Problem 52 - Prealgebra

Question

Determine whether each statement is sometimes, always, or never true. Explain by giving an example or a counterexample. An integer is a rational number.

Step-by-Step Solution

Answer

An integer is always a rational number.

Step 1: Understanding the Terms

First, we need to understand what it means for a number to be an integer and a rational number. Integers are whole numbers that can be positive, negative, or zero, such as -3, 0, and 5. Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero, such as \( \frac{2}{1} \) or \( \frac{-5}{3} \).

Step 2: Analyzing the Statement

The statement given is that 'an integer is a rational number.' We need to determine if this is sometimes true, always true, or never true. Since a rational number is defined as a number that can be expressed as a quotient of two integers, consider any integer \( n \), it can be expressed as \( \frac{n}{1} \).

Step 3: Providing a Example

To see this in practice, let's take an integer, like 3. We can write 3 as \( \frac{3}{1} \). This expression means 3 is composed by dividing the integer 3 by 1, a valid rational form. Another example would be the integer -5, which can be also written as \( \frac{-5}{1} \). Both examples maintain that each integer can be expressed in a rational form.

Step 4: Conclusion

Since we can express any integer \( n \) as \( \frac{n}{1} \) where 1 is a non-zero integer, integers are indeed rational numbers. Hence, the statement that 'an integer is a rational number' is always true.

Key Concepts

Integers

Integers are a fundamental part of mathematics, representing whole numbers that can be positive, negative, or zero. When we talk about integers, we refer to numbers like -3, 0, and 5. These numbers do not have fractional or decimal parts. They form the most basic number type and are used in everyday counting and simple arithmetic.

• Positive integers: 1, 2, 3, and so on.
• Negative integers: -1, -2, -3, and so on.
• Zero is also an integer.

Integers are part of the larger set of rational numbers, even though they do not have a fractional part like fractions or decimals do. This means that while integers are a subset of rational numbers, they serve as the building blocks for other numeric types in the number classification.

Number Classification

Numbers can be categorized into different groups based on their properties. Number classification helps us understand how various types of numbers relate to each other. Here are some main categories:

• Natural numbers: Starting from 1, these include all the counting numbers (1, 2, 3, ...).
• Whole numbers: Including all natural numbers plus zero (0, 1, 2, 3, ...).
• Integers: Consisting of whole numbers and their negative counterparts (-3, -2, -1, 0, 1, 2, 3, ...).
• Rational numbers: These include all numbers that can be expressed as a ratio of two integers, such as \( \frac{3}{4} \), \( \frac{-7}{1} \), and so forth. Every integer is a rational number because it can be expressed as itself divided by 1 (like \( \frac{5}{1} \)).
• Irrational numbers: Numbers that cannot be expressed as a simple fraction, such as \( \sqrt{2} \) or \( \pi \).

These categories help in the understanding and the logical organization of numbers, revealing how they are related and used in mathematics.

Mathematical Proofs

Mathematical proofs are logical arguments that demonstrate the truth of a mathematical statement. In this context, to determine that an integer is always a rational number, one can construct a proof through logical reasoning.
When we claim an integer is always a rational number, we rely on the definition: a rational number can be expressed as \( \frac{a}{b} \), where both \( a \) and \( b \) are integers with \( b eq 0 \). To prove this, for any integer \( n \), you can always write it as \( \frac{n}{1} \).
This representation satisfies the conditions of rational numbers:

• The numerator \( n \) is an integer.
• The denominator 1 is a non-zero integer.

These logical sequences form the core foundation of a mathematical proof, illustrating the power of logical reasoning in establishing facts within mathematics.