antallene

Antallene (the numbers) are abstract objects in mathematics used for counting, measuring, labeling, and ordering. They range from natural numbers used to count discrete objects to more general number systems employed in computation, geometry, and analysis. The study of antallene covers their properties, relationships, and the rules by which they can be combined.

Common classes include natural numbers, integers, rational numbers, real numbers, and complex numbers. Each class extends the previous: integers add negative values to the naturals; rationals include fractions; reals encompass irrational numbers as well; complexes extend the real numbers with an imaginary unit. In addition, other systems such as p-adic numbers and hyperreal numbers are used in specialized areas of mathematics and logic.

Foundations: The natural numbers are often introduced via axioms (for example, the Peano axioms), which define zero, succession, and basic operations like addition and multiplication. The rational, real, and complex numbers are built to satisfy field axioms, enabling arithmetic operations, order, and properties like completeness for the real numbers. The concept of number also connects to set theory, where numbers can be constructed from sets and ordinals.

History and use: Counting systems and numerals originated in ancient civilizations. The invention of zero in India and the widespread adoption of Hindu-Arabic numerals revolutionized calculation. In mathematics, antallene underpin arithmetic, algebra, analysis, and applied disciplines such as physics, computer science, and economics. Ongoing study explores properties such as divisibility, prime structure, numerical representations, and the cardinality of infinite sets.