Question

Prove that if \(r\) is rational and \(q\) is not, then \(r \pm q\) is irrational; so also are \(r q, q / r,\) and \(r / q\) if \(r \neq 0\) [Hint: Assume the opposite and find a contradiction.]

Short answer

If \( r \) is rational and \( q \) is irrational, then \( r \pm q \), \( r q \), \( \frac{q}{r} \), and \( \frac{r}{q} \) are all irrational.

Step-by-step solution

Define Rational and Irrational Numbers

A rational number can be expressed as the quotient of two integers, that is, \( r = \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b eq 0 \). An irrational number cannot be expressed in this form.

Express the Assumptions

We want to show that \( r \pm q \), \( r q \), \( \frac{q}{r} \), and \( \frac{r}{q} \) are irrational under given conditions. Assume, for the sake of contradiction, that \( r \pm q \) is rational.

Analyze Assumption for Addition/Subtraction

If \( r \pm q \) is rational, then \( r \pm q = \frac{c}{d} \) for integers \( c \) and \( d \), \( d eq 0 \). Then, rearrange to find \( q = (r \pm q) - r = \frac{c}{d} - \frac{a}{b} \). This implies \( q \) is rational, contradicting the assumption.

Contradiction for Addition/Subtraction

Since assuming \( r \pm q \) as rational leads to the conclusion that \( q \) is rational, which violates the initial condition that \( q \) is irrational, our assumption is incorrect. Thus, \( r \pm q \) must be irrational.

Analyze Assumption for Multiplication

Assume \( r q \) is rational, then \( r q = \frac{e}{f} \) where \( e \) and \( f \) are integers with \( f eq 0 \). Then \( q = \frac{1}{r} \times \frac{e}{f} = \frac{e}{f r} \), implying \( q \) is rational, contradicting the assumption.

Contradiction for Multiplication

Since assuming \( r q \) is rational implies \( q \) is rational, and this contradicts the fact \( q \) is irrational, \( r q \) must be irrational.

Analyze Assumption for Division by Rational

Consider \( \frac{q}{r} \). Assume it's rational, let \( \frac{q}{r} = \frac{g}{h} \), leading to \( q = r \times \frac{g}{h} \). This makes \( q \) rational, contradicting the given. Hence, \( \frac{q}{r} \) is irrational.

Analyze Assumption for Division by Irrational

Now, consider \( \frac{r}{q} \). Assume it's rational, so \( \frac{r}{q} = \frac{i}{j} \) where \( i \) and \( j \) are integers, implying \( r = q \times \frac{i}{j} \). This makes \( r \) irrational, a contradiction. Hence, \( \frac{r}{q} \) is irrational.

Key concepts

Rational Numbers

Rational numbers are numbers that can be written as a fraction of two integers. This means that any rational number can look like \( \frac{a}{b} \), where both \( a \) and \( b \) are integers (whole numbers) and \( b eq 0 \). Because they can be expressed as fractions, rational numbers are often more straightforward to work with. Some common examples include whole numbers like 3 (which is \( \frac{3}{1} \)), terminating decimals like 0.75 (which is \( \frac{3}{4} \)), and repeating decimals like 0.333... (which is \( \frac{1}{3} \)). Rational numbers are incredibly useful for representing quantities that can be divided evenly.

Irrational Numbers

Irrational numbers, unlike rational numbers, cannot be expressed as simple fractions with the numerator and denominator as whole numbers. Instead, these numbers go on forever without repeating and do not have an exact fractional equivalent. Examples of irrational numbers include \( \pi \) (pi) and \( \sqrt{2} \). These numbers present a unique challenge in calculations since their decimal forms are infinite and non-repeating.

• Pi (\( \pi \)) is approximately 3.14159 but continues indefinitely without any repeating pattern.
• The square root of 2 (\( \sqrt{2} \)) is around 1.41421 and also continues infinitely.

Understanding irrational numbers help us handle more complex problems in mathematics where dimension and angles involve numbers beyond fractions.

Contradiction Method

The contradiction method is a classic and powerful technique in mathematics used to prove statements by assuming the opposite is true. Then, through logical reasoning, you demonstrate this assumption leads to an illogical or false conclusion, hence the initial statement must be true. This method effectively shows that if the negated statement leads to an impossibility, the original statement holds.
Here's how it typically works:

• Assume the opposite of what you want to prove.
• Use logical steps to show this assumption results in a contradiction, meaning it clashes with known facts or leads to an impossibility.
• Conclude that the original statement must be true as assuming otherwise doesn't work.

This method not only strengthens understanding but also solidifies the foundation for many mathematical arguments.

Mathematical Proof

A mathematical proof is a logical argument that demonstrates the truth of a mathematical statement. In the world of mathematics, proving something means more than just showing it's likely true; it means showing that it must be true, without any doubt. Proofs are structured arguments that use axioms (basic truths), definitions, and previously established propositions to arrive at a conclusion.

• Proofs begin by clearly stating what you want to prove, known as the theorem or proposition.
• They employ logical steps, using rules and theorems, to build a case for why the statement is true.
• Each step in a proof must be justified by either logical deduction or mathematical laws.

The process of crafting a proof not only verifies the statement but also deepens understanding of the underlying mathematical structures and relationships. Emphasizing clarity and rigor, proofs help ensure that mathematical theories are reliably sound.