Problem 54 - Prealgebra

Question

Determine whether each statement is sometimes, always, or never true. Explain by giving an example or a counterexample. A whole number is not a rational number.

Step-by-Step Solution

Answer

The statement is never true; whole numbers are rational numbers.

Step 1: Understand the Definition of 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 \( b eq 0 \). This includes integers, fractions, and finite or repeating decimals.

Step 2: Analyze the Statement

The statement is "A whole number is not a rational number." To determine the truth of this statement, we need to assess whether whole numbers can be considered rational numbers based on the definition from Step 1.

Step 3: Check if Whole Numbers are Rational

Whole numbers like 2, 5, and 100 can be written as fractions with denominator 1, such as \( \frac{2}{1} \), \( \frac{5}{1} \), and \( \frac{100}{1} \), respectively. These fractions meet the definition of rational numbers.

Step 4: Conclusion Based on Analysis

Since whole numbers can be expressed as fractions where the denominator is 1, they are included in the set of rational numbers. Therefore, the statement "A whole number is not a rational number" is never true.

Key Concepts

Whole Numbers

Whole numbers are the building blocks of mathematics. They include all the natural numbers along with zero. So, the whole numbers are essentially:

• 0, 1, 2, 3, 4, 5, and so on.

These numbers do not include any decimals or fractions; they are simply the numbers we often use in everyday counting.
Think of them as the numbers you would use to count apples, cars, or books. They are straightforward — no tricky parts, just complete numbers.
It's also important to note that every whole number can be considered a rational number, since they can be written as fractions. For instance, 4 can be expressed as \( \frac{4}{1} \).
They're essential for basic arithmetic, providing a solid foundation for more complex mathematical concepts.

Fractions

A fraction represents a part of a whole, and it's expressed as \( \frac{a}{b} \). Here, \(a\) is the numerator and \(b\) the denominator, and \(b\) cannot be zero.
Fractions can be further classified into:

• Proper fractions: where the numerator is less than the denominator. Example: \( \frac{3}{4} \).
• Improper fractions: where the numerator is greater than or equal to the denominator. Example: \( \frac{5}{4} \).
• Mixed numbers: a combination of a whole number and a fraction. Example: 1\( \frac{1}{2} \).

Fractions are pivotal in mathematics because they enable us to express numbers between whole numbers. They describe how much of a whole we have.
When you divide a pizza among friends, each of those slices can be thought of as a fraction of the entire pizza.

Integers

Integers encompass all whole numbers as well as their negative counterparts. So, in the world of integers, you find numbers like:

• -3, -2, -1, 0, 1, 2, 3, and so on.

Think of integers as a number line extending indefinitely in both the positive and negative directions.
They serve an important role in math as they allow us to discuss gains and losses, profits and debts, or above and below zero temperatures.
Additionally, integers can also be classified as rational numbers since they can be written as fractions. For example, the integer -2 can be expressed as \( \frac{-2}{1} \), fulfilling the requirement for being a rational number.
Understanding integers is crucial in many real-world scenarios such as balancing checkbooks, calculating elevations, or assessing temperature changes.