One of histories most famous numbers is phi,which is approximately 1.618033…Tryout this video :https://www.youtube.com/watch?v=9mozmHgg9Sk. Be sure to skip the ads. As is mentioned, this number is found in art, music,, biology,human design, cosmology, mathematics, etc. Imagine a rectangle of length a and width b. Then define the sides such that phi=a/b=(a+b)/a =1+b/a=1+1/phi where a>b. phi^2-phi-1=0, and using the quadratic theorem, phi=(1+5^(1/2))/2. Consider the series 0+1=1,1+1=2,1+2=3,2+3=5,3+5=8,and by induction the following numbers are 13,21,34,55,89,144,233 etc.This is the fibonacci sequence.

f(n+2)=f(n+1)+f(n) for n an integer >-1.The limit of f(n+2)/f(n+1) as n approaches infinity is phi.Consider the ratios beginning with n=0.

2,1.5,1.67,1.666,1.6,1.625,1.61538,1.619,1.6176,1.61818,1.617978,1.618.

For large n, f(n+2)/f(n+1) near f(n+1)/f(n)=phi.

Watch https://www.youtube.com/watch?v=dREpRHgkjsg.

Pretend you could stack the earth and the moon on top of each other. Draw a line from the center of the earth to the center of the moon.Its’ length is the sum of their radii,In km, 6,378.1+1,735.97=8,114.07=phi^(1/2) if we define the radius of the earth as =1 and that of the moon as .272.Their sum is 1.618^(1/2) in that scale. Draw a line perpendicular to it from the center of the earth to its edge of length 1, and connect the endpoints to form a right triangle whose hypotenuse in that scale where the earth’s radius=1 is phi units.then phi+1=phi^2 by pythagorean theorem and phi =1.618. Amazing!!Their are other examples from cosmology.

Here is a cool multiplication: 111111111X111111111=12345678987654321.

Try this out for easy math:https://www.youtube.com/watch?v=02xI2jyPpQ4

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