Textbook
The real numbers, denoted $\mathbb R$, correspond to points on the number line. They include both rational and irrational numbers. An irrational real number cannot be represented as a ratio of integers with nonzero denominator; $\sqrt2$ is an example.
The number sets form the chain $\mathbb N\subseteq\mathbb Z\subseteq\mathbb Q\subseteq\mathbb R$. These inclusions are not equalities: $-1$, $1/2$, and $\sqrt2$ illustrate successive extensions of the allowed values. Positive infinity is not a real number; an infinite boundary describes an unbounded region rather than an additional point in $\mathbb R$.
A point on the line is determined by its position among numbers, not by the length of an available written representation. The exact expression $\sqrt2$ identifies the positive number whose square is 2, without listing decimal digits. Unlike a finite set of machine numbers, $\mathbb R$ has no immediate “next number” after a given one: between any $a<b$ lies $(a+b)/2$, for instance. A screen’s decimal grid only displays the line.
Definitions and further reading: Basic Analysis I.
In plain language
The real number line contains integers, fractions, and exact values such as $\sqrt2$, the diagonal length of a unit square. The decimal $1.414$ only approximates that root; the display’s last digit is not the end of the number. Zooming reveals positions that previously merged rather than creating new points. A physical ruler has limited precision, whereas the mathematical model does not; a finite calculator display cannot serve as an infinite printing press.
Example
The example locates the illustrative values −1, $1/2$, and $\sqrt2$. All are permitted in $\mathbb R$. The final step distinguishes the exact root from a decimal approximation.
Real Numbers: Points on the Number Line
Negative point
The value −1 lies left of zero and belongs to R.
Step 1 of 3: Negative point
Read all steps
Negative point
The value −1 lies left of zero and belongs to R.
- State
- −1current01
Fractional point
One half lies between zero and one.
- State
- 01/2current1
Exact and approximate
The root is a real number; 1.414 only approximates it.
- State
- 1.414current√2current1.415current
The final bound can be checked without trusting the drawing’s scale: $1.414^2=1.999396<2$, while $1.415^2=2.002225>2$. Both bounds are positive, so the root lies between them. This is more precise than saying “the root equals 1.414”: it specifies a containing interval instead of replacing the number with rounding. Greater accuracy requires tighter bounds while preserving the distinction between the exact object and its approximation.
Related concepts: Rational numbers, Real intervals.