numeri

Numeri, or numbers, are abstract objects used for counting, measuring, labeling, and performing arithmetic. In mathematics they appear in several related families, such as natural numbers, integers, rational numbers, real numbers, and complex numbers. They can be represented by symbols, written in different numeration systems, and manipulated according to defined rules.

Natural numbers are the nonnegative integers used for counting. They form the basis of arithmetic and are commonly described by axioms that establish successor, addition, and multiplication, as well as the principle of mathematical induction. Depending on convention, zero may be included or omitted.

Integers extend natural numbers to include zero and negative numbers, closed under addition and subtraction. Rational numbers are ratios of integers, forming a dense subset of the real numbers; every rational has a decimal expansion that either terminates or repeats. Real numbers include rationals and irrationals, providing a continuous line of values and enabling concepts such as limits and continuity. Complex numbers extend the real numbers by adjoining an imaginary unit i, with i^2 = -1; any complex number can be written as a + bi. The complex numbers form a field and are central in solving polynomial equations and in many areas of analysis.

Numeration and notation: numbers are often written in decimal positional notation, base 10, with digits 0–9. Other bases are used in computing (binary, octal, hexadecimal). Historic systems include Roman numerals. The study of numbers encompasses number theory, which investigates primes, divisibility, and the properties of integers.

Historically, ideas of number evolved from counting tools and tally marks to formal definitions. Contributions from various civilizations helped establish zero, negative numbers, and algebra. In modern mathematics, numbers underpin science, engineering, and computation, while philosophical questions about their nature continue to intrigue some scholars.