## Wednesday, November 12, 2014

### Preparing for Fibonacci Day: The (Other) Fibonacci Fraction

999,999,999,998,999,999,999,999 is an interesting number.  I could tell you about some of the reasons it is interesting, but I really only want to focus on one at a time.  There is one interesting property that you would not believe unless I showed you – so that is the one I am going to talk (write) about.
If you turn this number into a unit fraction (1/999,999,999,998,999,999,999,999), and then find its decimal expansion you first get a big funny look on your face – “I can’t get this number to fit into my calculator???"
I’ve got your back.  Go to the Wolfram Alpha website: www.wolframalpha.com.  At that site you can input the fraction shown above and it will perform this calculation for you.  It will show you that the answer is:
1.000000000001000000000002000000000003000000000005000000... × 10^-24
Convert from scientific notation to regular decimal notation and you get:
0.000000000000000000000001000000000001000000000002000000000003
000000000005000000...
Some of you may already notice the sequence 1, 1, 2, 3, 5 already – shhhh, don’t give it away yet.
If you request the “more digits” (several times) and you arrange the answer into 12 digit chunks, you will see the it is producing a famous sequence known as the Fibonacci sequence;
 ```1/999999999998999999999999 = 0.``` ```000000000000 000000000001 000000000001 000000000002 000000000003 000000000005 000000000008 000000000013 000000000021 000000000034 000000000055 000000000089 000000000144 000000000233 000000000377 000000000610 000000000987 000000001597 000000002584 000000004181 000000006765 000000010946 000000017711 000000028657 000000046368 000000075025 000000121393 000000196418 000000317811 000000514229 000000832040 000001346269 000002178309 000003524578 000005702887 000009227465 000014930352 000024157817 000039088169 000063245986 000102334155 000165580141 000267914296 000433494437 000701408733 001134903170 001836311903 002971215073 004807526976 007778742049 012586269025 020365011074 032951280099 053316291173 086267571272 139583862445 225851433717 ``` `Fibonacci Sequence` ` ` `0` `1` `1` `2` `3` `5` `8` `13` `21` `34` `55` `89` `144` `233` `377` `610` `987` `1597` `2584` `4181` `6765` `10946` `17711` `28657` `46368` `75025` `121393` `196418` ```317811 514229``` `832040` `1346269` `2178309` `3524578` `5702887` `9227465` `14930352` `24157817` `39088169` `63245986` `102334155` `165580141` `267914296` `433494437` `701408733` `1134903170` `1836311903` `2971215073` `4807526976` `7778742049` `12586269025` `20365011074` `32951280099` `53316291173` `86267571272` `139583862445` `225851433717`
The Fibonacci sequence is produced by starting with a 0, and a 1 (or a 1 and another 1), after that each new term is the sum of the two terms before it.
This fraction produces the first 56 terms of the Fibonacci sequence before the terms get too large to fit into a 12 digit format.  I think that is amazing.  I know there is a simple (or maybe not so simple) mathematical explanation about why this works out this way.  I also know that the stars in the night sky are really just giant ball of hydrogen gas undergoing nuclear fusion – but they are still amazing to look at.
Just one more thing:  Take a piece of paper and write down your current age, then write down the name of an animal that lives in Africa, then add 5 to your age, then write down the name of a swimming bird that lives in Antarctica.  The last number you wrote down is how old you will be in 5 years!  Amazing!

Don’t forget – Fibonacci day is 11/23/2014 (1, 1, 2, 3, …),

David