Why is 2 irrational

There's a shorter proof which requires unique factorization of integers, while ignoring the assumption that a and b have no common factors. That's the problem though: the proof through unique factorisation assumes the Fundamental Theorem of Arithmetic, which needs to be built from the ground up first. Either way, the irrationality of root 2 is a great piece of mathematics both for us and the ancient humans. Not really, you only have to indicate at the beginning of the proof that the numbers a, b they are not both even, otherwise you should simplify by dividing each one by 2.

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. A proof that the square root of 2 is irrational Here you can read a step-by-step proof with simple explanations for the fact that the square root of 2 is an irrational number.

Math Lessons menu. Hint: it has to do with a "recipe" that many math lessons follow. The do's and don'ts of teaching problem solving in math Advice on how you can teach problem solving in elementary, middle, and high school math. 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.

We have thus shown that both p and q are multiples of 2. But is that possible? Example 2: Thomas said to his friend Elle that the square root of 2 is an irrational number. Can you help her? Let's name the vertices of the square as shown. Keep the vertex O at 0. So place the unit square from 0 to 1 unit on the number line. We call such numbers " irrational ", not because they are crazy but because they cannot be written as a ratio or fraction.

And we say:. By the way, the method we used to prove this by first making an assumption and then seeing if it works out nicely is called "proof by contradiction" or "reductio ad absurdum". Reduction ad absurdum : a type of logical argument where one assumes a claim for the sake of argument and derives an absurd or ridiculous outcome, and then concludes that the original claim must have been wrong as it led to an absurd result. Many years ago around BC Greek mathematicians like Pythagoras believed that all numbers could be shown as fractions.

And they thought the number line was made up entirely of fractions, because for any two fractions we can always find a fraction in between them so we can look closer and closer at the number line and find more and more fractions.