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mathslover
How can we say that in golden ratio : \(\frac{a+b}{a} = \frac{a}{b} \) ??
Is their any proof for ^ the above equation?
that is simply how it is defined.
you might as well ask; is there any proof that green is green
You want to say that : (a+b)/a = a/b --> if a > b ^ that is : 1 + b/a = a/b , is universal ?
Sorry , I meant that : green = green is OK but a/b + 1 = b/a seems hard to agree.
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.
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\[\frac{1}{x}=\frac{x}{1-x}\]
by redefining the parts as x = a x-1 = b 1 = a+b we have your setup
sir, but what is the proof that : 1-x = x^2 ? Are we estimating this?
Estimation (in the following sense) : In this special case of golden ratio : (a+b)/b = a/b
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
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.
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"
OK thanks a lot sir!