radical 2 is irrational

Question 1

prove radical 2 is irrational - step 1 “lets suppose…”

Answer 1

Let’s suppose √2 were a rational number. Then we can write it √2 = a/b where a, b are whole numbers, b not zero. We additionally make it so that this a/b is simplified to the lowest terms, since that can obviously be done with any fraction.

Question 2

prove radical 2 is irrational - step 2 “it follows”

Answer 2

It follows that 2 = a2/b2, or a2 = 2 * b2. So the square of a is an even number since it is two times something. From this we can know that a itself is also an even number. Why? Because it can’t be odd; if a itself was odd, then a * a would be odd too. Odd number times odd number is always odd. Check if you don’t believe that!

Question 3

prove radical 2 is irrational - step 3 “if a”

Answer 3

if a itself is an even number, then a is 2 times some other whole number, or a = 2k where k is this other number. We don’t need to know exactly what k is; it won’t matter.

Question 4

prove radical 2 is irrational - step 4 “if we”

Answer 4

If we substitute a = 2k into the original equation 2 = a2/b2, this is what we get:
2	=	(2k)2/b2
2	=	4k2/b2
2*b2	=	4k2
b2	=	2k2.

Question 5

prove radical 2 is irrational - final “this means”

Answer 5

This means b2 is even, from which follows again that b itself is an even number!!!
WHY is that a contradiction? Because we started the whole process saying that a/b is simplified to the lowest terms, and now it turns out that a and b would both be even. So √2 cannot be rational.