foundations
The number system gives structure to all calculation.
The Number Hierarchy
Mathematics organizes numbers into sets, where each set typically expands upon
the previous one. This hierarchy is represented as:
\( \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} \)
Natural Numbers \( \mathbb{N} \)
The natural numbers are the counting numbers: \( \{1, 2, 3, 4, \dots\} \). They are the simplest numbers and are used for counting discrete objects.
Integers \( \mathbb{Z} \)
Integers include positive natural numbers, their negatives, and zero: \( \{\dots, -2, -1, 0, 1, 2, \dots\} \). They allow for the representation of direction and magnitude (e.g., debt or temperature).
Rational Numbers \( \mathbb{Q} \)
Rational numbers can be expressed as a fraction \( \frac{p}{q} \) where \( p, q \in \mathbb{Z} \) and \( q \neq 0 \). Examples include \( \frac{1}{2} \), \( -\frac{5}{7} \), and \( 0.75 \) (which is \( \frac{3}{4} \)).
Irrational Numbers
Irrational numbers cannot be expressed as a fraction of two integers. Their decimal expansions are non-repeating and infinite. Famous examples include \( \sqrt{2} \) and \( \pi \approx 3.14159 \).
Real Numbers \( \mathbb{R} \)
Real numbers are the union of rational and irrational numbers. They correspond to every single point on a continuous number line.
Complex Numbers \( \mathbb{C} \)
Complex numbers extend the real numbers by introducing the imaginary unit \( i \), defined by the property \( i^2 = -1 \). A complex number is written in the form \( a + bi \), where \( a \) and \( b \) are real numbers.
Key idea
The number system is not just a list of categories. It shows how mathematical objects can be organized by structure, and how different kinds of numbers respond to the operations we perform on them.