A number is an abstract mathematical object used to count, measure, order and label. The most basic examples are the counting numbers 1, 2, 3 and so on. Over thousands of years, mathematicians extended this idea to zero, negative numbers, fractions, irrational numbers and complex numbers. Each extension made it possible to solve problems the earlier system could not handle. A number is different from a numeral, which is a symbol or group of symbols that stands for a number. The number seven, for example, can be written as "7", "VII" or "七". Numbers are the subject of arithmetic and number theory, and they underlie nearly every quantitative discipline, from physics to economics.
Origins of counting
Counting is older than writing. People first kept track of quantities with fingers, pebbles and notches cut into wood or bone. The Ishango bone, found in the Ishango region of the present-day Democratic Republic of Congo, is dated to over 20,000 years old. Its notches are arranged in groups, and researchers do not agree on what they mean. Proposed explanations include a lunar calendar, a multiplication table or a simple tally, and there is no firm consensus. Tallies work by one-to-one correspondence, matching one mark to one object. That becomes unwieldy for large quantities, so people began grouping marks and inventing symbols for the groups. In Mesopotamia clay tokens recorded goods, while the Inca used the quipu, a knotted-string system that held complex numerical records.
Numeral systems
Early written numeral systems were often additive. The number system of ancient Egypt was additive, meaning that the position of a symbol did not matter. A scribe repeated the signs for units, tens, hundreds and so on as many times as needed. Roman numerals worked on a similar principle. The Babylonians developed a base-60 positional system, in which a symbol's value depends on where it stands. Traces of it survive today in the way we divide hours and angles into 60 minutes. In China, counting rods used during and before the Han dynasty followed a decimal place-value scheme, and the abacus later became widely used for calculation.
The decisive step was combining place value with a written symbol for zero. This happened in India during the first millennium CE. The resulting Hindu–Arabic numeral system spread through the scholars of the Abbasid Caliphate and reached Europe in the Middle Ages. By the Renaissance it had largely displaced Roman numerals for calculation. Base ten is a convention, probably based on the number of human fingers, and not a mathematical necessity. The binary system, which uses only 0 and 1, is the natural choice for the electronic circuits of a computer.
Kinds of numbers
Modern mathematics arranges the familiar number systems in a chain, with each one containing the one before it:
- Natural numbers (ℕ): 0, 1, 2, 3, … Some authors start at 1. They are used for counting and ordering. Prime numbers, which have no divisors other than 1 and themselves, are the multiplicative building blocks of the natural numbers.
- Integers (ℤ): the natural numbers together with their negatives. They allow subtraction without restriction. Negative numbers appear in Chinese texts such as The Nine Chapters on the Mathematical Art, but European mathematicians resisted them for centuries.
- Rational numbers (ℚ): ratios of integers such as ¾ or −2/5. They allow division by any number other than zero.
- Real numbers (ℝ): all rational numbers plus the irrational numbers, such as √2, π and e, which cannot be written as fractions. Real numbers correspond to the points on a continuous line and are the basic setting of calculus.
- Complex numbers (ℂ): numbers of the form a + bi, where i² = −1. Every polynomial equation has a solution within the complex numbers, a result known as the fundamental theorem of algebra. Complex numbers are indispensable in electrical engineering and quantum mechanics.
There are further systems beyond these. They include the quaternions, in which multiplication is no longer commutative, and Georg Cantor's transfinite cardinal and ordinal numbers, which measure the sizes and orderings of infinite sets.
Historical crises and extensions
Several of these extensions began as intellectual crises. According to tradition, mathematicians in ancient Greece found that the diagonal of a square cannot be expressed as a ratio of whole numbers when compared with its side. That discovery undermined the Pythagorean belief that all things can be measured by whole-number ratios. Greek mathematics responded by putting geometry ahead of arithmetic for a long time. Negative and imaginary quantities first appeared as awkward intermediate results in solving equations. Italian algebraists of the 16th century used them reluctantly, and they were fully accepted only after geometric interpretations, such as the complex plane, made them intuitive.
The development of calculus by Isaac Newton and Gottfried Wilhelm Leibniz depended on an intuitive idea of a continuum of real numbers, but it lacked a rigorous foundation. In the 19th century, Richard Dedekind and Cantor constructed the real numbers from the rationals, and Giuseppe Peano gave axioms for the natural numbers. These achievements moved numbers onto a firm logical basis.
Foundations and philosophy
In modern foundations, numbers are usually defined within set theory. A common construction defines zero as the empty set and each natural number as the set of all smaller natural numbers. From there, integers, rationals, reals and complex numbers are built step by step, and each construction can be checked by mathematical proof. Kurt Gödel's incompleteness theorems showed that no consistent, effectively axiomatized system rich enough to express the arithmetic of natural numbers can prove every true statement about them.
What numbers actually are is a long-standing question in philosophy and metaphysics. Platonists hold that numbers exist independently of minds. Formalists treat them as symbols manipulated according to rules. Intuitionists regard them as mental constructions. Logicists such as Gottlob Frege tried to reduce arithmetic to logic. Aristotle already argued that number is an abstraction from countable things rather than a separately existing entity.
Numbers in science and daily life
Numbers make measurement possible, and measurement in turn makes possible quantitative science, money, trade and administration. Probability and statistics use numbers to describe uncertainty and variation. Some numbers have special significance across many fields: π governs circles and waves, e governs continuous growth, and i governs rotation and oscillation. Euler's identity, e^(iπ) + 1 = 0, links all three together with 0 and 1. Cultures have also given particular numbers symbolic meanings, as in the association of 8 with prosperity in Chinese or the avoidance of 13 in parts of the West. These meanings belong to culture rather than mathematics.
References
- Tally marksen.wikipedia.org
- Tally marks emerge in Africa as humanity's first numeral systempeterschulte.org
- The Evolution of Number Systemflexbooks.ck12.org
- The Origin of Number Systems - Origin Traceorigin-trace.com