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## Fun Stuff

I was reading Kate Nowak’s blog f(t) yesterday and followed a link to the Phillips Exeter Academy website where they’ve posted a pretty amazing collection of problem sets for all levels of math from Pre-algebra to Calculus (and beyond!).

I’ve only just begun to work on some of these, but I’m having a good time so far.

Can you find a fraction so that the difference between the fraction and its reciprocal is exactly equal to 1?

That is $\frac ab - \frac ba=1$

The problem gave the example that $\frac85 - \frac58 = \frac{39}{40}$

D’oh – off by $\frac{1}{40}$

Then the question asks – Can you find another fraction that gets closer than this?

I approached this from a couple of different ways – first I took the original equation $\frac ab - \frac ba=1$ and created a common denominator to get $\frac{a^2 - b^2}{ab} = 1$ or $a^2 - b^2 = ab$ and $a^2 - ab - b^2 = 0$.

I didn’t pursue this past that point, but did come back to it later.

Next, I broke out the spreadsheet and set it up to take all the numbers from 1-30 and create fractions and their reciprocals from these and subtract them.

Looking at all that, I noticed a few places where the difference between 1 and the $\frac ab - \frac ba$ was smaller than $\frac{1}{40}$.

This happened for $\frac {13}{8} - \frac {8}{13}$ and $\frac {21}{13} - \frac {13}{21}$.  Then it was time for class.

The numbers in the fractions that were getting close to 1 had caught my eye yesterday and this morning when I came in, I sat down with it again and saw that they were all consecutive Fibonacci numbers.  So, I made a new spreadsheet with Fibonacci numbers and the fractions and reciprocals and noticed that the difference $\frac ab - \frac ba$ was approaching 1.

At some point yesterday afternoon I went to Wolfram Alpha and typed in $a^2 - ab - b^2 = 0$ just to see what I would get and it provided a relationship between the two variables that comes from treating one of the variables as a constant so that

$a=\frac b2 (1 \pm \sqrt{5})$

In other words a and b cannot both be integers!  And of course, some of you may have picked up on this sooner than I did – the number that produces the exact value of 1 is phi – The Golden Ratio.

$\phi - \frac {1}{\phi}=1$.