Problem 62 - Prealgebra

Question

Explain why each number is a rational number. $$\frac{10}{2}$$

Step-by-Step Solution

Answer

\( \frac{10}{2} \) is a rational number because it is a fraction of two integers with a non-zero denominator.

Step 1: Understanding the Characteristics of Rational Numbers

A rational number is any number that can be expressed as the quotient or fraction of two integers, where the numerator is any integer and the denominator is a non-zero integer.

Step 2: Introduction of the Number as a Fraction

The given number is \( \frac{10}{2} \). It is already presented as a fraction, with "10" as the numerator and "2" as the denominator.

Step 3: Verifying the Numerator and Denominator are Integers

Both 10 and 2 are integers, as 10 is a positive whole number and 2 is also a positive whole number. Additionally, the denominator, 2, is non-zero.

Step 4: Conclusion Based on Definition

Since \( \frac{10}{2} \) fits the definition of a rational number because it is composed of two integers and has a non-zero denominator, we can conclude that \( \frac{10}{2} \) is indeed a rational number.

Key Concepts

Understanding Fractions

Fractions are a significant part of mathematics, often representing parts of a whole. When you see a fraction, like \( \frac{10}{2} \), it is essentially a way of dividing 10 by 2. Fractions consist of two main parts:

• The top number, which is called the numerator.
• The bottom number, known as the denominator.

The fraction line indicates division. When you divide the numerator by the denominator, you get either a whole number or a decimal. Fractions show us how much of the whole we have or need. They can also be used to compare ratios between numbers, making them very versatile in various mathematical contexts.
An essential characteristic of fractions is that they can represent rational numbers when both the numerator and denominator are integers, and the denominator is not zero. This is because a rational number is defined by its ability to be expressed as a fraction with these properties.

Roles of Numerators

In fractions, the numerator plays a critical role. It tells us how many parts of a certain size we have. In the fraction \( \frac{10}{2} \), 10 is the numerator.

• The numerator can be any integer, which means it can be positive, negative, or zero.
• The larger the numerator, the more parts we have.
• If the numerator is the same as the denominator, the fraction equals 1, meaning one whole.

Think of the numerator as a way to count pieces. For example, if you have 10 apples and you are dividing them among 2 people, each person gets \( \frac{10}{2} = 5 \) apples. The numerator thus carries the quantity or the count in a fraction.

Importance of Denominators

Denominators are what make fractions significant in expressing parts of a whole. In the fraction \( \frac{10}{2} \), the denominator is 2. Here's why denominators are important:

• The denominator indicates the total number of equal parts the whole is divided into.
• It must be a non-zero integer to define a fraction as a rational number.
• The larger the denominator, the smaller each part becomes, as each fraction represents a smaller part of the whole.

The denominator affects how we interpret the numerator. With the denominator of 2, \( \frac{10}{2} \) can be visualized as 10 divided into 2 equal parts, which gives us 5 per part. As the rule goes, the denominator essentially sets the stage for how we look at the division of the whole.

Defining Integers

Integers are the foundation of many numerical concepts in mathematics. They encompass all whole numbers, whether positive, negative, or zero. This includes numbers like -3, 0, and 5. In the world of fractions and rational numbers, integers play a pivotal role:

• To form a proper rational number, both the numerator and the denominator must be integers.
• Integers make sure that fractions represent a clear, complete division of wholes into parts.
• Integers bring consistency and simplicity to mathematical operations, including addition, subtraction, multiplication, and division.

As seen with \( \frac{10}{2} \), both 10 and 2 are integers, perfectly aligning with the definition of a rational number. Integers ensure that when you deal with fractions, you're working with complete and concise numerical representations.