Numbers can be grouped into several nested number systems. Some sets sit inside larger sets.
For example, every integer is rational, every rational number is real, and every real number is complex.
1. Natural numbers
Natural numbers are the counting numbers. In many courses, they are written as:
\[
\begin{aligned}
1,\ 2,\ 3,\ 4,\ldots
\end{aligned}
\]
Some courses include \(0\) as a natural number. Because both conventions exist, the calculator lets the user choose the convention.
2. Whole numbers
Whole numbers are the nonnegative integers:
\[
\begin{aligned}
0,\ 1,\ 2,\ 3,\ 4,\ldots
\end{aligned}
\]
Every whole number is an integer, but not every integer is a whole number. For example, \(-3\) is an integer but not a whole number.
3. Integers
Integers include negative whole numbers, zero, and positive whole numbers:
\[
\begin{aligned}
\ldots,\ -3,\ -2,\ -1,\ 0,\ 1,\ 2,\ 3,\ldots
\end{aligned}
\]
Integers have no fractional part and no decimal part other than zeros.
4. Rational numbers
A rational number is any number that can be written as a ratio of two integers:
\[
\begin{aligned}
\frac{a}{b}
\quad\text{where }a,b\in\mathbb{Z}\text{ and }b\ne0.
\end{aligned}
\]
Examples:
\[
\begin{aligned}
22/7,\quad -5,\quad 0.25,\quad 0.\overline{3}
\end{aligned}
\]
All integers are rational because every integer \(n\) can be written as \(n/1\).
5. Irrational numbers
An irrational number is a real number that cannot be written as a ratio of two integers.
Its decimal expansion does not terminate and does not repeat.
Common examples are:
\[
\begin{aligned}
\sqrt{2},\quad \pi,\quad e.
\end{aligned}
\]
The number \(\sqrt{2}\) is irrational because \(2\) is not a perfect square.
6. Real numbers
Real numbers are all numbers that can be placed on the number line.
They include both rational and irrational numbers:
\[
\begin{aligned}
\mathbb{R}
&=
\{\text{rational numbers}\}
\cup
\{\text{irrational numbers}\}.
\end{aligned}
\]
Examples of real numbers include \(-4\), \(0\), \(22/7\), \(\sqrt{2}\), and \(\pi\).
7. Complex numbers
A complex number has the form:
\[
\begin{aligned}
a+bi
\end{aligned}
\]
where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit:
\[
\begin{aligned}
i^2&=-1.
\end{aligned}
\]
If \(b=0\), then \(a+0i=a\), so every real number is also complex.
If \(b\ne0\), then the number is non-real complex, such as \(3+4i\).
8. Nesting of number systems
The main nesting is:
\[
\begin{aligned}
\text{Natural}
\subset
\text{Whole}
\subset
\text{Integer}
\subset
\text{Rational}
\subset
\text{Real}
\subset
\text{Complex}.
\end{aligned}
\]
Irrational numbers are also real numbers, but they are not rational numbers.
Therefore, rational and irrational numbers are two different parts of the real number system.
9. Worked example: \(22/7\)
The number \(22/7\) is a ratio of two integers and the denominator is not zero:
\[
\begin{aligned}
\frac{22}{7}
&=
\frac{a}{b}
\quad\text{with }a=22,\ b=7.
\end{aligned}
\]
Therefore, \(22/7\) is rational. Since every rational number is real, and every real number is complex:
\[
\begin{aligned}
22/7\in\mathbb{Q},\quad
22/7\in\mathbb{R},\quad
22/7\in\mathbb{C}.
\end{aligned}
\]
It is not an integer, whole number, or natural number because it is not a whole counting value.
10. Worked example: \(\sqrt{2}\)
The number \(2\) is not a perfect square. Therefore, \(\sqrt{2}\) cannot be written as a ratio of two integers.
\[
\begin{aligned}
\sqrt{2}&\notin\mathbb{Q}.
\end{aligned}
\]
However, \(\sqrt{2}\) lies on the number line, so it is real:
\[
\begin{aligned}
\sqrt{2}&\in\mathbb{R}.
\end{aligned}
\]
Since every real number is complex, \(\sqrt{2}\) is also complex.
11. Worked example: \(3+4i\)
The number \(3+4i\) has imaginary part \(4\), which is not zero.
\[
\begin{aligned}
3+4i&\in\mathbb{C},\\
3+4i&\notin\mathbb{R}.
\end{aligned}
\]
Since it is not real, it is not rational or irrational. Rational and irrational are classifications inside the real number system.
12. Formula summary
This table uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
13. Common mistakes
- Thinking every decimal is irrational. Terminating and repeating decimals are rational.
- Forgetting that every integer is also rational.
- Forgetting that every real number is also complex because it can be written as \(a+0i\).
- Calling a non-real complex number rational or irrational. Rational and irrational are real-number categories.
- Forgetting that the convention for whether \(0\) is natural can vary by course.
Key idea: start with the most specific category, then move outward through the nested sets.
