How do you prove by contradiction the irrationality of surds. Use sqrt 2 as an example.
Proof by contradiction works by working of the assumption a statement is true, then showing it contradicts with a known truth. It's easy to rote learn the main ones required for Maths A Levels (irriationality of surds, infinity of primes, etc), but in the case you can't remember/don't recognise it, it's better to learn how to do it.By definition, a rational number can be expressed in the form a/b, where both a and b are integers and a/b is the simplest form of the fraction.So let's say root 2 is rational. It can thereore be expressed as a/b where both a and b are integers and a/b is the simplest form of the fraction.root 2 = a/bNow square both sides.2 = a2/b2 which can be rearranged to a2 = 2b2. From this we can infer that a must have a factor of 2. We can express it as a = 2c where c is an integer as well.Substitute this back into our equation.4c2 = 2b2 which can be simplified to 2b2 = c2. By the same logic from before, we can say that b must have a factor of 2.
This is where our contradiction is. If both a and b have a factor of 2, then a/b cannot be a fraction in its simplest form. #
Therefore root 2 is not rational.
