f}## with positive integer ##e## and ##f## such that ##f < b + d## is between ##\frac{a}{b}## and ##\frac{c}{d}##

## Homework Equations

n/a

## The Attempt at a Solution

[/B]

Two points first:

1) Please point out flaws in presentation as well as errors in math; I'm studying at home as an older adult without classroom assistance. I've chosen Gelfand's book to help me recover and improve on whatever I've forgotten from high school algebra, now more than 40 years ago.

2) This problem has been previously posted on the forum - however that thread trailed off in a manner not helpful to me, so I will not refer to it again: Problem 42 on Gelfand's Algebra (on neighbor fractions)

Back to the problem. I think I've got answers for (a) and (b) - see below - but have no one to review them, so please give corrections or hints. As for (c), I am stumped despite several tries & would appreciate clues or hints so that I can get further.

**(a) Prove that neither fraction can be simplified:**

Assume that for ##\frac{a}{b}##, a common factor

*f*exists such that the fraction is actually ##\frac{af}{bf}## .

Therefore the test for whether the two fractions are neighbors becomes ##\frac{afd - bfc}{bfd}## and the numerator test for establishing neighbors becomes

*afd – bfc*= ##\pm 1##; we can now express

*afd – bfc*as

*f*(

*ad – bc*).

If the neighbor test is already true for ##ad - bc = \pm 1##, there is no

*f*other than 1 for

*f*(

*ad – bc*) that would not change the neighbor test to produce a result other than 1; therefore there can be no common factor

*f*for a neighbor fraction.

(

**b) Prove that if if ##\frac{a}{b}## and ##\frac{c}{d}## are neighbor fractions, then ##\frac{a + c}{b + d}## is between them and is a neighbor fraction for both ##\frac{a}{b}## and ##\frac{c}{d}##:**

Surprisingly (at least to me, if I have got this right) this turns out to be a simple algebraic simplification. Let us check ##\frac{a}{b}## and ##\frac{a+c}{b+d}## to see if they can be shown to be neighbors. We disregard the multiplication of the two denominators and just examine whether the difference between the numerators when the fractions are given a common denominator is either +1 or -1:

*becomes*

a(b+d) - b(a+c)

a(b+d) - b(a+c)

*ab + ad - ba - bc*

*ab*cancels

*ba*and this leaves the original

*ad - bc*which we already know is = ± 1, given that we have already established ##\frac{a}{b}## and ##\frac{c}{d}## as neighbor fractions. Essentially the same proof can be done for ##\frac{a+c}{b+d}## and ##\frac{c}{d}##.

**(c) Prove that no fraction ##\frac{e}{f}## with positive integer ##e## and ##f## such that ##f < b + d## is between ##\frac{a}{b}## and ##\frac{c}{d}##**

Here I haven't gotten very far at all; I don't know how to attack the problem. I don't usually try working out equations until I have a sense of what I am trying to achieve. The way the question is posed, it seems the issue is not whether

*e/f*is itself a neighbor fraction, but rather, is there room for

*any*fraction here that is not a neighbor fraction? Given the stipulation that

*f < b + d*, it seems to me that this also requires

*e < a + c*; if we then attempt to give

*a/b*and

*e/f*a common denominator, the difference between numerators would have to be less than 1, thus zero (

*a = e*); but this seems absurd. Likewise if we attempted to give

*a/b*and

*e/f*a common numerator, then the same thing occurs; the difference between denominators would have to be less than 1, thus zero (

*f =*

*b*); and again this seems absurd.

To me I am going around in circles here: I have developed a rather sketchy verbal argument but don't know how to translate it into equations, and without equations I can't determine its validity. So as I say, hints would be welcome. But also let me know if you think I should demonstrate more of an attempt to solve the problem; or if mistakes made in my attempts at (a) and (b) may be preventing me from understanding (c).

P.S. One of the issues with Gelfand is that solutions are provided for only about half the problems, which makes it harder for someone doing self-study - this problem #42 is a good example of that. On the other hand, it's a very enjoyable text and I have heard it praised. Should I try & find another text where all the problems have solutions provided? I've tried Googling to learn more about "neighbor fractions", but the only hits I get are links to the Gelfand book; so it seems to be a concept that he & Shen developed, at least by that name. But Gelfand usually seems to give problems for sound pedagogical reasons, so obviously there is something the student is intended to learn here about either fractions or problem-solving.

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