Problem 64 - College Algebra

Question

Determine whether the statement is true or false: The product of a rational and irrational number is always irrational.

Step-by-Step Solution

Answer

False; there are exceptions, e.g., when the rational number is zero.

Step 1: Define Rational and Irrational Numbers

A rational number can be expressed as the ratio of two integers, where the denominator is not zero. An irrational number cannot be expressed as a simple fraction—its decimal goes on forever without repeating.

Step 2: Consider the Product of Any Rational and Irrational Numbers

If we multiply a rational number \( a \) with an irrational number \( b \), generally, the result \( a \cdot b \) is irrational. For example, if 2 (rational) is multiplied by \( \sqrt{2} \) (irrational), the result is \( 2\sqrt{2} \), which is still irrational.

Step 3: Consider a Special Case

There exists a crucial exception: If the rational number is zero (e.g., \( a = 0 \)), the product \( 0 \times b = 0 \), which is a rational number, not irrational.

Step 4: Conclusion

The statement is: "The product of a rational and irrational number is always irrational." Since we found a special case where the product is rational (when one of them is zero), the statement is false.

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction or ratio of two integers, such as \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \). This means rational numbers include all whole numbers, fractions, and repeating or terminating decimals. For example:

• 5 can be written as \( \frac{5}{1} \).
• \( \frac{3}{4} \) is a rational number because it's already a fraction.
• 0.75 is rational because it can be expressed as \( \frac{3}{4} \).

It's essential to recognize that rational numbers have a predictable, repeated pattern or terminate when converted to decimals.

Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction. Their decimal representation goes on forever without repeating. This intriguing nature of their decimals makes them unique. Famous examples include:

• \( \pi \), which is approximately 3.14159, continues indefinitely with no repeating pattern.
• \( \sqrt{2} \), whose decimal equivalent is approximately 1.414213, also extends endlessly without repetition.
• The golden ratio \( \phi \), appearing as 1.6180339... and so on.

These numbers occur frequently in different areas of mathematics, art, and nature. Their complexities lead to fascinating problems, like determining products with rational counterparts.

Number Properties

Understanding number properties helps us navigate through arithmetic and algebra. It's all about recognizing patterns and rules, which aid in predicting outcomes. Let's dive into some properties:

• Associative Property: Changing the grouping of numbers in addition or multiplication doesn't change the result. For example, with addition: \( (1 + 2) + 3 = 1 + (2 + 3) \).
• Commutative Property: The order in which you add or multiply numbers does not affect the outcome. For instance, with multiplication: \( a \times b = b \times a \).
• Distributive Property: This describes the way multiplication is distributed over addition, shown as \( a(b + c) = ab + ac \).

These properties are particularly useful when combining rational and irrational numbers. Knowing these rules will help you explore the exceptions in mathematical operations, such as when multiplying zero (a rational number) with any number, always resulting in zero, which is rational.