General Discussion

So I looked it up and it's *around* 1.618033, but the actual Golden Ratio is irrational like pi, therefore it's unending and non-repeating, and cannot be expressed as integer/integer.

The characteristic of numbers in the Fibonacci sequence are that any two successive numbers approach "the golden ratio" IIF the (L)arger number divided by a (S)maller number i.e. (L/S) is CLOSE to equal to the ratio of their sum divided by the larger number. So if L/S approximately equals (L+S)/L, the two numbers approach the Golden Ratio with one another.

One example of successive Fibonacci numbers are 21 (F8 i.e the 8th number in the Fibonacci sequence) and 34 (F9), so in our equation S=21 and L=34. So 34/21 = (34+21)/34 (not equal though, just really close). In this case, the left side = 1.6190 and the right side = 1.6176 (both are trimmed here to four places, but they both go on forever). Those numbers are both close to the Golden Ratio.

If you did this with two Fibonacci numbers that are a lot of digits, instead of 2, such as 701408733 (F44) and 1134903170 (F45)? You get 1.618033988749894849 and 1.618033988749894847 on either side of the equation (both are trimmed here to this number of digits, but they actually go on forever).

The larger the two numbers in the Fibonacci sequence are, the closer their ratio becomes to the Golden Ratio, but it can never be equal to it because it's an irrational number. There does not exist two numbers that are EXACTLY in the Golden Ratio, so in a sense the Golden Ratio is theoretical, but the larger the two Fibonacci numbers are, the more they approach it.

It's a similar number to Pi, wherein Pi = Circumference/Diameter of a circle in that they are both irrational constants that cannot be expressed as one integer over another, no matter how large those numbers are