Problem 8 - Elementary Algebra

Question

Determine whether the following real numbers are integers, rational, or irrational. $$ -3 $$

Step-by-Step Solution

Answer

-3 is an integer and a rational number.

Step 1: Identify the Type of Real Number

The given number is \(-3\). First, we need to determine if it is an integer. Integers are whole numbers without any fractional or decimal part. They include negative numbers, zero, and positive numbers. Since \(-3\) is a whole number and has no fractions or decimals, it is an integer.

Step 2: Determine if the Number is Rational

A rational number can be expressed as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b eq 0\). Since \(-3\) can be written as \(\frac{-3}{1}\), it is a rational number.

Step 3: Check for Irrationality

Irrational numbers cannot be written as a simple fraction and typically have non-repeating, non-terminating decimals. Examples include \(\sqrt{2}\), \(\pi\), etc. Since \(-3\) can be expressed as a fraction, it is not irrational.

Key Concepts

Integers

Integers are a fundamental part of real numbers. They include both positive and negative whole numbers, as well as zero.
So, when you think about integers, remember:

• They don't have fractional components.
• They include numbers like -3, 0, 4, and so on.

The number (-3) is a perfect example because it sits on the number line with no decimal or fractional part. Understanding this helps in identifying other integers easily. Consider any whole number without fractions or decimals, and you're dealing with an integer.

Rational Numbers

Rational numbers might sound complex, but they are straightforward. A rational number is any number that can be expressed as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b \, eq 0\).
Here's what to keep in mind:

• They can be whole numbers themselves, like (-3), written as \(\frac{-3}{1}\).
• They include positive and negative fractions and decimals that can be converted to fractions.

When you see a number like \(-3\), you can write it as a fraction with the denominator of 1, which confirms it's rational. Rational numbers are everywhere around us, in objects like half a pie or 3/4 of a gallon. This makes them incredibly practical in math and everyday life.

Irrational Numbers

Irrational numbers are the real numbers that can't be written as simple fractions.
They have specific characteristics:

• They include numbers like \(\sqrt{2}\) and \(\pi\), which cannot be exactly expressed as fractions.
• They have non-repeating, non-terminating decimal expansions.

Examples of irrational numbers are fascinating. Take \(\pi\), the ratio of a circle's circumference to its diameter, which continues indefinitely without repeating. Another example is \(\sqrt{2}\), the diagonal of a square with side length 1. These numbers have unique properties that distinguish them from rational numbers. Understanding irrationality opens up new areas of exploration in geometry and calculus, making them integral to higher-level mathematics.