Problem 60 - Introductory Algebra for College Students

Question

List all the integers in this set: $$\left\\{-73,-\frac{2}{3}, 0, \frac{3}{1}, \frac{3}{2}, \frac{\pi}{1}\right\\}$$

Step-by-Step Solution

Answer

The integers in the given set are -73, 0, and 3.

Step 1: Go through the set

Inspect each number in the set one by one to determine if it's an integer. The numbers in the set are -73, -2/3, 0, 3/1, 3/2, and π/1.

Step 2: Identify the integers

Identify which numbers are considered integers. Here, -73 is a negative integer, 0 is considered an integer (neither positive nor negative), and 3/1 is also an integer because the division of any number by 1 gives the same number. So, 3/1 is actually an integer 3.

Step 3: Exclude non-integers

Exclude numbers that are not considered integers. Here, -2/3, 3/2 and π/1 are not integers. -2/3 and 3/2 are fractions and not whole numbers, while π/1 is an irrational number, as π is a non-terminating non-repeating decimal.

Key Concepts

Integers

Integers are whole numbers that can be positive, negative or zero. They do not include fractions, decimals or any form of a "part" of a number. For instance, -73 and 0 are integers because they meet these characteristics.

It's important to remember that dividing a number by one does not change its value—this is why \( \frac{3}{1} \) is also considered an integer as it simplifies to 3. Overall, integers can be thought of as "complete" numbers that stand alone in counting scenarios. They make up the number line which extends infinitely in both directions from zero.

• Examples of integers: -5, 0, 12, 100
• Non-examples: \( \frac{1}{2} \), 2.5, \( \frac{\pi}{1} \)

Fractions

Fractions are numbers that represent a part of a whole or, more generally, any number of equal parts. They are usually written in the form \( \frac{a}{b} \), where \( a \) is the numerator (the top number) and \( b \) is the denominator (the bottom number), and \( b eq 0 \).

In the set we are examining, \(-\frac{2}{3}\) and \(\frac{3}{2}\) are fractions because they show parts of a whole number. These are not integers since their numerators or denominators are not equal to one where they would signify whole numbers. In understanding and utilizing fractions, they help in situations needing more precision than what integers provide.

• Common fraction examples: \( \frac{1}{2} \), \( \frac{3}{4} \), \( \frac{7}{8} \)
• Non-fraction examples would be whole numbers like 4 or -3

Rational Numbers

Rational numbers encompass all numbers that can be expressed as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b eq 0 \). This means that integers are indeed a category of rational numbers because they can be written with a denominator of 1.

For example, 0 can be expressed as \( \frac{0}{1} \), and therefore it is part of this category along with \(-\frac{2}{3}\) and \(\frac{3}{2}\). Understanding rational numbers is crucial because they allow us to represent quantities that aren't whole, like everyday measurements or probabilities.

• Examples: \(-3, \frac{2}{1}, \frac{1}{4} \)
• Non-examples: numbers that cannot be expressed as fractions with integer numerators and non-zero denominators, such as \( \sqrt{2} \) or \( \pi \).

Irrational Numbers

Irrational numbers are real numbers that cannot be written as a simple fraction. They are numbers with non-terminating, non-repeating decimal expansions.

No matter how you slice it, numbers like \( \pi \) and \( \sqrt{2} \) cannot be precisely expressed as fractions involving integers, making them irrational. In the given set, \( \frac{\pi}{1} \) is exactly such a number because \( \pi \) continues infinitely without recurring. Recognizing irrational numbers is important as they often appear naturally in geometry, such as in calculations involving circles (\( \pi \)) and right-angled triangles (\( \sqrt{2} \)).

• Common irrational numbers: \( \pi, \sqrt{2}, e \)
• Non-examples: values that can form a neat fraction, like \( \frac{3}{4} \) or 0.5