Problem 1 - Prealgebra and Introductory Algebra

Question

Determine which of the numbers are a. integers, b. rational numbers, c. irrational numbers, and d. real numbers. List all that apply. $$-\frac{15}{2}, 0,-3, \pi, 2 . \overline{33}, 4.232232223 \ldots, \frac{\sqrt{5}}{4}, \sqrt{7}$$

Step-by-Step Solution

Answer

Integers: -3; Rational Numbers: \( -\frac{15}{2}, 0, -3, 2 . \overline{33}\); Irrational Numbers: \(\pi, \frac{\sqrt{5}}{4}, \sqrt{7}\); Real Numbers: \( -\frac{15}{2}, 0, -3, \pi, 2 . \overline{33}, 4.23223223 \ldots , \frac{\sqrt{5}}{4}, \sqrt{7}\)

Step 1: Identify Integers

By identifying numbers without any fraction or decimal parts, -3 is the only integer here.

Step 2: Identify Rational Numbers

Rational numbers can be expressed as a ratio of two integers. So, \(-\frac{15}{2}\), 0, -3 and \(2 . \overline{33}\) are rational numbers. Note that \(2 . \overline{33}\) is a repeating decimal, which can be expressed as a ratio of two integers, so it's a rational number.

Step 3: Identify Irrational Numbers

\(\pi\), \(\frac{\sqrt{5}}{4}\), and \(\sqrt{7}\) cannot be expressed as a ratio of two integers. Thus, they are irrational numbers.

Step 4: Identify Real Numbers

All the numbers listed here are real numbers because they can all be located on the number line.

Key Concepts

Integers

Integers are a fundamental part of real numbers. They include all whole numbers, both positive and negative, as well as zero. An integer does not have fractions or decimal parts. This means any number like 1, 2, 3, -1, -2, -3, and so on, is an integer. For example, in the given problem, we only have one integer, which is

• -3

When looking at integers, remember there are no decimal points involved, making them simple and easy to identify. Integers are essential in mathematics because they serve as building blocks for other types of numbers.

Rational Numbers

Rational numbers are numbers that can be written as a fraction of two integers. The definition can be broken into simpler parts:

• They must have a numerator and a denominator that are integers.
• The denominator cannot be zero.

This includes numbers with repeating or terminating decimals. For instance, 0, even though it does not look like a fraction, can be expressed as \( \frac{0}{1} \). In our exercise, the numbers

• -15/2
• 0
• -3
• 2 . \overline{33}

are all rational numbers. This is because they either have whole numbers or repeating decimal patterns that allow them to be converted into fractions. Understanding rational numbers is crucial as it extends the basic concept of fractions to a broader range of decimals and integers.

Irrational Numbers

Irrational numbers are those that cannot be expressed as a simple fraction (the ratio of two integers). This includes numbers with non-repeating, non-terminating decimal parts. They cannot be neatly expressed in a fractional form like rational numbers.

• Examples include \( \pi \) and \( \sqrt{7} \).
• \( \frac{\sqrt{5}}{4} \) is also irrational because \( \sqrt{5} \) is not rational.

These numbers continue infinitely without repeating, making them unique compared to rational numbers. Exploring irrational numbers helps in understanding the "gaps" between rational numbers on the number line and provides insight into the complexities of real numbers.