Number Sets in Mathematics

This page provides an overview of various number sets, presenting them in both German and English. It covers the progression from basic natural numbers to more complex number sets, illustrating the expanding scope of mathematical concepts.

Definition: Number sets are collections of numbers with specific properties and rules.

The page outlines the following number sets:

1. Natural Numbers (N): In German, "Natürliche Zahlen". These are the counting numbers starting from 1 (or sometimes including 0).

Example: N = {1, 2, 3, ...} or {0, 1, 2, 3, ...}

2. Whole Numbers (W): In German, "Ganze Zahlen positiv". These include all natural numbers and zero.

Example: W = {0, 1, 2, 3, ...}

3. Integers (Z): In German, "Ganze Zahlen". This set includes all positive and negative whole numbers, and zero.

Example: Z = {..., -2, -1, 0, 1, 2, ...}

4. Rational Numbers (Q): In German, "Rationale Zahlen". These are numbers that can be expressed as a fraction of two integers.

Example: Q includes numbers like 3, 1/7, 2.27

5. Irrational Numbers (I): Numbers that cannot be expressed as a simple fraction and have non-repeating, non-terminating decimal expansions.

Example: I includes numbers like √2, π

6. Real Numbers (R): In German, "Reelle Zahlen". This set includes all rational and irrational numbers.

Example: R includes numbers like 11, √2, 3, π

7. Complex Numbers (C): In German, "Komplexe Zahlen". These include all real numbers and imaginary numbers.

Highlight: Complex numbers are often used in advanced mathematics and physics.

The page also provides additional information:

Vocabulary:

• "Divisionsberechnung" (German) = division calculation
• "Dezimalzahl" (German) = decimal number

Note: Imaginary numbers are described as numbers that "don't really exist," with √-19 given as an example.

This comprehensive overview of number sets provides a foundation for understanding the structure and relationships between different types of numbers in mathematics.