Problem 79 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that the sum of two rational numbers is rational.

Step-by-Step Solution

Answer

The sum of two rational numbers is rational because it results in another fraction form with integer numerator and non-zero integer denominator.

Step 1: Define Rational Numbers

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

Step 2: Express Two Rational Numbers

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

Step 3: Add the Two Rational Numbers

The sum of the two rational numbers can be expressed as \(\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}\).

Step 4: Verify Result is a Rational Number

The numerator \(ad + bc\) is an integer because the sum and product of integers are also integers. The denominator \(bd\) is also an integer and not zero because neither \(b\) nor \(d\) is zero. Therefore, \(\frac{ad + bc}{bd}\) is a rational number.

Key Concepts

Addition of Rational Numbers

Adding rational numbers is akin to putting together fractions. Each rational number can be written as a fraction where both the numerator and the denominator are integers, with the denominator not equal to zero. When you add two rational numbers, the formula used is \(\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}\). This might look complicated initially, but it simplifies the process of addition by establishing a common denominator.

Here's a simple way to visualize it: think of the denominators \(b\) and \(d\) as different units, like meters and centimeters. Before you can add them, you need to convert everything into one consistent unit. That consistent unit here is the product \(bd\).

This method ensures that you handle rational numbers uniformly, keeping them within the realm of fractions and thus rational numbers once more. It's all about respecting the structural rules of rational numbers.

Properties of Integers

Integer properties play a vital role in dealing with rational numbers. Simply because rational numbers are built from integers. Here’s what makes integers reliable building blocks:

• Closure Property: The sum of any two integers results in another integer. This means when you add, subtract, multiply, or even zero any integers, the outcome remains an integer.
• Commutative and Associative Properties: These allow flexibility in calculations; you can rearrange integers during addition or multiplication without altering the result.
• Identity Elements: Zero is the identity for addition, meaning \(a + 0 = a\), and one is the identity for multiplication, i.e., \(a \times 1 = a\).

Understanding these properties shows why mathematical structures, like our formula for adding rational numbers, hold together so well—it's because of these foundational properties provided by integers.

Fraction Simplification

Simplifying fractions is a vital part of working with rational numbers. Upon getting a result from the addition or multiplication of fractions, it's important to simplify it to its lowest terms. This makes the fraction easier to understand and use.

To simplify a fraction, you'll need to follow these steps: first, find the greatest common divisor (GCD) of the numerator and the denominator. Then divide both by this divisor. For example, if you have \(\frac{8}{12}\), the GCD is 4. Dividing both by 4 simplifies \(\frac{8}{12}\) to \(\frac{2}{3}\).

Remember, simplifying a fraction doesn't change its value, just the way it's expressed. It's like cleaning up and organizing a room; the space is the same, but it’s easier to navigate. Simplifying is crucial because it provides the most concise representation of the number without altering its value.