Table of Contents
What is an Irrational Number?
An irrational number is a real number that cannot be expressed as a rational number. This means that it cannot be expressed as a fraction of two integers. Irrational numbers are found in nature and are used in mathematics to help explain certain concepts.
Euclid Square Root 2 Irrational Proof
Mathematicians have been trying to find a proof for the irrationality of the square root of two for centuries. Euclid was the first to come up with a proof that is still used today.
Euclid’s proof is based on the following theorem:
If a and b are two positive integers and m is a positive integer, then the following are equivalent:
1. a + b = m
2. a = m – b
3. b = m – a
Euclid used this theorem to show that the square root of two is irrational. He began by assuming that the square root of two is rational. This means that there is a rational number r such that r2 = 2. He then showed that this assumption leads to a contradiction.
To see how this works, let’s look at a specific example. Suppose that r = 2 and a = 1. This means that b = 1 and m = 3. According to the theorem, this would mean that a + b = 3 and a = 2. But this is not the case. a + b is 1, not 3, and a is 1, not 2.
This contradiction shows that the assumption that the square root of two is rational is false. This means that the square root of two is irrational.
