## mathslover 3 years ago How can we say that in golden ratio : $$\frac{a+b}{a} = \frac{a}{b}$$ ??

1. mathslover

Is their any proof for ^ the above equation?

2. amistre64

that is simply how it is defined.

3. amistre64

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

4. mathslover

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

5. mathslover

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

6. 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.

7. 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$\frac{1}{x}=\frac{x}{1-x}$

8. amistre64

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

9. mathslover

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

10. mathslover

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

11. 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

12. 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.

13. 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"

14. mathslover

OK thanks a lot sir!