Question

what is the definition of phi?

Answer 1

Euler's totient? It counts the number of natural numbers less or equal to than \(n\) that are coprime to \(n\).

Answer 2

https://en.wikipedia.org/wiki/Euler's_totient_function

Answer 3

The twenty-first letter of the Greek alphabet (Φ, φ), transliterated as ‘ph.’.

Answer 4

@rizwan_uet pi and phi are two different letters ^_^

Answer 5

do you mean the number?
1 + √5
Φ = ------
2

Answer 6

It's the golden ratio
http://en.wikipedia.org/wiki/Golden_ratio

Answer 7

A good explanation of phi and examples

Answer 8

http://goldenratiomyth.weebly.com/mathematical-definitions-of-phi.html

Answer 9

phi can also be used to generate fibonacci numbers

Answer 10

in abstract and number theory it is Euler's totient, in other areas it has many norms.

Answer 11

phi(P^k) = P^k - P^(k-1)
and since the function is multiplicative phi(ab) = phi(a)phi(b) so we can do huge numbers. It is very cool imho

Answer 12

knowing what numbers are relative prime to other is very crucial in abstract algebra and this theorem is used in many proofs:)

Answer 13

let ϕ and φ be the roots of the quadratic equation x^2-x-1=0

ϕ = (1+√5)/2
φ = (1-√5)/2

let the nth fibonacci number be expressed as F(n)

F(n) = (ϕ^n - φ^n)/√5

Answer 14

^^^that is sweet as heck as well:)

Answer 15

Tool has a song with the Fibonacci embedded in it.

Answer 16

$$\varphi=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}=\frac1{1+\frac1{1+\frac1{1+\dots}}}$$

Answer 17

well, where is asker? :D

Answer 18

sqrt(2)?

Answer 19

sqrt(3)

Answer 20

nope! \(\varphi=\dfrac12\left(1+\sqrt5\right)\)

Answer 21

lol @mukushla sorry this one got the nerds going:)

Answer 22

ahh

Answer 23

$$\varphi=\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}=\sqrt{1+\varphi}\\\varphi^2=1+\varphi\\\varphi^2-\varphi-1=0\\\varphi=\frac{1+\sqrt5}2\text{ the other root is extraneous since }\varphi>0$$

Answer 24

whats the other solution expansion and if phi = phi+1 then 1=0:)

Answer 25

$$\varphi=1+\frac1{1+\frac1{1+\dots}}=1+\frac1\varphi\\\varphi^2=\varphi+1\\\varphi^2-\varphi-1=0\\\varphi=\frac{1+\sqrt5}2$$

Answer 26

The angle of declination from the Z axis! (Minimum: 0, Maximum: pi)