Arithmetic is the oldest and most basic branch of mathematics. It studies numbers and the operations used to combine them, mainly addition, subtraction, multiplication and division. Exponentiation and root extraction are often included as well. In everyday use, the word means practical calculation with whole numbers, fractions and decimals. In a more advanced sense, "higher arithmetic" is an older name for number theory, which studies the properties of the integers. Arithmetic is the base on which algebra, geometry and most quantitative science are built. It has been needed for trade, taxation, surveying and money since the first states.
Basic operations and properties
The four elementary operations come in two pairs of inverses. Subtraction undoes addition, and division undoes multiplication. Multiplication of natural numbers can be understood as repeated addition, and exponentiation as repeated multiplication. Addition and multiplication follow several general laws:
- Commutativity: a + b = b + a, and a × b = b × a.
- Associativity: (a + b) + c = a + (b + c), and the same holds for multiplication.
- Distributivity: a × (b + c) = a × b + a × c.
- Identity elements: zero is the identity for addition, and one is the identity for multiplication.
Subtraction and division are neither commutative nor associative. Division by zero is undefined. Rules for the order of operations say that powers come before multiplication and division, which come before addition and subtraction. These rules let written expressions be read without ambiguity.
Arithmetic covers several number systems. Each one was introduced so that an operation always has an answer. The natural numbers are closed under addition and multiplication. The integers add negative numbers so that subtraction always works. The rational numbers allow division by any non-zero number. Real numbers and complex numbers carry these operations further, into areas usually treated under analysis and algebra.
Numeral systems and computation
How easy calculation is depends a great deal on how numbers are written. Additive systems such as Egyptian hieroglyphic numerals and Roman numerals made large multiplications awkward. In positional (place-value) systems, the value of a digit depends on where it stands. This makes standard written algorithms possible for long addition, long multiplication and long division. The Babylonians of Mesopotamia used a positional system in base 60, which still survives in how we measure time and angles. The decimal Hindu–Arabic numeral system has ten digits including zero, and it became the global standard. Modern computers do arithmetic in binary, with circuits that build every operation from simple logic gates.
Calculation tools developed alongside these notations. Counting boards, counting rods and the abacus let people calculate quickly long before written algorithms became common. Mechanical calculators followed in the seventeenth century, including machines designed by Blaise Pascal and Gottfried Wilhelm Leibniz. Then came Charles Babbage's plans for the Difference and Analytical Engines, and finally electronic computers.
History
Ancient civilizations
Tally marks and similar artifacts suggest that people counted long before writing existed. The earliest detailed mathematical texts come from ancient Egypt and Mesopotamia. The Rhind Mathematical Papyrus, dated to about 1650 BCE, is the largest surviving Egyptian mathematical papyrus. It was named after A. Henry Rhind, who bought it in 1858. The Egyptians wrote fractions as sums of distinct unit fractions, and they multiplied by repeated doubling and adding.
In China, the Nine Chapters on the Mathematical Art was compiled by the time of the Han dynasty. It describes procedures with fractions, proportions and calculations done with counting rods, including work with negative quantities. In ancient Greece, thinkers separated arithmetike, the theoretical study of numbers, from logistike, practical calculation. Euclid's Elements contains early results in number theory, including a proof that there are infinitely many prime numbers.
India, the Islamic world and Europe
Indian mathematicians developed a complete decimal place-value system. By the 5th century, Aryabhata described a positional system that needed a placeholder to tell values apart. In his Brahmasphutasiddhanta (628), Brahmagupta treated zero as a number in its own right. He also set out rules for calculating with zero and with negative numbers.
Scholars of the Abbasid Caliphate took up these methods. Al-Khwarizmi wrote a treatise on Indian reckoning in the 9th century, and it later circulated in Latin as Algoritmi de numero Indorum. The word "algorithm" comes from his name. In 1202 Leonardo of Pisa (Fibonacci) published Liber Abaci, which introduced the Hindu–Arabic numerals and their methods to Europe. They still spread slowly. Roman numerals and the abacus stayed common until expanding commerce and printed arithmetic textbooks made the new numerals practical during the Renaissance and after.
Foundations
For most of history, facts such as x + y = y + x seemed obvious and needed no formal justification. In 1861 Hermann Grassmann showed that such laws could be derived from more basic facts about the successor operation ("add one") and mathematical induction. Richard Dedekind gave a rigorous account in 1888. In 1889 Giuseppe Peano published the Peano axioms in Arithmetices principia, nova methodo exposita. Their first-order form, called Peano arithmetic, became the standard axiomatic system for the natural numbers and a central object of study in mathematical logic.
David Hilbert hoped to prove that arithmetic is consistent using only finite methods. In 1931 Kurt Gödel showed that this was impossible in a strong sense. His incompleteness theorems prove that any consistent formal system containing basic arithmetic has true statements it cannot prove. They also show that such a system cannot prove its own consistency. These results changed the philosophy of mathematics and later influenced computer science and the theory of computability.
Teaching and applications
Arithmetic is the first formal mathematics most children learn. Together with reading and writing it is a core part of basic education and literacy. The curriculum usually moves from counting and number facts to written algorithms, fractions, decimals, percentages and ratios. Numeracy, the ability to use arithmetic in everyday situations, underlies personal finance, measurement and the reading of quantitative information.
Outside the classroom, arithmetic is everywhere: in accounting and banking, in engineering calculations, in statistics and probability, and in cryptography. Modular arithmetic ("clock arithmetic") works with remainders after division. It is the basis of public-key encryption systems and error-detecting codes. Computers represent real numbers with floating-point arithmetic, which uses finite precision. This brings rounding errors, and the field of numerical analysis studies how to control them.
References
- Rhind Papyruscut-the-knot.org
- Note (a) for Implications for Mathematics and Its Foundations: A New Kind of Sciencewolframscience.com
- On Non-Standard Models of Peano Arithmetic and Tennenbaum's Theoremarxiv.org
- Giuseppe Peano and the Axiomatization of Mathematicsmathshistory.st-andrews.ac.uk
- Zerotranslate.google.com
- Brahmaguptaen.wikipedia.org
- K A R P R E S E A R C H R E P O R T Origins of Abstract Numberingsouldriver.com.au