A Little about Irrationals

Irrational numbers were so interesting for Pythagoreans “a sort of organization interested in doing math and leaded by the famous figure of Pythagoras” since a head of time. The very first appearance of an irrational number was in the very well-known Pythagoras Theorem itself. This theorem works smoothly for various triangle side lengths such as 3,4 and 5 or 5, 12, 13. However, it was absurd, at least for Pythagoras, for some other lengths such as 1,1, where the result is square root of 2. During that time, the square root of 2 was non existent!

This discovery leads Pythagoreans to further explore that number and study its properties. However, it was very hard for them to study a number that they cannot even write it down! All numbers known to them were either whole numbers or a fraction of two whole numbers. Nevertheless, this kind of numbers, as they suggests, cannot be written either ways. By the way, this is the exact definition of an irrational number. They, actually, had to spend some time finding out why.

As you guessed, Pythagoreans were successful in finding such a proof for this problem. It was a nice elegant and elementary one, however. The proof they conducted was by contradiction. The idea of this kind of proofs is to start assuming that a given argument is initially correct. However, its correctness leads to a definitely wrong conclusion. Thus, this argument must be false.

proof:

This type of argument can also be used to proof the irrationality of many other numbers such as square root of 3, 5, 7 and so on.

As you saw earlier, the proof was too smooth. But a little bit long. A more elegant one can be also conducted using the Fundamental Theorem of Arithmetic. This theorem states that any integer can be written as a unique factorization of prime numbers. Meanwhile, if two numbers have the same prime decomposition, they are equal, otherwise, they are not. This is a key point to this proof. The proof is also done by contradiction where we assume that the fraction a/b is written in its most simplest form. it is as follows:

The proof is also more elegant and elementary. It can, moreover, be used to proof the irrationality of the square root of any non square integer. By square integers, I am referring to any integer of the form n² where n is an integer; For example, 4 is a square number while 5 is not.

Nevertheless, can we generalize the fact that the square root of any non square integer is irrational? This is a very good question to ask. And the answer is fortunately available.

The argument we are going to proof here is:

The proof is also a consequence of the Fundamental Theorem of Arithmetic.

Proof:

This generalization is more than helpful to prove more general instances of this problem. For example, the proof that (square root of 3 + square root of 2) is irrational. You may do the proof as a useful exercise to this post.

This is a list of resources I find them useful to read. The proofs also are extracted from them.

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