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The LAST Digit of Pi. A Smartass Answer to a Smartass Question.
I recently was asked the other day to produce the last known digit of pi. Pi, as you might know is an irrational number and therefore can't be represented as a fraction. Here's a proof I found at http://pi314.at/math/irrational.html:
What can we infer from this? Well, technically this means there is NO last digit of pi, only a last known digit of pi based on the most accurate calculation. The person was at least kind enough to make note of that, but what he was basically asking me to do is look up the largest calculation of pi, trust that it was done so accurately, and pull off the last digit. This irritated me to no end, so I leaned on a little known mathematical trick to arbitrarily set the last digit of pi to a single number, 0.
Numbers can be represented in any number of ways. 10 base 2 is 1010, 8 base 4 is 20, etc. What you can also use number bases which are non integers and even irrational numbers! One commonly known trick is setting a number to base phi or the golden ratio (references: link link2). So what I did was set pi to base pi!
Pi in base pi is 10 for 1 * pi^1 + 0 * pi^0. What's interesting is the only subset of numbers which can be rationally represented in base pi are certain multiples of pi, powers of pi, 3, 2, and 1. After that in order to get to 4 you have to add powers of pi multiplied by integer digits from 1-3 (all integers less than pi).
So next time someone you know asks you what the last digit of pi is, simply blurt out 0. It IS the last digit when pi is represented in base pi and if they want it in decimal and not base pi, you can just wait for the next largest calculation which ends in zero to say you were right anyway.
What can we infer from this? Well, technically this means there is NO last digit of pi, only a last known digit of pi based on the most accurate calculation. The person was at least kind enough to make note of that, but what he was basically asking me to do is look up the largest calculation of pi, trust that it was done so accurately, and pull off the last digit. This irritated me to no end, so I leaned on a little known mathematical trick to arbitrarily set the last digit of pi to a single number, 0.
Numbers can be represented in any number of ways. 10 base 2 is 1010, 8 base 4 is 20, etc. What you can also use number bases which are non integers and even irrational numbers! One commonly known trick is setting a number to base phi or the golden ratio (references: link link2). So what I did was set pi to base pi!
Pi in base pi is 10 for 1 * pi^1 + 0 * pi^0. What's interesting is the only subset of numbers which can be rationally represented in base pi are certain multiples of pi, powers of pi, 3, 2, and 1. After that in order to get to 4 you have to add powers of pi multiplied by integer digits from 1-3 (all integers less than pi).
So next time someone you know asks you what the last digit of pi is, simply blurt out 0. It IS the last digit when pi is represented in base pi and if they want it in decimal and not base pi, you can just wait for the next largest calculation which ends in zero to say you were right anyway.
**SPECIAL NOTE!!!**
Buy T-Shirts with Pi=10! :D Link -> CafePressPage
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