Tölur

Tölur is the Icelandic term for numbers, the abstract objects used to count, measure, order, and label quantities. In mathematics, tölur form a hierarchy of number systems, including natural numbers, integers, rational numbers, real numbers, and complex numbers, among others. Natural numbers are used for counting, integers extend them to negative values, rational numbers include fractions, real numbers describe continuous quantities and include irrational numbers, and complex numbers combine real and imaginary parts and are central in many areas of mathematics and physics.

The historical development of tölur spans many cultures. Early counting used natural numbers; the invention of zero and a place-value system in India greatly influenced later mathematics. The concepts of negative numbers, fractions, and later real and complex numbers emerged gradually across different regions and periods, culminating in frameworks that support algebra, analysis, and geometry today.

Key properties of tölur include closure under addition and multiplication in many systems, as well as commutativity and associativity of these operations. Division by zero is undefined, and order relations allow comparison of values. Real numbers enable precise measurement of continuous quantities, while complex numbers extend the real line into a two-dimensional plane and underpin many computational methods and signal processing techniques.

Representations of tölur typically use numeral systems. The decimal system is positional, using base ten and digits 0–9, while other systems such as binary, octal, or hexadecimal are used in computing. Numbers serve as labels, quantities, and parameters across science, engineering, economics, and beyond, and they underpin theories in number theory, analysis, and mathematical logic.