Problem 29 - Calculus Single Variable

Question

Explain why the sum and product of two rational numbers are always rational.

Step-by-Step Solution

Answer

The sum and product of two rational numbers are rational because they result in quotients of integers.

Step 1: Define Rational Numbers

A rational number is any number that can be expressed as the quotient or fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \).

Step 2: Represent Two Rational Numbers

Consider two rational numbers \( \frac{a}{b} \) and \( \frac{c}{d} \), where \( a, b, c, d \) are integers and \( b, d eq 0 \).

Step 3: Calculate the Sum

The sum of two rational numbers \( \frac{a}{b} + \frac{c}{d} \) can be computed by finding a common denominator, leading to:\[\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}\]Since \( ad, bc, \) and \( bd \) are all integers (products and sums of integers), \( \frac{ad + bc}{bd} \) is a rational number.

Step 4: Calculate the Product

The product of the two rational numbers \( \frac{a}{b} \times \frac{c}{d} \) is given by:\[\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\]Since \( ac \) and \( bd \) are integers (products of integers), \( \frac{ac}{bd} \) is a rational number.

Step 5: Conclude with Observation

Both the sum \( \frac{ad + bc}{bd} \) and the product \( \frac{ac}{bd} \) of two rational numbers are of the form \( \frac{x}{y} \) where \( x \) and \( y \) are integers and \( y eq 0 \). Therefore, they are both rational numbers.

Key Concepts

Sum of Rational Numbers

Rational numbers are numbers that can be expressed as fractions, specifically as \( \frac{a}{b} \), where both \( a \) and \( b \) are integers, and \( b eq 0 \). When it comes to adding rational numbers, the process involves ensuring both fractions have the same denominator.

Consider two rational numbers \( \frac{a}{b} \) and \( \frac{c}{d} \). To find their sum, we find a common denominator, which is typically the product of the two denominators \( b \) and \( d \). The formula for the sum of these two rational numbers becomes:

• Convert them to have the common denominator: \( \frac{a}{b} = \frac{ad}{bd} \) and \( \frac{c}{d} = \frac{bc}{bd} \)
• Add the numerators: \( \frac{ad + bc}{bd} \)

This new fraction \( \frac{ad + bc}{bd} \) is also a rational number because it is still a ratio of two integers, and the denominator is not zero. Thus, no matter which rational numbers you choose, their sum will always result in another rational number.

Product of Rational Numbers

When multiplying rational numbers, things are slightly more straightforward than addition. The product of two fractions is simply the product of the numerators divided by the product of the denominators.

For two rational numbers, \( \frac{a}{b} \) and \( \frac{c}{d} \), their product is calculated as follows:

• Multiply the numerators: \( ac \)
• Multiply the denominators: \( bd \)
• The resulting product is \( \frac{ac}{bd} \)

Since both \( ac \) and \( bd \) are integers, and \( bd eq 0 \), \( \frac{ac}{bd} \) is a rational number. This shows that the product of two rational numbers always results in another rational number.

Integer Operations

Understanding operations with integers is essential, as they form the backbone of rational numbers. An integer is a whole number, which can be positive, negative, or zero, but never includes fractions or decimals.

When performing arithmetic operations:

• Addition: Adding two integers results in another integer. For instance, \( 3 + (-2) = 1 \)
• Multiplication: Similarly, multiplying two integers yields an integer. \( 4 \times (-3) = -12 \)

The relevance of integer operations in rational numbers is pivotal because when adding or multiplying the numerators or denominators of two rational numbers, we are employing integer operations.

This underlying principle assures us that the operations on rational numbers, like their sum and product, remain rational because they rely solely on integer results, keeping the fractions well-defined.