Numbers Types: Part One: Irrational Numbers Make Sense?

Think about what it means to be an irrational number: its value cannot be expressed as the ratio of one whole number value to another, that is, it cannot be written as a fraction without using roots greater than 1. Irrational, not a ratio. We use irrational in common English to describe someone who is lacking reason, not making sense, speaking gibberish, or unbalanced.

Fractional values allow you to work with absolutely perfect values – that is, 1/3 can be multiplied and worked with perfectly, however it’s decimal equivalent of .333333333 will never be a perfect value. No matter how far out I take the decimal places it will always be an approximation (this is why the discovery of C/d for pi was so important).

Irrational numbers were to be the great crisis of the Pythagoreans – who believed that every number could be expressed as the ratio of two other numbers. At the time, Pythagoras had discovered that if a string had a length of 634mm (as an example) and tuned to the note C, then by cutting the string in exactly half (317mm) he would again find the same note C – just an octave higher.

The Pythagoreans’ assumption that all numbers could be expressed geometrically as the relationship between two whole number values would be for naught. The demonstration of applied mathematics has always served as a point of validity for mathematicians, and certainly Pythagoras was seeking to demonstrate the relevance of mathematics by, at the very least, perfecting the musical scale.

Of course, the discovery of the octave left the Pythagoreans scratching their heads. After all, how do you exactly measure a distance whose decimal values extend forever? To obtain a perfect pitch they discovered that frets on a guitar, as an example, needed to be placed in such a way that the string length played on each fret would diminish by successive powers of the twelfth root of 2 – which is an irrational value.

In this case, if L₀ is an E note, then L₁ is the next note on the musical scale: F, L₂ is an F#, L₃ is a G, L₄ is a G#, and so on.

According to legend, the Pythagoreans put one of their own to death for revealing to the outside world that the square root of two is an irrational value and, thus, that irrational numbers did in fact exist.

It is amazing to think that mathematics, irrational numbers to be more specific, could have such a valuable place in the things we love the most. Anyone who has enjoyed music, art, and nature has been blessed by the discovery and application of irrational numbers, which would never have been discovered in the first place were it not for the hard work and dedication put forth by history’s mathematicians.