You probably know what a ratio is in general. It's a measure of proportion. If there are 100 boys in a school and 50 girls, then the ratio of boys to girls is 100:50, which can be thought of as the fraction 100/50 = 2/1. So you can also say the ratio is 2:1 ("two to one").
Suppose that you have a piece of rope, and you cut it in some place so that you now have two pieces of rope. If you cut it in the middle, then the two pieces are equal, so their lengths are in a 1:1 ratio. If you cut the rope 1/3 of the way down from one end, then the bigger piece will be twice as long as the smaller piece, so their lengths are in a 2:1 ratio.
In theory, it's possible to cut the rope so that the ratio of the larger piece to the smaller piece, is the same ratio as the original uncut rope length to the bigger piece. THIS special ratio is defined as the golden ratio. There are lots of equivalent ways of defining it, but that's the formal definition.
Unfortunately, there's no neat way of writing the number. You can't write it out as the ratio of two whole numbers. So like pi, it's an irrational number. But expressed as a decimal it's approximately equal to 1.618. You can use algebra to show that it's exact value is (1+√5)/2.
It turns out that the golden ratio has lots of other interesting properties. It shows up in a lot of Greek and Roman architecture, because golden rectangles (a rectangle whose length to width are in the golden ratio) are believed to be most aesthetically pleasing. It also appears in nature. Also, take the Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21....
This is the sequence of numbers where you start with "1, 1", and keep adding the two previous numbers to get the next number (1+1 = 2, 1+2 = 3, 2+3 = 5, 3+5 = 8, etc.). It turns out that if you keep writing this out, the ratios of consecutive numbers get closer and closer to the golden ratio:
1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13...
BS & MS in mathematics. I've been studying the golden ratio on the side for over 15 years.
· 1 decade ago