# Proving that there is no rational number whose square is two

In summary, the conversation discusses the proof that no rational number squared can be equal to 2. The approach is to assume that the rational number is in its lowest form and one of the numbers is odd. This is done through the concept of "without loss of generality." By assuming this, it is shown that if a rational number squared is equal to 2, then it leads to a contradiction and therefore, no rational number squared can be equal to 2.

## Homework Statement

The question is to prove that no rational number squared is = 2

## The Attempt at a Solution

I want to understand why for (a/b)^2 = 2, we assume one of the numbers is odd.

Is this because, from approximation we know that root 2 is not a whole number, and If they were both even, we would end up with a whole number?

If both numbers are even, we would cancel the common factors of 2 until one of them is odd.

RPinPA
As @phyzguy said, the assumption is that (a/b) is a fraction reduced to lowest form, and that means that there are no common factors of 2.

This is an example of something you'll see in a lot of mathematical proofs called "without loss of generality..." (WLOG). We can always assume that a rational number can be represented as (a/b) with one of them odd, because so long as both are even you can reduce the fraction by dividing both numerator and denominator by 2. So assuming any rational number at all means there's a rational number with that property.

So the mathematician would say, "Assume that ##\sqrt 2## is rational and WLOG that ##\sqrt 2 = a/b## where either a or b or both are odd."

if you know the lowest term form of a fraction is unique, you are almost done. i.e. if a/b is in lowest terms, i.e. a and b have no common prime factors, then so is a^2/b^2 in lowest terms, since the same prime factors occur here. But then a^2/b^2 = 2/1, and both sides are in lowest terms, hence tops and bottoms are equal, so a^2 = 2 and b^2 = 1. But no integer a can have a^2 = 2. done.

## 1. What does it mean for a number to be rational?

A rational number is any number that can be expressed as a ratio of two integers, where the denominator is not equal to zero. In other words, it is a number that can be written in the form of p/q, where p and q are integers.

## 2. Why is proving that there is no rational number whose square is two important?

The proof that there is no rational number whose square is two is important because it helps to establish the concept of irrational numbers, which cannot be expressed as a ratio of two integers. This proof also has applications in various mathematical fields, such as algebra and number theory.

## 3. How can we prove that there is no rational number whose square is two?

The proof involves assuming that √2 is a rational number and then using logical reasoning to show that this leads to a contradiction. This is known as a proof by contradiction. The proof was first discovered by the ancient Greek mathematician, Pythagoras.

## 4. Can you provide an example of a proof that there is no rational number whose square is two?

Sure, here is a simple example: Assume that √2 is a rational number and can be expressed as p/q, where p and q are integers. Without loss of generality, we can assume that p and q are coprime (i.e. they have no common factors). Then, √2 = p/q can be rewritten as 2 = p^2/q^2. This means that 2q^2 = p^2, which implies that p^2 is an even number. Since the square of an even number is always even, this means that p must also be an even number. But if p is even, then it can be expressed as 2k, where k is an integer. Substituting this into the equation 2 = p^2/q^2, we get 2 = 4k^2/q^2, which simplifies to 1 = 2k^2/q^2. This means that q^2 must also be an even number. However, this contradicts our initial assumption that p and q are coprime, since both p and q are now divisible by 2. Therefore, our assumption that √2 is a rational number leads to a contradiction, and thus √2 must be an irrational number.

## 5. Are there any other proofs that there is no rational number whose square is two?

Yes, there are many different proofs of this statement, including geometric proofs and algebraic proofs. Some of these proofs are more complex and require a deeper understanding of mathematical concepts, while others are more accessible to a wider audience. Ultimately, all of these different proofs lead to the same conclusion - that there is no rational number whose square is two.

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