Why radical 2 is irrational

This means that is even, which again means that is even. But then, both and are even, which contradicts the assumption that they contain no common factor: if they are both even, then they have a common factor of. This contradiction implies that our original assumption, that can be written as a fraction must be false.

Therefore, is 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. 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.

How to set up algebraic equations to match word problems Students often have problems setting up an equation for a word problem in algebra. This article explains some of those relationships. Seven reasons behind math anxiety and how to prevent it Mental math "mathemagic" with Arthur Benjamin video Keeping math skills sharp in the summer Geometric vanish puzzles Science resources Short reviews of the various science resources and curricula I have used with my own children. Keep the vertex O at 0.

So place the unit square from 0 to 1 unit on the number line. Using a compass with center O and radius OB, draw an arc intersecting the number line at point P.

But, later we find out that exist no co-prime integers m and n, so our assumption was wrong. The other method that could be used is the long division method. If it is, then it is an irrational number as per the properties of irrational numbers. When we multiply any rational number to an irrational number, the product is always an irrational number. It states that x 2 is a multiple of 2, which also means that x is also a multiple of 2 [as when a prime number is a factor of a number, let's say, p 2 , it is also a factor of p].

Now, we can write x as 2c, as we just found that x is a multiple of 2. It means that y is also a multiple of 2. When both x and y have a common multiple 2, it means they are not co-prime numbers. So, our assumption was wrong. It states that q 2 is a multiple of 2, which also means that q is a multiple of 2 [as when an integer is a multiple of a prime number, then its square root is also a multiple of that prime number].

It means that p is also a multiple of 2, which contradicts our assumption.