There are a couple of errors I noticed. When the author says: Even if we start with two numbers that are completely unrelated in the Fibonacci sequence – say,

There are a couple of errors I noticed. When the author says: Even if we start with two numbers that are completely unrelated in the Fibonacci sequence – say, 8 and 41 – the simple way that we pick the next number of the Fibonacci sequence will cause us to approximate the golden ratio after only a few iterations: 8 / 41 = 0.1951219 (8 + 41 = 49) / 8 = 6.125 (49 + 8 = 57) / 49 = 1.16326531 (57 + 49 = 106) / 57 = 1.85964912 (106 + 57 = 163) / 106 = 1.53773585 Why is that? Well, because of the definition of the golden ratio. He mis-adds in the third step 8+41 ought to be 41+49.. But that's not all. He says "if we start with [any] two numbers...in the Fibonacci sequence" but in fact you can start with ANY two numbers WHETHER OR NOT they are fibonacci numbers.. and perform the Fibonacci operation and divide adjacent numbers and it will converge to the golden ratio. E.g. 8 10 1.25 18 1.8 28 1.555555556 46 1.642857143 74 1.608695652 120 1.621621622 194 1.616666667 314 1.618556701