Question

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

Answer 1

Is their any proof for ^ the above equation?

Answer 2

that is simply how it is defined.

Answer 3

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

Answer 4

You want to say that : (a+b)/a = a/b --> if a > b

^ that is : 1 + b/a = a/b , is universal ?

Answer 5

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

Answer 6

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.

Answer 7

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)the golden ratio is defined as the value such that the ratio\[\frac{1}{x}=\frac{x}{1-x}\]

Answer 8

by redefining the parts as

x = a
x-1 = b
1 = a+b

we have your setup

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

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.

Answer 13

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"

Answer 14

OK thanks a lot sir!