Problem 58 - 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. On a number line, consecutive integers do not have any other integers between them.

Step-by-Step Solution

Answer

The statement 'On a number line, consecutive integers do not have any other integers between them.' is true.

Step 1: Understand the Statement

The statement reads: 'On a number line, consecutive integers do not have any other integers between them.' To analyze this statement, it is crucial to understand the definition of consecutive integers. Consecutive integers are integers that follow one after the other, for example, -1, 0, 1, 2, and so on.

Step 2: Evaluate the Statement

On a number line, consecutive integers follow each other without any additional integers between them. To elaborate, there is no integer between 1 and 2, neither between 2 and 3, and this continues for positive consecutive integers. The same holds for negative integers, there is no integer between -2 and -1, for example. Therefore, the statement that describes consecutive integers having no other integers between them on a number line seems correct.

Step 3: Finalize the Answer

As per the analysis and evaluation in previous steps, it is discerned that the statement, 'On a number line, consecutive integers do not have any other integers between them' is indeed true. Hence, there are no necessary changes needed to make this statement true.

Key Concepts

Number Line

A number line is a visual representation of numbers laid out in a straight line, usually with equal spacing between each number. At the center of a number line, you will find zero, with positive numbers extending to the right and negative numbers extending to the left. This uninterrupted stretch of numbers makes the concept of consecutive integers very clear.

On a number line, each point corresponds to a unique integer. For example, the points labelled 1, 2, 3, and so on, are single units apart. This unit gap represents the fundamental nature of consecutive integers – there is a consistent increase (or decrease) by one as you move from one point to the next along the number line. Understanding the layout of a number line is crucial in visualizing mathematical concepts and relationships between numbers, such as addition, subtraction, and, in this case, the succession of integers.

Integers

Integers are the set of whole numbers that include all positive natural numbers, zero, and the negatives of the natural numbers. It’s important to clarify that while all integers are whole numbers, not all whole numbers are integers since whole numbers are only the non-negative integers (0, 1, 2, ...).

Of particular interest to our exercise are consecutive integers. These are integers that come one after another, without any gap of a fractional or decimal value in between. If we pick an integer 'n', the consecutive integer immediately after 'n' would be 'n+1', and the one before it would be 'n-1'. For example, 4 and 5 are consecutive integers, so are -3 and -2. The term 'consecutive' depicts their uninterrupted sequence on the number line. This understanding is key when dealing with series, patterns, and mathematical proofs involving sequential whole numbers.

True or False Statements

In mathematics, true or false statements are used to assert the validity of a particular proposition. Each statement is given a truth value: 'true' if the proposition is correct, and 'false' if it's not. This type of logical assertion is binary — it cannot be both true and false at the same time. For students to successfully navigate through true or false questions, they must have a solid understanding of the related mathematical concepts.

When evaluating whether a statement about numbers is true or false, look for counterexamples to quickly disprove a false claim. In the absence of counterexamples, rely on definitions and properties to validate the truth. In our exercise, we applied the definition of consecutive integers to confirm the truth of the statement. Developing the ability to critically assess these statements is a valuable skill in mathematics and helps strengthen logical reasoning.