Irrational Numbers

Learnt this story from the lectures of Professor Starbird of University of Texas at Austin.

In ancient Greece, there were a group of people who people refer to as Philosophers. One particular set of these Philosophers were called Pythagoreans. The Pythagoreans were involved in discovering numbers. Numbers were very valuable commodity in the Greek society, mainly fractions. Fractions were required as people needed dividing things all the time. The Greeks went for a war and come back with the loot. The loot had to be divided among the people in different proportions. Now, the loot would have various characteristics. Sometimes it would be precious metal; some other time it would be articles made of metal; some other times it would be slaves, who were people captured from the loosing rivals; and sometimes it would be beautiful women; etc. So, number of methods were required for dividing the loot. No single mechanism would suffice. As Greek ventured more to other parts of the World (like Alexander the Great came up to India), the commodities collected were of more diverse nature. This meant more demand for fractions.

The Greeks were very happy with Rational Numbers. Rational Numbers could be expressed as (a/b), where both a and b were Natural Numbers. So, the loot of a units needed being divided among b number of people. This also meant that they had no need for the number “Zero“.

Also, the Greeks noticed that these Rational Numbers could be defined precisely on a line. Every Rational Number they found could be plotted on a linear scale. Plotting on a linear scale was important as this gave a sense of “more” or “less”. The farther the number was on the right hand side, the number was larger. The farther the number was on the left hand side, the number was smaller. This was an easy way of convincing people who were not conversed with numbers.

One day, the Pythagoreans found that there was a number which had a place on the Number Line, but it could not be expressed as a fraction. This number was the “square root of 2“. It was found by taking one unit on the X-axis and one unit on the Y-axis. The resulting hypotenuse had a defined length which could be plotted on the Number Line. But the resulting Hypotenuse had a length equal to “square root of 2” and they just could not find a fraction notation for this number. So, the Pythagoreans concluded that there were numbers which were not Rational. They called it the “Irrational Numbers“. Now, for the first time, we had 2 sets of number beyond the known Rational Numbers. So, the super set was the set of Real Numbers.

The Pythagoreans had a rule in their society. The rule stated that all the work of the society had to be kept a secret. No member of the Society could leak any of the findings of the Pythagorean Society to the remaining community. It so happened that one Pythagorean told someone about the Irrational Numbers. When the Pythagoreans found this out, they took this person in a boat and drowned him in the middle of the lake.

The practice of the Pythagoreans continues till this date. There are a lot of people who know a lot of things and do not tell others about it. It is most likely that they fear being drowned in a lake by the people of their secret societies.

Pythagoras Theorem

Image taken from Google Images


  1. The question is why did the Pythagoreans have this code of secrecy in their society. It would not have been for nothing that they made this code. It must have come from some experience. If this could be determined then it is possible finding out whether the concerns of the Pythagoreans apply to the modern world. If it does, then what people do must be right. Else, people need changing their ways.

    I think this mechanism is better than just following a code blindly.

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