Problem 113 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Some whole numbers are not integers.

Step-by-Step Solution

Answer

The statement is false. The correct statement is 'All whole numbers are integers.'

Step 1: Identify the Definition of Whole Numbers

Whole numbers are all the counting numbers including zero, i.e., 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and so on. They do not include any fractional or decimal parts.

Step 2: Identify the Definition of Integers

Integers are all whole numbers and their negatives i.e., ..., -3, -2, -1, 0, 1, 2, 3, ..., they also do not include any fractional or decimal parts.

Step 3: Determine if the Statement is True or False

By comparing the definitions of whole numbers and integers, we can see that the statement 'Some whole numbers are not integers' is false. All whole numbers are integers.

Step 4: Correct the False Statement

Since the statement is false, the corrected statement should be 'All whole numbers are integers.'

Key Concepts

Whole Numbers

Whole numbers are a fundamental part of mathematics, representing numbers without fractions or decimals. They include the set of numbers starting from zero and moving upward without an end: 0, 1, 2, 3, 4, and so forth.
Whole numbers are considered the building blocks of basic arithmetic. They form the basis for counting and measuring. Some key characteristics of whole numbers include:

• Non-negative: Every whole number is either positive or zero.
• Infinite: There is no largest whole number; they continue indefinitely in a positive direction.
• Discrete: Each whole number is distinct and separate from the next.

Understanding whole numbers is crucial, as they set the stage for more complex concepts in mathematics, such as integers and rational numbers.

Integers

Integers expand the concept of whole numbers by including negative numbers as well. They are essential in understanding the full scope of numerical value in mathematics. An integer can be negative, zero, or positive and includes the set of all whole numbers and their negative counterparts: ..., -3, -2, -1, 0, 1, 2, 3, ...
Key points about integers:

• Comprehensive: Includes all whole numbers and their opposites (negatives).
• Zero: Acts as a neutral integer, central in recognizing opposites.
• Non-fractional: Like whole numbers, integers do not include fractions or decimals.

Recognizing and distinguishing integers is vital for solving equations and understanding real-world scenarios involving gains and losses or temperature changes. The knowledge that all whole numbers are part of the integer set is essential in confirming statements about these number systems.

True or False Statements

Determining the truthfulness of a statement is a common exercise in logical reasoning and mathematics. It involves evaluating assertions based on definitions and established facts.
When given a statement such as "Some whole numbers are not integers," we analyze it by considering fundamental definitions: 1. **Whole Numbers** - Numbers from 0 upwards without fractional or decimal parts. 2. **Integers** - A set that includes all whole numbers as well as their negative counterparts. Since every whole number is inherently part of the integer set, the statement "Some whole numbers are not integers" is false. All whole numbers can indeed be found within the integer set. In such exercises, if a statement is false, we correct it by identifying the accurate relationship or fact, like saying, "All whole numbers are integers." Finding and understanding these truths is crucial for accurate mathematical reasoning and communication.