Prove that square root of 5 is irrational
To prove that square root of 5 is irrational, we will use a proof by contradiction. What is a proof by contradiction?.
Suppose we want to prove that a math statement is true. Simply put, we assume that the math statement is false and then show that this will lead to a contradiction.
If it leads to a contradiction, then the statement must be true
To show that
√5
is an irrational number, we will assume that it is rational
Then, we need to find a contradiction when we make this assumption
If we are going to assume that
√5
is rational, then we need to understand what it means for a number to be
rational
Basically, if square root of 5 is rational, it can be written as the ratio of two numbers as shown below:
Square both sides of the equation above
Multiply both sides by y
^{2}
5 × y
^{2} =
x^{2}
/
y^{2}
× y
^{2}
We get 5 × y
^{2} = x
^{2}
{
Another important
concept before we finish our proof: Prime factorization
Key question: is the number of prime factors for a number raised to the second power an even or odd number?
For example, 6
^{2}, 12
^{2}, and 15
^{2}
6
^{2} = 6 × 6 = 2 × 3 × 2 × 3 (4 prime factors, so even number)
12
^{2} = 12 × 12 = 4 × 3 × 4 × 3 = 2 × 2 × 3 × 2 × 2 × 3 (6 prime factors, so even number)
15
^{2} = 15 × 15 = 3 × 5 × 3 × 5 = (4 prime factors, so even number)
There is a solid pattern here to conclude that any number squared will have an even number of prime factors
In order words, x
^{2} has an even number of prime factors
}
Let's finish the proof then!
5 × y
^{2} = x
^{2}
Since 5 × y
^{2} is equal to x
^{2}, 5 × y
^{2} and x
^{2} must have the
same number of prime factors
We just showed that
x
^{2} has an even number of prime factors
y
^{2} has also an even number of prime factors
5 × y
^{2} will then have an odd number of prime factors.
The number 5 counts as 1 prime factor, so 1 + an even number of prime factors is an odd number of prime factors
5 × y
^{2} is the same number as x
^{2}. However, 5 × y
^{2} gives an odd number of prime factor while x
^{2} gives an even number of prime factors
This is a contradiction since a number cannot have an odd number of prime factors and an even number of prime factors at the same time
The assumption that square root of 5 is rational is wrong. Therefore, square of 5 is irrational