 The earliest discovered example, found by the Pythagoreans, is the ratio of the length of the diagonal to the length of a side of a square, which is the square root of 2, formally formula_8 Another example is the ratio of a circle's circumference to its diameter, which is called , and is not just an algebraically irrational number, but a transcendental irrational.Also well known is the golden ratio of two (mostly) lengths and , which is defined by the proportion
Taking the ratios as fractions and formula_11 as having the value , yields the equation
which has the positive, irrational solution formula_14
Thus at least one of "a" and "b" has to be irrational for them to be in the golden ratio.An example of an occurrence of the golden ratio in math is as the limiting value of the ratio of two consecutive Fibonacci numbers