Question

79\. Show that the sum of two rational numbers is rational.

Short answer

The sum of two rational numbers is another rational number.

Step-by-step solution

Define Rational Numbers

A rational number can be expressed as \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b eq 0 \). Similarly, another rational number is \( \frac{c}{d} \) where \( c \) and \( d \) are also integers and \( d eq 0 \).

Obtain Common Denominator

To add the two rational numbers \( \frac{a}{b} \) and \( \frac{c}{d} \), find a common denominator. A common denominator is \( bd \), the product of the two denominators.

Rewrite Fractions with Common Denominator

Rewrite both rational numbers using the common denominator \( bd \):\[ \frac{a}{b} = \frac{a \, (d)}{b \, (d)} = \frac{ad}{bd} \] and \[ \frac{c}{d} = \frac{c \, (b)}{d \, (b)} = \frac{bc}{bd} \].

Add the Fractions

Add the two fractions: \[ \frac{ad}{bd} + \frac{bc}{bd} = \frac{ad + bc}{bd} \].

Prove Result is Rational

Since \( a \), \( b \), \( c \), and \( d \) are integers, \( ad + bc \) is an integer, and \( bd eq 0 \). Therefore, \( \frac{ad + bc}{bd} \) is a rational number. This shows that the sum of two rational numbers is rational.

Key concepts

Adding Rational Numbers

Adding rational numbers is a foundational skill in mathematics. Let's break it down simply. A rational number can be represented as a fraction, where both the numerator and the denominator are integers, and the denominator is not zero. When you add two rational numbers, you are essentially adding two fractions.
To add them, the key step is to ensure that both fractions have the same denominator, known as a common denominator, to facilitate straightforward addition.
Once the fractions are rewritten with a common denominator, you can add the numerators while keeping the denominator the same. This preserves the rational quality of the sum.

Common Denominator

Finding a common denominator is essential when adding two fractions, as it aligns their denominators to be the same. This step is crucial because it simplifies the process of addition. To find a common denominator, multiply the two denominators together.
For example, if you have fractions with denominators 3 and 4, their common denominator would be 12 because 3 multiplied by 4 equals 12.
Rewriting each fraction with this new denominator involves multiplying the numerator and the denominator of each fraction by the necessary factor to reach the common denominator. This makes it easy to add the fractions and ensures you maintain a uniform equation.

Integer Properties

Understanding integer properties helps us work effectively with rational numbers. Integers are whole numbers that can be positive, negative, or zero. When dealing with rational numbers in fraction form, the numerators and denominators are integers.
A key property of integers is that they are closed under addition and multiplication, which means adding or multiplying two integers always yields another integer.
This property is crucial when dealing with rational numbers because it guarantees that operations like adding or multiplying the numerators or denominators do not introduce any non-integer values, preserving the integrity of the rational numbers involved.

Number Theory

Number theory is the branch of mathematics that studies the properties and relationships of numbers. It provides deep insights into rational numbers and their behaviors.
In number theory, numbers are often categorized, and rational numbers fall under the category that can be expressed as the quotient of two integers.
When analyzing the sum of rational numbers, number theory explains why the sum is always rational. Since the operations on numerators and denominators involve only integers, the resultant sum also remains a rational number. Thus, understanding basic concepts in number theory helps confirm the validity and accuracy of operations performed on rational numbers.