Fibonacci ratio

[Daniel Radcliffe]

Fibonacci. A/B = B/A+B. ∴ A/B = ???

If A > B , then A/B = ɸ , ie 1.618033988749895etc.

But how is this value { ɸ } calculated just from the equation A/B = B/A+B. ?

 

[Dr.Peterson]

Fibonacci. A/B = B/A+B. ∴ A/B = ???

If A > B , then A/B = ɸ , ie 1.618033988749895etc.

But how is this value { ɸ } calculated just from the equation A/B = B/A+B. ?

Given [imath]\displaystyle\frac{A}{B}=\frac{B}{A+B}[/imath], which appears to be what you mean, one way to start is to cross-multiply. That gives you [imath]A(A+B)=B^2[/imath].

Distribute the LHS, then divide every term by [imath]B^2[/imath], and the result will be a quadratic equation you can solve for [imath]\displaystyle\frac{A}{B}[/imath].

Another way is to first divide the numerator and denominator of the RHS of the original equation by B, giving [imath]\displaystyle\frac{A}{B}=\frac{1}{\frac{A}{B}+1}[/imath]. That is, [imath]\displaystyle\phi=\frac{1}{\phi+1}[/imath]. Now solve that for [imath]\phi[/imath].

Or just read here. That will also cause you to check your equation.