Zahlen

Zahlen, or numbers, are abstract objects used for counting, measuring, and ordering. In mathematics they are organized into a hierarchy of sets: natural numbers N (often including zero), integers Z (positive, negative, and zero), rational numbers Q (fractions of integers), real numbers R (all decimal expansions, including irrational numbers), and complex numbers C (numbers with a real and an imaginary part). Each class extends the previous, while preserving arithmetic operations such as addition and multiplication.

Rational numbers can be written as fractions; irrational numbers cannot be expressed as fractions. Real numbers form a continuum and support limits and calculus. Complex numbers, written as a + bi, extend real arithmetic to two dimensions and are essential in many areas of analysis and applied sciences. Notation uses digits, symbols, and algebraic rules that govern these sets.

Historical overview: Counting relied on primitive tally marks and later more elaborate numeral systems. Babylonian and Egyptian numerals illustrate early approaches to written numbers. The Hindu–Arabic numeral system introduced decimal place value and zero, transforming arithmetic and computation. Through scholarly transmission, these numerals spread widely and became the standard in many parts of the world.

Formal foundations and uses: Natural numbers can be defined by Peano axioms, real numbers by limits or Dedekind cuts, and complex numbers as ordered pairs. Numbers underpin measurement, modeling, and data processing in science and engineering. In computing, numbers are stored in binary form and approximated with floating-point representations, enabling digital computation across applications.