Given that root two is irrational, prove that five plus three root two is also irrational.
Now to begin with, let’s just clarify what is meant by that term “irrational.” If a number is rational, this means that it can be expressed as a quotient or fraction: 𝑎 over 𝑏, where 𝑎 and 𝑏 are both integers and 𝑏 is not equal to zero. For a number to be irrational, it means that it cannot be expressed as a fraction in this way.
To answer this question, we’re going to use a method called proof by contradiction. So we’re going to assume instead that five plus three root two is rational and therefore can be written as a fraction 𝑎 over 𝑏. We want to show that this leads to a contradiction. And the contradiction that we’re looking for is that root two is rational. As we know this to be false, this will mean that our original assumption is also false.
So we begin with the assumption that five plus three root two is equal to some fraction 𝑎 over 𝑏. And we’re going to rearrange this to give an expression for root two. The first step is to subtract five from each side, giving three root two is equal to 𝑎 over 𝑏 minus five. Now, 𝑎 over 𝑏 minus five can be combined if we think of five as five 𝑏 over 𝑏. So we have 𝑎 over 𝑏 minus five 𝑏 over 𝑏. And as the denominators of these two fractions are the same, we can subtract the numerators. And it simplifies to 𝑎 minus five 𝑏 all over 𝑏.
We have then three root two is equal to 𝑎 minus five 𝑏 all over 𝑏. And the next step in rearranging to give an expression for root two is to divide both sides of this equation by three. Dividing by three just leaves root two on the left of this equation and the three joins the denominator on the right of the equation. So we have root two is equal to 𝑎 minus five 𝑏 all over three 𝑏. So we found our expression for root two in terms of 𝑎 and 𝑏.
Now, here’s where our contradiction comes in. If 𝑎 and 𝑏 are integers, which they were when we defined them at the start of the question, then 𝑎 minus five 𝑏 and three 𝑏 are also integers. If we allow 𝑎 minus five 𝑏 to be the integer 𝑝 and three 𝑏 to be the integer 𝑞, then we’ve written root two as the fraction 𝑝 over 𝑞.
However, by our definition of irrational number, this would mean that root two is rational, which is a contradiction, as of course root two is irrational. So by assuming that five plus three root two is a rational number, this led to a contradiction of the fact that root two is irrational. This means that our original assumption was false. And therefore, five plus three root two is irrational.