What is an irrational number? All real numbers other than rational numbers are irrational numbers. Now, the question is what rational numbers are. A rational number is a real number that can be expressed in terms of a fraction p/q of two integers. And the fraction should have a non-zero denominator. It is pretty easy to tell if a number is an integer, rational, or natural by looking at a number. But, there might be additional effort when it comes to an irrational number. Pi, Euler's number, and the golden ratio are a few widely used irrational numbers. We are going to restrict the scope of discussion to the irrationality of the square root of two.
The question of square root two irrationality is generally discussed because of its one of the methods of proof. It is a classic example of proof by contradiction. So what is proof by contradiction? Let us look at how we can prove that the square root of two is an irrational number. We require to assume that the contradictory statement is true.
Assumption : Assume that the square root of two is a rational number.
By the definition of a rational number, the square root of two can be expressed in the form of p/q. The p and q are integers with the condtion denominator q as nonzero.
Express p/q such that p and q are relatively prime. GCD of p and q is 1.
i.e. √2 = p/q Now we have to prove two contradictory assertions inferred from the above assumption.
Squaring both sides, we get equation 1,
p^2 = 2 * q^2
Since 2*q^2 is even, p^2 is also even. And thus, p is also an even number.
Since p is even number, we can express p as p = 2*k.
Substituting p = 2*k in equation 1,
4 * k^2 = 2 * q^2
2 * k^2 = q^2
Since 2*k^2 is even. q^2 is even. And thus, q is also an even number.
Since both p and q are even numbers, p and q have a common facor of 2. GCD of p and q is not 1.
This contradicts our initial assumption. We cannot express it in terms of fraction with two relatively prime integers as numerator and denominator.
Thus, the square root of the two is an irrational number. The idea of showing two assertions contradictory at the same time is the principle. This principle is found in one of the metaphysical thoughts by Aristotle, called the law of noncontradiction. The law of noncontradiction states that contradictory statements cannot both simultaneously be true in the same sense. "Statement A is true and is not true" is not possible at the same time. Since we now understand the law of noncontradiction, let us discuss the concept of proof by contradiction.
The proof by contradiction is also known as indirect proof. As mentioned before, the idea for indirect proof originated during the period of Greek philosophy. The concept is known as reductio ad absurdum. Reductio ad absurdum translates as a reduction to the absurdity. These arguments endeavor to show that contrary scenarios would lead to meaninglessness and absurdity. Socrates and Plato used this technique to win philosophical debates and arguments. They would start with assuming false statements and prove the results meaningless. In turn, this would force the contestants to agree that the initial assumption was wrong. Euclid and Archimedes employed this technique to prove mathematical results at the time.
Greek and Buddhist philosophy practiced reductio ad absurdum to construct many philosophical arguments. The technique is then formalized and called proof by contradiction in mathematics. The famous English mathematician G. H. Hardy's comment about Reductio ad absurdum in his book called 'A Mathematician's apology' captures it as poetic as possible. He says, "Reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or a piece, but a mathematician offers the game."
To summarize:
Principle : The law of noncontradiction Origin : Reductio ad absurdum (Reduction to absurdity)Technique for indirect proof:
Given we have to prove that proposition A is true,
1) Assume that proposition A is not true. Call it proposition B.
2) Prove that proposition B indicates two mutually contradictory assertions simultaneously.
3) Since this voids the law of noncontradiction, our assumption in step 1 is wrong. Thus, proposition A must be true.
Comments