Table of Contents
Understanding Real Numbers
Real numbers are the numbers we use to measure and count along a continuous line, called the number line. In earlier chapters you meet specific types of numbers: natural numbers, integers, rational numbers, and irrational numbers. The real numbers gather all of these into one large, unified set.
In this chapter, the focus is on what makes real numbers special as a whole, beyond each individual type.
The Number Line and Real Numbers
Imagine a straight horizontal line that stretches without end to the left and right. We mark a point in the middle as $0$. To the right we have positive numbers; to the left, negative numbers.
Every real number can be shown as a point on this line, and every point on the line corresponds to exactly one real number.
Examples:
- $3$ is three units to the right of $0$.
- $-1.5$ is one and a half units to the left of $0$.
- $\sqrt{2}$ is somewhere between $1$ and $2$, closer to $1.4$.
- $\pi$ is a bit more than $3$, about $3.14$.
Key idea:
The number line is a picture of the real numbers. Thinking of real numbers as points on a line helps when talking about distance, ordering, and size.
What Belongs to the Real Numbers
The set of real numbers, usually denoted by $\mathbb{R}$, includes:
- All natural numbers: $1, 2, 3, \dots$
- All integers: $\dots, -2, -1, 0, 1, 2, \dots$
- All rational numbers (fractions of integers, like $\dfrac{2}{3}$, $-5$, $\dfrac{7}{1}$).
- All irrational numbers (numbers that cannot be written as a fraction of integers, like $\sqrt{2}$, $\pi$).
Symbolically, you can think of:
$$
\mathbb{R} = \{\text{all rational numbers}\} \cup \{\text{all irrational numbers}\}.
$$
So every rational and every irrational number is a real number.
Infinite and Dense: No Gaps Between Real Numbers
Two important features of real numbers are:
1. There are infinitely many real numbers
Between any two different real numbers, no matter how close they are, there are infinitely many other real numbers.
Example:
Between $1$ and $2$ you can find $1.1, 1.11, 1.111, \dfrac{3}{2}, \sqrt{2}, \pi - 1$, and infinitely many more.
2. The real numbers are dense
A set of numbers is called dense on the number line if between any two numbers in the set, there is always another number from the set.
The real numbers are dense:
- Between any two real numbers $a$ and $b$ with $a < b$, there is another real number, such as $\dfrac{a+b}{2}$ (the midpoint).
In fact, both rational and irrational numbers are also dense in $\mathbb{R}$:
- Between any two real numbers there is a rational number.
- Between any two real numbers there is also an irrational number.
This density is part of why the number line looks “solid,” with no gaps.
Ordering Real Numbers
Real numbers can be compared and ordered. For any two real numbers $a$ and $b$, exactly one of the following is true:
- $a < b$ (a is less than b),
- $a = b$ (they are equal),
- $a > b$ (a is greater than b).
On the number line:
- If $a < b$, then $a$ lies to the left of $b$.
- If $a > b$, then $a$ lies to the right of $b$.
Examples:
- $-3 < 2$ because $-3$ is to the left of $2$.
- $1.4142 < \sqrt{2} < 1.5$ (using approximate decimal values).
- $-1.2 > -3.4$ because $-1.2$ is to the right of $-3.4$.
Ordering lets us talk about intervals and ranges of real numbers.
Intervals of Real Numbers
An interval is a continuous chunk of the number line. Instead of listing every number, we describe it using boundary points.
Common types:
- Closed interval $[a, b]$:
All real numbers between $a$ and $b$, including $a$ and $b$.
$$
[a, b] = \{x \in \mathbb{R} \mid a \le x \le b\}.
$$ - Open interval $(a, b)$:
All real numbers between $a$ and $b$, but not including $a$ and $b$.
$$
(a, b) = \{x \in \mathbb{R} \mid a < x < b\}.
$$ - Half-open intervals:
- $(a, b]$: includes $b$ but not $a$.
- $[a, b)$: includes $a$ but not $b$.
- Unbounded intervals:
- $(a, \infty)$: all real numbers greater than $a$.
- $(-\infty, b)$: all real numbers less than $b$.
- $(-\infty, \infty)$: all real numbers.
These notations are used constantly in algebra, calculus, and beyond to describe domains, solution sets, and ranges.
Absolute Value on the Real Line
For real numbers, absolute value measures distance from $0$ on the number line.
The absolute value of a real number $x$ is written $|x|$ and defined by:
- $|x| = x$ if $x \ge 0$,
- $|x| = -x$ if $x < 0$.
So:
- $|5| = 5$,
- $|-5| = 5$,
- $|0| = 0$.
Geometric interpretation:
- The distance between $a$ and $b$ on the number line is $|a - b|$.
Examples:
- Distance between $3$ and $-2$ is $|3 - (-2)| = |5| = 5$.
- Distance between $1.5$ and $1.2$ is $|1.5 - 1.2| = 0.3$.
Absolute value expressions often appear in inequalities and distance problems.
Approximating Real Numbers with Decimals
Many real numbers cannot be written as exact finite decimals, especially irrational numbers. However, we can approximate them.
Example:
- $\pi \approx 3.14$ or $\pi \approx 3.1416$.
- $\sqrt{2} \approx 1.41$ or $\sqrt{2} \approx 1.414$.
Two ideas:
- Finite decimal approximation:
A short decimal like .14$ is close to $\pi$ but not equal. - Infinite decimal representation:
Some real numbers have infinite decimal expansions: - Rational example: $\dfrac{1}{3} = 0.3333\ldots$ (repeating pattern).
- Irrational example: $\pi = 3.14159265\ldots$ (non-repeating, non-ending).
Real numbers can be thought of as numbers with possibly infinite decimal expansions.
Completeness of the Real Numbers (Informal Idea)
The real numbers fill the number line completely, with no gaps. This idea is sometimes called “completeness.”
Informally, this means:
- If you have a sequence of real numbers that gets closer and closer to some value, that limiting value is also a real number.
- If you have a non-empty set of real numbers that is bounded above (it doesn’t go off to infinity), then it has a least upper bound (a smallest number that is greater than or equal to everything in the set), and that least upper bound is a real number.
You do not need the formal definition at this level, but the key point is: the real numbers are designed so that limits and “approaching” ideas (used heavily in calculus) make sense and stay inside $\mathbb{R}$.
Summary
- Real numbers $\mathbb{R}$ include all rational and irrational numbers.
- They can all be represented as points on a single number line.
- Between any two real numbers, there are infinitely many other real numbers; the set is dense.
- Real numbers can be ordered: each pair can be compared as less than, equal, or greater than.
- Intervals describe continuous stretches of real numbers.
- Absolute value measures distance on the real line.
- Many real numbers require infinite decimal expansions; finite decimals often give approximations.
- The real numbers form a complete system, filling the number line with no gaps.