Problem 3 - Prealgebra

Question

Name all of the sets of numbers to which each real number belongs. Let \(\mathbf{N}=\) natural numbers, \(\mathbf{W}=\) whole numbers, \(\mathbf{Z}=\) integers, \(\mathbf{Q}=\) rational numbers, and I = irrational numbers. $$-\frac{3}{4}$$

Step-by-Step Solution

Answer

The number \(-\frac{3}{4}\) is a rational number (\mathbf{Q}).

Step 1: Analyzing the Negative Fraction

We begin by identifying the characteristics of the given number, \(-\frac{3}{4}\), which is a negative fraction.

Step 2: Determine if the Number is a Natural Number (N)

Natural numbers (N) are positive integers starting from 1 (i.e., 1, 2, 3, ...). Since \(-\frac{3}{4}\) is not positive and is not a whole number, it is not a natural number.\[N = \{1, 2, 3, \ldots\}\]

Step 3: Check if the Number is a Whole Number (W)

Whole numbers (W) are non-negative integers starting from 0 (i.e., 0, 1, 2, 3, ...). Since \(-\frac{3}{4}\) is negative, it cannot be classified as a whole number.\[W = \{0, 1, 2, 3, \ldots\}\]

Step 4: Check if the Number is an Integer (Z)

Integers (Z) include all whole numbers and their negatives without any fractional part (i.e., ..., -2, -1, 0, 1, 2, ...). \(-\frac{3}{4}\) has a fractional part, so it is not an integer.\[Z = \{\ldots, -2, -1, 0, 1, 2, \ldots\}\]

Step 5: Determine if the Number is a Rational Number (Q)

Rational numbers (Q) are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. \(-\frac{3}{4}\) fits this definition as it can be expressed as a fraction. Therefore, it is a rational number.\[Q = \left\{ \frac{a}{b} \mid a, b \in \u007fZ, b eq 0 \right\}\]

Step 6: Examine if the Number is an Irrational Number (I)

Irrational numbers cannot be expressed as a simple fraction of two integers. Since \(-\frac{3}{4}\) is already expressed as a fraction, it is not irrational.\[I = \text{numbers like } \pi, \sqrt{2}, \text{ etc., that cannot be written as } \frac{a}{b}\]

Key Concepts

Rational Numbers

Rational numbers are a significant part of the real number system. These numbers can be expressed as the fraction of two integers. This means you can write them as \( \frac{a}{b} \), where both \(a\) and \(b\) are integers, and \(b\) is never zero.

• Examples include fractions like \(-\frac{3}{4}\) and whole numbers like 5 (which can be seen as \(\frac{5}{1}\)).
• Even repeating decimals such as 0.333... or 1.666... are rational because they can be transformed into fractions: \(\frac{1}{3}\) and \(\frac{5}{3}\), respectively.

Rational numbers are incredibly useful because they provide a way to express quantities that are not whole numbers.
If a number can be written in the form of a fraction with two integer components, then it is within this versatile group.

Natural Numbers

Natural numbers are the building blocks of numbers that we start learning about when we first encounter mathematics. They are defined as the set of positive integers starting from 1, moving upward through 2, 3, 4, etc.

• The sequence is continuous and infinite, beginning at 1 without any fractional or decimal parts.
• Natural numbers are used in everyday counting and ordering.

It's important to recognize what constitutes a natural number, especially because they are the base subset from which more complex sets like integers and rational numbers are derived.
Simply put, if you are counting objects, you are dealing with natural numbers!

Integers

Integers expand upon natural numbers by including zero and negative numbers. They encompass all whole numbers, both positive and negative, and provide a comprehensive set without any fractional parts.

• Examples of integers include \(-2, -1, 0, 1, 2\).
• They are crucial for depicting situations where values can go below zero, like elevations below sea level or financial debts.

Integers are particularly handy in practical applications where fluctuations occur, such as gains and losses or temperature changes.
They bridge the gap between strictly positive natural numbers and the more nuanced fractional world of rational numbers.