Why is sq root of 2 irrational

Okay, if a itself is an even number, then a is 2 times some other whole number. We don't need to know what k is; it won't matter. Soon comes the contradiction. This means that b 2 is even, from which follows again that b itself is even. And that is a contradiction!!! WHY is that a contradiction? See also What is a mathematical proof? Irrational Numbers; Rational Square Roots How can you tell whether root 10 is a terminating or repeating decimal, or an irrational number?

Are some square roots rational? More proofs that square root of 2 is irrational Provided by Cut-the-Knot. Is root 2 an irrational number? Now let us take a look at the detailed discussion and prove that root 2 is irrational. The square root of a number is the number that gets multiplied to itself to give the original number. Any number that has a non-terminating and non-repeating decimal expansion is always an irrational number.

In this method, we start with an assumption that is contrary to what we are actually required to prove. Then, using a series of logical deductions from this assumption, we reach an inconsistency — a mathematical or logical error — which enables us to conclude that our original assumption was incorrect. We also make the assumption that p and q have no common factors. As even if they have common factors we would cancel them to write it in the simplest form.

So, let us assume that p and q are co-prime numbers here having no common factor other than 1. We note that the right-hand side of the equation is multiplied by 2, which means that the left-hand side is a multiple of 2. So, we can say that p 2 is a multiple of 2. This further means that p itself must be a multiple of 2, as when a prime number is a factor of a number, let's say, m 2 , it is also a factor of m. Thus, we can assume that,. Now, the right-hand side is a multiple of 2 again, which means that the left-hand side is a multiple of 2, which further means that q is a multiple of 2, i.

If it is a fraction, then we must be able to write it down as a simplified fraction like this:. You can see we really want m 2 to be twice n 2 is about twice Can you do better? Because whenever we multiply by an even number 2 in this case the result is an even number. Like this:. And if m 2 is even, then m must be even if m was odd then m 2 is also odd. And all even numbers are a multiple of 2, so m is a multiple of 2 , so m 2 is a multiple of 4.

But hang on