## Friday, December 21, 2018

### My Favorite Irrational Number

I have always been interested in numbers, especially special numbers like Pi (3.1415. . . ).

Normal whole numbers, like 1, 2, 3, etc (and their negative counter parts) are amazing when you consider there is no end to how far you can go to find a larger number, just by adding one. Within that very large range of numbers, there are many numbers which have their own special properties, for example the "Prime Numbers". As with all whole numbers, there is no Largest Prime Number, there is always a larger one to be found.

The whole numbers (plus/minus) and fractions that can be written using them are known a "Rational Numbers" - meaning that they can be written as a ratio (a fraction), for example:  1/1, 1/2, 1/3, etc.

But there is another class of numbers that can NOT be written as a ratio of two whole numbers, they are called "Irrational Numbers", for example the value of "Pi".  Pi can be written with many-many non-repeating digits without end, but Pi could never be written as an exact fraction (a ratio) of two whole numbers. There are many such "Irrational Numbers", many are used in science and found in Nature.

My favorite Irrational Number can be computed as simple as: (1+sqrt(5))/2

This can be computed with as many digits that you like with a simple Unix command (change that scale as desired for the number of digits):

echo "scale=1000; (1+sqrt(5))/2" | bc

This is a very special number known as "phi".

A very interesting property of phi is:  1/phi = phi-1

No other number has this simple property.

Phi is found in many places in Nature, for example it is the way seeds are arranged withing a Sun Flower, the size of chambers of the Nautilus fish,  etc.

Phi is the name given to what is also called the "Golden Ratio" or "Golden Rectangle".

Here are the first 1000 digits of Phi:

1.618033988749894848204586834365638117720309179805762862135448622705
26046281890244970720720418939113748475408807538689175212663386222353
69317931800607667263544333890865959395829056383226613199282902678806
75208766892501711696207032221043216269548626296313614438149758701220
34080588795445474924618569536486444924104432077134494704956584678850
98743394422125448770664780915884607499887124007652170575179788341662
56249407589069704000281210427621771117778053153171410117046665991466
97987317613560067087480710131795236894275219484353056783002287856997
82977834784587822891109762500302696156170025046433824377648610283831
26833037242926752631165339247316711121158818638513316203840052221657
91286675294654906811317159934323597349498509040947621322298101726107
05961164562990981629055520852479035240602017279974717534277759277862
56194320827505131218156285512224809394712341451702237358057727861600
86883829523045926478780178899219902707769038953219681986151437803149
97411069260886742962267575605231727775203536139362