Euclid's Proof that √2 is Irrational
Euclid proved that √2 (the square root of 2) is an irrational number.
He used a proof by contradiction.
First Euclid assumed √2 was a rational number.
A rational number is a number that can be in the form p/q where p and q are integers and q is not zero.
He then went on to show that in the form p/q it can always be simplified.
But we can't go on simplifying an integer ratio forever, so there is a contradiction.
So √2 must be an irrational number.
We will go into the details of his proof, but first let's take a look at some useful facts:
Rational Numbers and Even Numbers
First, let's look at some interesting facts about even numbers and rational numbers:
Any integer multiplied by 2 gives an even number.
- 2×3 = 6, 6 is an even number.
- 2×16 = 32, 32 is an even number.
- etc
The square of an even number is always an even number (multiplying two even numbers gives an even number).
Likewise if a number is even and is a square of an integer, then its square root must be even.
In fact the square root of 256 is 16.
Rational numbers or fractions must have a simplest form.
We can go further and simplify it to 7/25.
But it cannot be simplified further since 7 and 25 have no common factors.
The Proof
Euclid's proof starts with the assumption that √2 is equal to a rational number p/q.
p2 must be even (since it is 2 multiplied by some number).
Since p2 is even, then p is also even (square root of a perfect square is even).
Since p is even, it can be written as 2m where m is some other whole number (because an even number can be written as 2 multiplied by a whole number).
Substituting p=2m in the above equation:
q2 is an even number (since it is written as 2 multiplied by some number).
So q is an even number.
Since q is even, it can be written as 2n where n is some other whole number.
Now we have p = 2m and q = 2n and remember we assumed that √2 = p/q:
But now we can repeat the whole process again using m/n and simplify it to something else (say g/h).
We can then do that again ... and again ... and again ... !
But a rational number cannot be simplified forever. There must eventually be a simplest rational number, but in our case there is not: we have a contradiction!
So something is definitely wrong here. √2 cannot be written as p/q or it can be simplified forever.
So √2 cannot be rational and so must be irrational.