Problem 16 - 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. $$-\frac{24}{8}$$

Step-by-Step Solution

Answer

The number -3 belongs to the sets \(\mathbf{Z}\) and \(\mathbf{Q}\).

Step 1: Simplify the Fraction

Simplify \(-\frac{24}{8}\). Divide both the numerator and the denominator by their greatest common divisor, which is 8.\[-\frac{24}{8} = -3\].

Step 2: Identify the Type of Number

Determine which sets the number \(-3\) belongs to. Since \(-3\) is a negative whole number, it is an integer.

Step 3: Classify as a Rational Number

Recognize that any number that can be expressed as a fraction of integers is a rational number. Since \(-3 = \frac{-3}{1}\), it is a rational number \(\mathbf{Q}\).

Step 4: Exclude Other Sets

Verify and exclude other number sets. \(-3\) is not a natural number or a whole number as it is negative. It is also not an irrational number because it can be expressed as the fraction \(\frac{-3}{1}\).

Key Concepts

Real Numbers

Real numbers are all the numbers that exist on the number line. This includes both rational and irrational numbers. To understand real numbers, think of any number you know, like 2, -5, or even \(\sqrt{2}\). These all fall into the category of real numbers.
The beauty of real numbers lies in how comprehensive they are. They include everything from positive and negative numbers, whole numbers, decimals, and even numbers that can't be quite written down fully, like \( \pi \) or \( \sqrt{2} \). Such diverse numbers fit into real numbers because they satisfy the need to represent points on the number line and solve equations that require them.

• Rational numbers: Yes, they're part of real numbers.
• Irrational numbers: Yes, these join in too!

In short, if you can find it on the number line, it's a real number.

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction of two integers. This means any number \( q = \frac{a}{b} \,b eq 0 \) is considered rational. So, whether \( a \) and \( b \) are positive or negative doesn't matter.
A clear way to understand rational numbers is by looking at common fractions, like \( \frac{1}{2} \), or even whole numbers like 4, which can be written as \( \frac{4}{1} \). Both these examples can be written as a fraction of two integers, making them rational. The number \( -3 \) from the exercise is rational because you can express it as \( \frac{-3}{1} \).
Different sets of numbers:

• Decimal numbers that repeat, like \( 0.333... \), are rational because they can be represented as fractions.
• Whole numbers, fractions, and zero are all rational numbers.

If you can write it as a neat fraction, it belongs to the rational number set.

Integers

Integers are whole numbers and their negative counterparts. This means they include: zero, positive numbers (1, 2, 3...), and negative numbers (-1, -2, -3...). Importantly, integers do not include fractions, decimals, or irrational numbers.
One way to think of integers is as the complete set of whole numbers, stretching out infinitely in both positive and negative directions on the number line. They are fundamental in basic math operations because they are complete numbers without fractions or decimals.

• Positive integers are also called natural numbers.
• Zero is a special integer, acting as the neutral point between positive and negative.
• Negative integers count downwards in the opposite direction from zero.

The exercise number \( -3 \) fits perfectly as an integer because it doesn't have any fractional or decimal part. It's just a straight-up whole number, albeit a negative one.