Question

Show that the product of a rational number (other than 0 ) and an irrational number is irrational. Hint: Try proof by contradiction.

Short answer

The product of a rational (non-zero) and an irrational number is always irrational.

Step-by-step solution

Understanding the Given

We are asked to prove that the product of a rational number (other than 0) and an irrational number is irrational. We'll use proof by contradiction for this.

Define the Rational Number

Let the rational number be \( r = \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \). This rational number is different from 0.

Define the Irrational Number

Let \( x \) be an irrational number. By definition, \( x \) cannot be expressed as a ratio of two integers.

Assume the Contradiction

Assume that the product \( rx = y \) is rational, where \( y = \frac{c}{d} \) and \( c \) and \( d \) are integers with \( d eq 0 \).

Manipulate the Equation

From \( rx = y \), we have \( x = \frac{b}{a} \times \frac{c}{d} = \frac{bc}{ad} \). Since \( bc \) and \( ad \) are both integers and \( ad eq 0 \), this implies \( x \) is rational.

Reach the Contradiction

The assumption leads to \( x \) being rational, which contradicts our original statement that \( x \) is irrational. This contradiction means our assumption that \( rx \) is rational must be false.

Conclusion

Since assuming the product of a rational and irrational number is rational led to a contradiction, the original statement holds true: The product \( rx \) is irrational.

Key concepts

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the numerator is an integer and the denominator is a non-zero integer. Mathematically, for a number to be considered rational, it must take the form \( \frac{a}{b} \), where both \( a \) and \( b \) are integers and \( b eq 0 \).

• Rational numbers include integers (e.g., 2, -3) because they can be written as fractions with a denominator of 1 (e.g., \( 2 = \frac{2}{1} \)).
• They also consist of simple fractions like \( \frac{1}{2} \) or repeating decimals such as 0.333... since these can be converted into a fractional form.
• Importantly, 0 is rational, as it can be expressed by \( \frac{0}{1} \), but we exclude 0 in some proofs due to its unique properties as a multiplicative identity.

Understanding rational numbers is crucial when exploring proofs about their interactions with other types of numbers.

Irrational Numbers

Irrational numbers are those that cannot be expressed as a fraction of two integers. They have non-repeating, non-terminating decimal expansions. An example is \( \pi \), where its full decimal form cannot be expressed as a simple fraction.

• Numbers like \( \sqrt{2} \) are irrational since no fraction can precisely represent this value.
• The decimal form of an irrational number never repeats or terminates, which fundamentally differentiates them from rational numbers.
• Historically, the discovery of irrational numbers challenged mathematicians’ understanding of number systems, broadening the concept of completeness in real numbers.

Distinguishing irrational numbers from rational ones is critical in many mathematical proofs and real-life applications.

Product of Rational and Irrational

When you multiply a rational number by an irrational number, the result is usually irrational. But why is this the case? Consider the rational number \( r = \frac{a}{b} \) and the irrational number \( x \).

• The result of their product is \( rx \). If this product were rational, \( x \), too, could be expressed as a fraction, contrary to its definition as irrational.
• If you hypothetically write \( rx = \frac{c}{d} \), where both \( c \) and \( d \) are integers, this implies that \( x = \frac{c \cdot b}{d \cdot a} \), which would make \( x \) rational, violating our assumption.
• Thus, this interaction helps illustrate how irrational numbers behave under multiplication, ensuring \( x \)'s irrational nature remains intact, provided the rational number is not zero.

This phenomenon provides an interesting insight into the diversity and complexity of number types.

Proof Techniques

Proof by contradiction is a fascinating mathematical technique used to demonstrate the truth of a statement by assuming the opposite and showing that this leads to a logical inconsistency.

• The method starts with assuming that the statement is false. For example, assuming the product of a rational and an irrational number results in a rational number.
• Logical steps are then followed to show that this assumption leads to a contradiction, such as an irrational number appearing rational.
• Since this contradiction arises, it confirms the original statement, proving it must be true.
• This technique is widely used in many branches of mathematics because it efficiently confirms the truth of complex propositions by calculating indirect results.

Understanding proof by contradiction deepens our logical reasoning skills and allows us to engage with advanced mathematical concepts with a clear strategy.