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:
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