Number Systems

The Extension Chain

\[ \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} \]

Each extension solves a problem that was unsolvable in the previous number system.

Natural Numbers (ℕ)

\[ \mathbb{N} = \{0, 1, 2, 3, \ldots\} \]

Suitable for: counting, addition, multiplication.

Problem: The equation \(x + 3 = 1\) has no solution in \(\mathbb{N}\).

Integers (ℤ)

\[ \mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\} \]

Extension of \(\mathbb{N}\) by negative numbers. Subtraction is unrestricted.

Problem: The equation \(2x = 3\) has no solution in \(\mathbb{Z}\).

Rational Numbers (ℚ)

\[ \mathbb{Q} = \left\{\frac{a}{b} : a \in \mathbb{Z},\; b \in \mathbb{Z} \setminus \{0\}\right\} \]

Extension of \(\mathbb{Z}\) by fractions. Division (except by \(0\)) is unrestricted.

Rational numbers have either a terminating or a repeating decimal expansion: \(\frac{1}{4} = 0.25\), \(\frac{1}{3} = 0.\overline{3}\).

Problem: The equation \(x^2 = 2\) has no solution in \(\mathbb{Q}\) (since \(\sqrt{2}\) is irrational).

Real Numbers (ℝ)

\(\mathbb{R}\) comprises all points on the number line: rational and irrational numbers.

Irrational numbers have a non-repeating, infinite decimal expansion: \(\sqrt{2} = 1.41421\ldots\), \(\pi = 3.14159\ldots\)

\(\mathbb{R}\) is complete: every Cauchy sequence converges in \(\mathbb{R}\).

Problem: The equation \(x^2 = -1\) has no solution in \(\mathbb{R}\).

Complex Numbers (ℂ)

\[ \mathbb{C} = \{a + bi : a, b \in \mathbb{R}\}, \quad i^2 = -1 \]

Extension of \(\mathbb{R}\) by the imaginary unit \(i\). Every polynomial equation has a solution in \(\mathbb{C}\) (Fundamental Theorem of Algebra).

Example. \(x^2 + 1 = 0\) has solutions \(x = i\) and \(x = -i\).

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Summary

Number System	Symbol	New Property	Unsolvable Problem
Natural numbers	\(\mathbb{N}\)	Counting	\(x + 3 = 1\)
Integers	\(\mathbb{Z}\)	Subtraction	\(2x = 3\)
Rational numbers	\(\mathbb{Q}\)	Division	\(x^2 = 2\)
Real numbers	\(\mathbb{R}\)	Completeness	\(x^2 = -1\)
Complex numbers	\(\mathbb{C}\)	Algebraically closed	—

References

• Ebbinghaus, H.-D. et al.: Numbers. Springer, 1991.
• Courant, Richard; Robbins, Herbert: What Is Mathematics? Oxford University Press, 2nd edition, 1996. Chapter 2.