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- mathslover

How can we say that in golden ratio : \(\frac{a+b}{a} = \frac{a}{b} \) ??

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- mathslover

How can we say that in golden ratio : \(\frac{a+b}{a} = \frac{a}{b} \) ??

- jamiebookeater

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- mathslover

Is their any proof for ^ the above equation?

- amistre64

that is simply how it is defined.

- amistre64

you might as well ask; is there any proof that green is green

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- mathslover

You want to say that : (a+b)/a = a/b --> if a > b
^ that is : 1 + b/a = a/b , is universal ?

- mathslover

Sorry , I meant that : green = green is OK but a/b + 1 = b/a seems hard to agree.

- mathslover

usually when we are to prove x = y then it is agreed that YES IT IS LIKE GREEN = GREEN but not exactly, it is like : color of leaf = green (TO PROVE) .
I hope you are getting my point sir.

- amistre64

take a line of unit length (1); take some part of it and define it as "x"; which leaves us with the rest of it as (1-x)|dw:1350393541608:dw|the golden ratio is defined as the value such that the ratio\[\frac{1}{x}=\frac{x}{1-x}\]

- amistre64

by redefining the parts as
x = a
x-1 = b
1 = a+b
we have your setup

- mathslover

sir, but what is the proof that : 1-x = x^2 ? Are we estimating this?

- mathslover

Estimation (in the following sense) :
In this special case of golden ratio : (a+b)/b = a/b

- amistre64

when 1-x = x^2 the ratio of the parts to the whole IS the golden ratio.
this is along the same line of thought as: the ratio of the circumference of a circle to its diameter defines pi. How would you prove the C/d = pi ?
its not a matter of proof, but of definition

- mathslover

So can we regard that as , axiom ? postulate? Well I think it can be defined well as 'definition' , correct?
btw, a nice example given by you sir.

- amistre64

id say "definition" is a good term to use :)
we can prove that the value of the golden ratio is what it is by solving for "x"

- mathslover

OK thanks a lot sir!

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