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

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

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

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