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

5 =
x2 / y2

Multiply both sides by y2

5 × y2 =
x2 / y2
× y2

We get 5 × y2 = x2


{

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, 62, 122, and 152

62 = 6 × 6 = 2 × 3 × 2 × 3 (4 prime factors, so even number)

122 = 12 × 12 = 4 × 3 × 4 × 3 = 2 × 2 × 3 × 2 × 2 × 3 (6 prime factors, so even number)

152 = 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, x2 has an even number of prime factors

}

Let's finish the proof then!

5 × y2 = x2

Since 5 × y2 is equal to x2, 5 × y2 and x2 must have the same number of prime factors

We just showed that

x2 has an even number of prime factors

y2 has also an even number of prime factors

5 × y2 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 × y2 is the same number as x2. However, 5 × y2 gives an odd number of prime factor while x2 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





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