golden ratio?

Question

does anyone know how to fully prove the golden ratio. i've been to tons of sites and they all seem to skip some steps i need. i know that a+b/a = a/b and that eventually it comes out to 1+sqrt[5]/2 but how do you get to that?

how does everyone keep getting sqrt[5]?
i've tried wiki's way a couple of times and i end up at the quadratic part and get stumped.

The expression is actually (a + b) /a = a / b,
not forgetting the brackets.
This gives : 1 + b / a = a / b

'b' is usually given the value of 1.
So, 1 + b / a = a / b becomes 1 + 1 / a = a

Now multiply through by a to get : a + 1 = a^2.
Rearranging gives : a^2 - a - 1 = 0

This is a quadratic equation, which can be
solved by using the quadratic formula.

Thus, a = {1 ± sqrt[1^2 - 4(1)(-1)]} / (2*1)
right here is where i get screwed up in this problem
by the way he set this up for the quadratic equation, a=a, b=a, c=1
where does he get "1" for the [-b] in the quadratic equation?, shouldn't it be set up
a = {a ± sqrt[-a^2 - 4(a)(-1)]} / (2*a)?

Answer 1

The expression is actually (a + b) /a = a / b,
not forgetting the brackets.
This gives : 1 + b / a = a / b

'b' is usually given the value of 1.
So, 1 + b / a = a / b becomes 1 + 1 / a = a

Now multiply through by a to get : a + 1 = a^2.
Rearranging gives : a^2 - a - 1 = 0

This is a quadratic equation, which can be
solved by using the quadratic formula.

Thus, a = {1 ± sqrt[1^2 - 4(1)(-1)]} / (2*1)

= [1 ± sqrt(5)] / 2

The larger value, [1 + sqrt(5)] / 2,
not forgetting the brackets again,
is usually given as the value of the golden ratio.

Answer 2

(a+b)/a = a/b
(a/b + b/b)/(a/b)=a/b
(a/b + 1)/(a/b)=a/b
a/b+1 = (a/b)²
(a/b)² - a/b = 1
(a/b)² - a/b + 1/4 = 5/4
(a/b - 1/2)² = 5/4
a/b - 1/2 = √5/2
a/b = (1+√5)/2

Edit (for completeness): Note that the last steps could have also gone:
a/b - 1/2 = -√5/2
a/b = (1-√5)/2

However, while this solution satisfies the equation, the golden ratio is a ratio of lengths, which are always positive, so φ must also be positive. Since this solution is negative, it is not considered to be the golden ratio.

Edit 2: "how does everyone keep getting sqrt[5]"

From taking the square root of both sides in the third to last step.

Edit 3: No, the quadratic formula uses the coefficients of the variables, not the variuables themselves. In the equation a²-a-1=0
The coefficients are 1, -1, and -1, respectively. Perhaps you will be less confused if you relabel the free variable:

x²-x-1=0

x=(-1±√5)/1

Now are you less confused?

Answer 3

Yes. If you define it as the
- it is the limiting ratio of successive terms in the Fibonacci series
- you can define it directly, geometrically by 1 + 1/r = r

Either way you get the difference equation:
1 + 1/r = r
which gives you the quadratic:
r^2 -r -1 = 0
which has two roots: (1+√5) / 2 ~= 1.618 (Golden Mean),
(1-√5) / 2 ~= -0.618 = 1 - 1/r1 (conjugate GM)

Here's the derivation from Fibonacci series:
F(n) = x^n + y^n
y<0, so for large n, F(n) ~= x^n
F(n+1) ~= x^(n+1)
F(n+2) ~= x^(n+2) = F(n+1) + F(n)
Therefore x^(n+2) = x^(n+1) + x^n
x^2 = x + 1 , giving same quadratic as before.
r = Lim n->inf F(n+1)/F(n) or also Lim n->inf F(n)^(1/n)

Answer 4

The golden ratio is oftentimes referred to as the divine ratio, because it is likewise the ratio of stability. while you're making a cellular, you will use the golden ratio to verify the place to place the subsequent point, etc. So, if this is the ratio of stability, the two bodily and artistically, i could desire to verify the place superstitious minds could see it as some thing religious or perhaps godlike. only like casting out 9s.

Answer 5

I believe your starting point is wrong, it should be:

1+b/a=a/b

the solution starts with a=Rb where R is the ratio.

putting it in the equation above you get

(R+1)/R=R,

R^2-R-1=0

and solving the quadratic equation you get

(1+sqrt(5))/2, not 1+(sqrt(5)/2)

Answer 6

Draw a perfect pentagram (5-point star).

You will find the Golden Ratio consistent in incredible ways throughout.

You will discover uniquely consistent trigonometrics within 54 & 108 degrees, as well as the contained pentagon.

Answer 7

I'd do it dierecty, but Y!A wouldn't be too readable.

Answer 8

let f=golden ratio.
1/f-1=f
1-f=f^2
f^2+f-1=0
f=(-1+/-√(1^2-4*1*(-1)))/2
f=(-1+/-√(1+4))/2
f=(-1+√5)/2
f=(-1-√5)/2
f=(-1-√5)/2=-1.6180339887498948482045868343656
f=(-1+√5)/2=0.61803398874989484820458683436564