The golden number might have been the first irrational number known to the Greeks. When the Pythagoreans discovered that irrational numbers existed, i.e. that they could not be written as the quotient of two whole numbers, they were dismayed, as this fact broke many of their philosophical theories. That is why they decided to keep this discovery a secret.

Theano of Crotone, a Pythagorean mathematician, was the first woman to carry out these divisions, confirming thus the existence of irrational numbers. As a good Pythagorean, she believed and defended that all material objects were composed of natural numbers, so that the measure of anything could be expressed with an exact measure. However, she was also the first to posit the existence of the golden ratio as the essence of the universe.

The golden number, or golden ratio, is represented with the Greek letter Φ (Phi), honouring Phidias. Let's divide any segment into two parts a and b so that a/b is the golden ratio.

• Draw a segment with points at the ends A and B. Let's call it AB.
• Through point B, draw a perpendicular that measures the same as AB.
• On this perpendicular we find the midpoint, which we will call D, and it joins with A, forming the right-angled triangle ABD.
• With the compass, we prick on D and open it up to point B. We draw an arc that cuts the hypotenuse of the triangle. This cutting point is E.
• With the compass, we prick on A and open it up to point E. We draw a new arc that cuts the original AB segment. This last point is C.
• We have managed to divide segment AB into two parts, AC and BC, the bigger part is AC and the smaller BC, both lengths are in golden ratio.
• Measure the segments AB and AC and calculate the quotient AB/AC. Similarly, if we divide the length of the segments AC and BC we find the same number.
• In both cases the same result is obtained, the golden number: "The whole is to the part as the part is to the remainder".

$$\frac{AB}{AC} = \frac{AC}{CB}\;,\qquad \Phi = \frac{1+\sqrt{5}}{2}$$

Phi is the golden number, also known as the "divine proportion".