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.
The first one that caught my eye was this:
Can you find a fraction so that the difference between the fraction and its reciprocal is exactly equal to 1?
The problem gave the example that
D’oh – off by
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 and created a common denominator to get or and .
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 was smaller than .
This happened for and . 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 was approaching 1.
At some point yesterday afternoon I went to Wolfram Alpha and typed in 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
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.