anonymous
  • anonymous
what is the definition of phi?
Mathematics
  • Stacey Warren - Expert brainly.com
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katieb
  • katieb
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anonymous
  • anonymous
Euler's totient? It counts the number of natural numbers less or equal to than \(n\) that are coprime to \(n\).
anonymous
  • anonymous
https://en.wikipedia.org/wiki/Euler's_totient_function
anonymous
  • anonymous
The twenty-first letter of the Greek alphabet (Φ, φ), transliterated as ‘ph.’.

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anonymous
  • anonymous
@rizwan_uet pi and phi are two different letters ^_^
anonymous
  • anonymous
do you mean the number? 1 + √5 Φ = ------ 2
anonymous
  • anonymous
It's the golden ratio http://en.wikipedia.org/wiki/Golden_ratio
RANE
  • RANE
A good explanation of phi and examples
RANE
  • RANE
http://goldenratiomyth.weebly.com/mathematical-definitions-of-phi.html
anonymous
  • anonymous
phi can also be used to generate fibonacci numbers
zzr0ck3r
  • zzr0ck3r
in abstract and number theory it is Euler's totient, in other areas it has many norms.
zzr0ck3r
  • zzr0ck3r
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
zzr0ck3r
  • zzr0ck3r
knowing what numbers are relative prime to other is very crucial in abstract algebra and this theorem is used in many proofs:)
anonymous
  • anonymous
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
zzr0ck3r
  • zzr0ck3r
^^^that is sweet as heck as well:)
zzr0ck3r
  • zzr0ck3r
Tool has a song with the Fibonacci embedded in it.
anonymous
  • anonymous
$$\varphi=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}=\frac1{1+\frac1{1+\frac1{1+\dots}}}$$
anonymous
  • anonymous
well, where is asker? :D
zzr0ck3r
  • zzr0ck3r
sqrt(2)?
zzr0ck3r
  • zzr0ck3r
sqrt(3)
anonymous
  • anonymous
nope! \(\varphi=\dfrac12\left(1+\sqrt5\right)\)
zzr0ck3r
  • zzr0ck3r
lol @mukushla sorry this one got the nerds going:)
zzr0ck3r
  • zzr0ck3r
ahh
anonymous
  • anonymous
$$\varphi=\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}=\sqrt{1+\varphi}\\\varphi^2=1+\varphi\\\varphi^2-\varphi-1=0\\\varphi=\frac{1+\sqrt5}2\text{ the other root is extraneous since }\varphi>0$$
zzr0ck3r
  • zzr0ck3r
whats the other solution expansion and if phi = phi+1 then 1=0:)
anonymous
  • anonymous
$$\varphi=1+\frac1{1+\frac1{1+\dots}}=1+\frac1\varphi\\\varphi^2=\varphi+1\\\varphi^2-\varphi-1=0\\\varphi=\frac{1+\sqrt5}2$$
anonymous
  • anonymous
The angle of declination from the Z axis! (Minimum: 0, Maximum: pi)

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