Problem 11 - A Primer of Real Analysis

Question

Show that there does not exist a rational number \(s\) with the property that \(s^{2}=6\)

Step-by-Step Solution

Answer

No rational number \(s\) exists such that \(s^2 = 6\).

Step 1: Define the Conditions for Rational Numbers

A number is called rational if it can be expressed as a fraction \( \frac{p}{q} \), where \(p\) and \(q\) are integers and \(q eq 0\). Furthermore, \(p\) and \(q\) should be coprime, meaning they have no common factors other than 1.

Step 2: Set Up the Equation

Assume that there exists a rational number \(s\) such that \(s^2 = 6\). We can write \(s\) as \( \frac{p}{q} \), so \(s^{2} = \left(\frac{p}{q}\right)^2 = \frac{p^2}{q^2} = 6\). This gives us the equation \(p^2 = 6q^2\).

Step 3: Analyze the Implications for Integer Squareness

The equation \(p^2 = 6q^2\) implies that \(p^2\) is a multiple of 6. Hence, \(p\) must also be a multiple of 6. Let \(p = 6k\) for some integer \(k\). Substituting, we get \((6k)^2 = 6q^2\).

Step 4: Simplify and Derive a Contradiction

Substitute \(p = 6k\) into the equation to get \(36k^2 = 6q^2\). Simplifying, we have \(6k^2 = q^2\). This implies \(q^2\) is a multiple of 6, therefore \(q\) must also be a multiple of 6. But this contradicts the condition that \(p\) and \(q\) are coprime since both are multiples of 6.

Step 5: Conclude with the Result

Since assuming \(s = \frac{p}{q}\) leads to a contradiction with the fact that \(p\) and \(q\) are coprime, our initial assumption that \(s^2 = 6\) for a rational \(s\) must be false. Therefore, no such rational number \(s\) exists.

Key Concepts

Irrational Numbers

Numbers play an interesting role in mathematics, and among them are irrational numbers. Unlike rational numbers, which can be expressed as fractions like \(\frac{p}{q}\), irrational numbers cannot be written as a simple fraction. They are non-repeating and non-terminating when written in decimal form. For example, the square root of 2 (\(\sqrt{2}\)) is an irrational number. It cannot be exactly written as any simple fraction, and its decimal representation goes on forever without a repeating pattern. Another common example of an irrational number is \(\pi\), the famous constant for the ratio of a circle's circumference to its diameter. Irrational numbers are an essential concept in the real number system and a fundamental building block in understanding number theory and algebra. When someone claims that a number like \(\sqrt{6}\) is rational, we can prove this wrong by showcasing an impossibility in assuming it can be expressed as \(\frac{p}{q}\), no matter what integers \(p\) and \(q\) are chosen, provided \(q eq 0\). This is a useful strategy to classify numbers as irrational.

Proof by Contradiction

Mathematicians often employ a clever technique known as proof by contradiction to establish the truth of a statement. It involves assuming the opposite of what you want to prove, and then showing that this assumption leads to a logical inconsistency or contradiction.Consider the exercise where we assume a rational number \(s\) exists such that \(s^2 = 6\). We express \(s\) as \(\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Under this assumption, the logical steps follow until a contradiction arises, specifically that both \(p\) and \(q\) must be divisible by 6, despite being coprime. This inconsistency means our initial assumption was incorrect.Proof by contradiction is a powerful method because if assuming the opposite of what you want to prove leads to an absurdity, then the original statement must be true. It's a direct way to demonstrate impossibility or falsity, and it’s widely utilized across different areas of mathematics.

Coprime Integers

Coprime integers, sometimes known as relatively prime numbers, are pairs of integers that only have 1 as their common factor. This means their greatest common divisor (GCD) is 1.A classic example of coprime integers are 8 and 15. Despite neither number being prime themselves, they share no divisors other than 1. This property of being coprime is crucial in problems involving rational numbers. In the context of proving that \(\sqrt{6}\) is irrational, \(p\) and \(q\) represent the numerator and denominator of a rational number respectively and must be coprime to reinforce our mathematical assumptions and constructions. By starting with the assumption that \( \frac{p}{q} \) is reduced to its simplest form when proving statements, we prepare for exploiting contradictions involving supposed factors such as divisibility by a certain number.The coprime condition is fundamental in number theory and helps maintain the validity and logical progression in proofs involving rational numbers.