3.22.2006

More φ

Mark wanted me to say a little more about φ (phi - my favorite number) because I didn't mention anything specific before. It is irrational, so its decimal place goes on forever with no
discernable pattern. It has been calculated out to many thousands of decimals. The first few are:

1·61803...

A simple definition of its unique property is that to square it, you just add one:
φ² = φ +1 This can be "simplified" to φ = 1 /[φ - 1]. -- equally strange.

(Which makes φ the smallest existing solution to the quadratic equation and the simplest continued fraction.)

It is explicitly defined as the positive solution to the quadratic formula (from the equation above):

[sqrt(5) + 1]/2 - The negative solution has all the same freaky properties, though.

It cannot be written as exact fractions of integers, but there is a sequence of fractional "fibonacci" numbers that approximate it sequentially better as it progresses farther down the sequence.

For more, check out some basic facts or more thorough information.

1 comment:

Aqua water said...

THat was so cute. But the last picture looked freaky like they were tring to kill eachother!!!