How to Prove That the Square Root of 2 Is Irrational
What is an irrational number?
In mathematics, any real number can be described as either rational or irrational.
A rational number is any number that can be written as a fraction of two whole numbers. For example, 2.5 can be written as 5/2, while 0.333333... can be written as 1/3, hence both of these numbers are rational.
Any real number that is not rational, hence can't be written as a fraction of whole numbers, is irrational. Famous examples of these include π, e and √2.
Something useful to note is that when written in decimal form, rational numbers either terminate or recur, while irrational numbers go on forever without repetition or pattern.
A rational number is any number that can be written as a fraction of two whole numbers. For example, 2.5 can be written as 5/2, while 0.333333... can be written as 1/3, hence both of these numbers are rational.
Any real number that is not rational, hence can't be written as a fraction of whole numbers, is irrational. Famous examples of these include π, e and √2.
Something useful to note is that when written in decimal form, rational numbers either terminate or recur, while irrational numbers go on forever without repetition or pattern.
Proof by contradiction
We have mentioned that √2 is an irrational number, but how can we prove this?
In this article, we are going to use a proof by contradiction based on work by the Greek mathematician Euclid (mid-4th century BC). Proof by contradiction works by assuming the opposite of what you want to prove is true and then working through the mathematical steps until you come to a contradiction of your original assumption.
In this article, we are going to use a proof by contradiction based on work by the Greek mathematician Euclid (mid-4th century BC). Proof by contradiction works by assuming the opposite of what you want to prove is true and then working through the mathematical steps until you come to a contradiction of your original assumption.
Statue of Euclid, Oxford University Museum of Natural History, Oxford, UK -
Photograph by Mark A. Wilson courtesy of Wikimedia Commons
Proving that the square root of 2 is irrational
Let's assume that √2 is rational and, therefore, can be written as a fraction in lowest terms p/q, where p and q are integers and q ≠ 0.
√2 = p/q
Square both sides
2 = p^2 / q^2
Multiply both sides by q^2
2q^2 = p^2
As p^2 is equal to two times a whole number, it must be even. This further implies that p is even, hence, it can be written as 2n for some whole number n.
Therefore 2q^2 = (2n)^2
2q^2 = 4n^2
q^2 = 2n^2
q^2 is also two times a whole number, hence also even. Again, this implies that q is also even.
We have now shown that p and q are both even, but if this is true, then p/q cannot be in its simplest form as we would be able to divide the numerator and denominator by 2. We, therefore, have a contradiction with our original statement; hence, this original statement is wrong, and so √2 must be irrational.
√2 = p/q
Square both sides
2 = p^2 / q^2
Multiply both sides by q^2
2q^2 = p^2
As p^2 is equal to two times a whole number, it must be even. This further implies that p is even, hence, it can be written as 2n for some whole number n.
Therefore 2q^2 = (2n)^2
2q^2 = 4n^2
q^2 = 2n^2
q^2 is also two times a whole number, hence also even. Again, this implies that q is also even.
We have now shown that p and q are both even, but if this is true, then p/q cannot be in its simplest form as we would be able to divide the numerator and denominator by 2. We, therefore, have a contradiction with our original statement; hence, this original statement is wrong, and so √2 must be irrational.
Comments
What do you think of this proof? Do you have a different way of proving that the square root of 2 is irrational?
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