The Real Number System

The real number system is the foundation of algebra. It consists of all the numbers that can be found on a number line. To understand real numbers, it helps to look at the "family tree" or hierarchy of numbers, starting from the simplest counting numbers up to decimals that go on forever.

The Hierarchy of Numbers

The real number system is divided into two main categories: Rational Numbers and Irrational Numbers. Within the rational numbers, there are several smaller nested sets.

1. Natural Numbers

These are the basic counting numbers starting from $1$.

• Set: $\{1, 2, 3, 4, \dots\}$

2. Whole Numbers

Whole numbers include all the natural numbers, plus zero.

• Set: $\{0, 1, 2, 3, 4, \dots\}$

3. Integers

Integers include all whole numbers and their negative counterparts. They do not include fractions or decimals.

• Set: $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$

4. Rational Numbers

A rational number is any number that can be written as a fraction $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.

• Examples: $\frac{2}{3}$, $-\frac{5}{1}$ (which is $-5$), $0.75$ (which is $\frac{3}{4}$).
• Note: If a rational number is written as a decimal, it either terminates (ends, like $0.5$) or repeats (like $0.333\dots$).

5. Irrational Numbers

Irrational numbers cannot be written as a simple fraction. When written as decimals, they go on forever without ever repeating a pattern.

• Examples: $\pi$ (approx. $3.14159\dots$), $\sqrt{2}$, $\sqrt{7}$.

Together, all Rational and Irrational numbers combine to form the Real Numbers. Every single real number has an exact, unique location on the number line.

Example Problems

Example 1: Classifying Numbers

Question: Classify the following numbers into their most specific number sets: $-3$, $0$, $\sqrt{2}$, $\frac{2}{3}$, $\pi$.

Solution:

• $-3$: It is an Integer (and therefore also a rational and real number).
• $0$: It is a Whole Number (and also an integer, rational, and real number).
• $\sqrt{2}$: It is an Irrational Number (and a real number). It cannot be written as a fraction.
• $\frac{2}{3}$: It is a Rational Number (and a real number).
• $\pi$: It is an Irrational Number (and a real number).

Example 2: True or False

Question: True or false: Every integer is a rational number.

Solution: True. Any integer can be written as a fraction by placing it over $1$. For example, the integer $-7$ can be written as $\frac{-7}{1}$, which perfectly fits the definition of a rational number.