φ can be easily found using the quadratic formula, as you pointed out.
It also makes an interesting irrational base. I don't think it has much practical use, but it does have at least one terminating expansion for every integer and repeating expansion for every rational number. For example, 6 = 1010.0001φ.
The golden ratio is more interesting, though, for its relationship to the Fibonacci numbers. In particular,
Where F(k) is the kth Fibonacci number (F(0) = 0, F(1) = 1). This follows directly from Binet's formula for Fibonacci numbers.
Actually, the golden ratio is pretty interesting in general. You should read the rest of that article; there's a lot in there.
Without using the Internet, can anyone answer this middle of the road Mathematical Equation?
Two quantities a and b are said to be in the golden ratio φ if:
This equation unambiguously defines φ.
The fraction on the left can be converted to
Multiplying through by φ produces
which can be rearranged to
The only positive solution to this quadratic equation is.......?
If your answer is correct. I will believe you.
Although there is a hidden meaning that if solved would be far beyond the above average thought process.
Are you game?
George1st