How To Prove That the Square Root of 2 is Irrational

In ancient times, most mathematicians believed that all numbers were rational, which means they could be written as a ratio: one whole number divided by another whole number. The first person to disprove this (at least according to my 9th grade geometry teacher) was a mathematician named Hypatia. Here is a brief history of her life:

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Although preceded by Pandrosion, another Alexandrine female mathematician, she is the first female mathematician whose life is reasonably well recorded. Hypatia was renowned in her own lifetime as a great teacher and a wise counselor.

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Ancient sources record that Hypatia was widely beloved by pagans and Christians alike and that she established great influence with the political elite in Alexandria. Towards the end of her life, Hypatia advised Orestes, the Roman prefect of Alexandria, who was in the midst of a political feud with Cyril, the bishop of Alexandria. Rumors spread accusing her of preventing Orestes from reconciling with Cyril and, in March 415 AD, she was murdered by a mob of Christians led by a lector named Peter.

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The key to her proof was that she started by assuming that the square root of 2 is rational.

Let the square root of 2 = a/b, where a and b are whole numbers with no common factors. That means the fraction a/b is fully reduced.

sqrt(2) = a/b

We square both sides to get:

2 = a^2/b^2

Multiply both sides by b and divide by a to get:

2b/a = a/b

We have a contradiction here, because we said a/b is fully reduced, but this is telling us that a/b is twice b/a. Therefore, the square root of two cannot be written as one whole number divided by another, and so it must be irrational.

Mathematicians in prior centuries suspected there were irrational numbers because pi, the ratio of a circle's circumference to its diameter, also cannot be written as one whole number divided by another. An early and common approximation was 22/7. The great mathematician Archimedes improved that greatly. He said that pi is more than 3 and 1/7 and less than 3 and 10/71. In other words, pi is more than 3.1408 and less than 3.1429.

Another way of estimating pi (which I discovered) is the formula n*cos(a/2), where n is the number of sides of a regular polygon and a is one of its interior angles. Using this formula for a regular polygon with 100,000 sides, a = ((100,000 - 2)*180)/(100,000) = 179.9964 degrees.

100,000*cos(179.9964/2) =  3.14159265

Those are indeed the first 10 digits of pi.