Number

A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers: 1, 2, 3, 4, 5, and so forth. Individual numbers can be represented in language with number words or by dedicated symbols called numerals; for example, "five" is a number word and "5" is the corresponding numeral. As only a limited list of symbols can be memorized, a numeral system is used to represent any number in an organized way. The most common representation is the Hindu–Arabic numeral system, which can display any non-negative integer using a combination of ten symbols, called numerical digits. Numerals can be used for counting (as with cardinal number of a collection or set), labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, a numeral is not clearly distinguished from the number that it represents.

In mathematics, the notion of number has been extended over the centuries to include zero (0), negative numbers, rational numbers such as one half ${\displaystyle \left({\tfrac {1}{2}}\right)}$, real numbers such as the square root of 2 ${\displaystyle \left({\sqrt {2}}\right)}$, and π, and complex numbers which extend the real numbers with a square root of −1, and its combinations with real numbers by adding or subtracting its multiples. Calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic, a term which may also refer to number theory, the study of the properties of numbers.

Viewing the concept of zero as a number required a fundamental shift in philosophy, identifying nothingness with a value. During the 19th century, mathematicians began to develop the various systems now called algebraic structures, which share certain properties of numbers, and may be seen as extending the concept. Some algebraic structures are explicitly referred to as numbers (such as the p-adic numbers and hypercomplex numbers) while others are not, but this is more a matter of convention than a mathematical distinction.

Bones and other artifacts have been discovered with marks cut into them that many believe are tally marks. Some historians suggest that the Lebombo bone (dated about 43,000 years ago) and the Ishango bone (dated about 22,000 to 30,000 years ago) are the oldest arithmetic artifacts but this interpretation is disputed. These tally marks may have been used for counting elapsed time, such as numbers of days, lunar cycles or keeping records of quantities, such as of animals. A perceptual system for quantity thought to underlie numeracy, is shared with other species, a phylogenetic distribution suggesting it would have existed before the emergence of language.

A tallying system has no concept of place value (as in modern decimal notation), which limits its representation of large numbers. Nonetheless, tallying systems are considered the first kind of abstract numeral system.

The earliest unambiguous numbers in the archaeological record are the Mesopotamian base 60 (sexagesimal) system (c. 3400 BC); place value emerged in the 3rd millennium BCE. The earliest known base 10 system dates to 3100 BC in Egypt. A Babylonian clay tablet dated to 1900–1600 BC provides an estimate of the circumference of a circle to its diameter of ${\textstyle 3{\frac {1}{8}}}$ = 3.125, possibly the oldest approximation of π.

Numbers should be distinguished from numerals, the symbols used to represent numbers. The Egyptians invented the first ciphered numeral system, and the Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. (However, in 300 BC, Archimedes first demonstrated the use of a positional numeral system to display extremely large numbers in The Sand Reckoner.) Roman numerals, a system that used combinations of letters from the Roman alphabet, remained dominant in Europe until the spread of the Hindu–Arabic numeral system around the late 14th century, and the Hindu–Arabic numeral system remains the most common system for representing numbers in the world today. The key to the effectiveness of the system was the symbol for zero, which was developed by ancient Indian mathematicians around 500 AD.

The first known recorded use of zero as an integer dates to AD 628, and appeared in the Brāhmasphuṭasiddhānta, the main work of the Indian mathematician Brahmagupta. He is usually considered the first to formulate the mathematical concept of zero. Brahmagupta treated 0 as a number and discussed operations involving it, including division by zero. He gave rules of using zero with negative and positive numbers, such as "zero plus a positive number is a positive number, and a negative number plus zero is the negative number". By this time (the 7th century), the concept had clearly reached Cambodia in the form of Khmer numerals, and documentation shows the idea later spreading to China and the Islamic world. The concept began reaching Europe through Islamic sources around the year 1000.