# Dictionary Definition

arithmetic adj : relating to or involving
arithmetic; "arithmetical computations" [syn: arithmetical] n : the
branch of pure mathematics dealing with the theory of numerical
calculations

# User Contributed Dictionary

## English

### Etymology

From arsmetike, from arismetique, from arithmetica, from Ancient Greek sc=polytonic (sc=polytonic), from sc=polytonic. Used in English since 13th Century.# Extensive Definition

Arithmetic or arithmetics
(from the Greek word
αριθμός = number) is the oldest and most elementary branch of
mathematics, used by almost everyone, for tasks ranging from simple
day-to-day counting to advanced science and business calculations. In
common usage, the word refers to a branch of (or the forerunner of)
mathematics which
records elementary properties of certain operations on numbers. Professional mathematicians sometimes
use the term (higher) arithmetic when referring to number
theory, but this should not be confused with elementary
arithmetic.

## History

The prehistory of arithmetic
is limited to a very small number of small artifacts indicating a
clear conception of addition and subtraction, the best-known being
the Ishango bone
from
central Africa, dating from somewhere between 18,000 and 20,000
BC.

It is clear that the Babylonians had
solid knowledge of almost all aspects of elementary arithmetic by
1800 BC, although historians can only guess at the methods utilized
to generate the arithmetical results - as shown, for instance, in
the clay tablet Plimpton
322, which appears to be a list of Pythagorean
triples, but with no workings to show how the list was
originally produced. Likewise, the Egyptian
Rhind
Mathematical Papyrus (dating from c. 1650 BC, though evidently
a copy of an older text from c. 1850 BC) shows evidence of
addition, subtraction, multiplication, and division being used
within a unit
fraction system.

Nicomachus (c.
AD60 - c.
AD120)
summarised the philosophical Pythagorean
approach to numbers, and their relationships to each other, in his
Introduction
to Arithmetic. At this time, basic arithmetical operations were
highly complicated affairs; it was the method known as the "Method
of the Indians" (Latin "Modus Indorum") that became the arithmetic
that we know today. Indian arithmetic was much simpler than Greek
arithmetic due to the simplicity of the Indian number system, which
had a zero and
place-value
notation. The 7th century
Syriac
bishop Severus Sebhokt mentioned this method with admiration,
stating however that the Method of the Indians was beyond
description. The Arabs learned this new method and called it hesab.
Fibonacci
(also known as Leonardo of Pisa) introduced the "Method of the
Indians" to Europe in 1202. In his book
"Liber
Abaci", Fibonacci says that, compared with this new method, all
other methods had been mistakes. In the Middle Ages,
arithmetic was one of the seven liberal arts
taught in universities.

Modern algorithms for arithmetic
(both for hand and electronic computation) were made possible by
the introduction of Arabic
numerals and decimal
place notation for numbers. Arabic numeral based arithmetic was
developed by the great Indian mathematicians Aryabhatta,
Brahmagupta and
Bhāskara
I. Aryabhatta tried different place value notations and
Brahmagupta added zero to the Indian number system. Brahmagupta
developed modern multiplication, division, addition and subtraction
based on Arabic numerals. Although it is now considered elementary,
its simplicity is the
culmination of thousands of years of mathematical development. By
contrast, the ancient mathematician Archimedes
devoted an entire work, The Sand
Reckoner, to devising a notation for a certain large integer.
The flourishing of algebra in the medieval Islamic world and
in Renaissance
Europe was
an outgrowth of the enormous simplification of computation through decimal notation.

## Decimal arithmetic

Decimal
notation constructs all real numbers from the basic digits, the
first ten non-negative integers 0,1,2,...,9. A decimal numeral
consists of a sequence of these basic digits, with the
"denomination" of each digit depending on its position with respect
to the decimal point: for example, 507.36 denotes 5 hundreds (10²),
plus 0 tens (101), plus 7 units (100), plus 3 tenths (10-1) plus 6
hundredths (10-2). An essential part of this notation (and a major
stumbling block in achieving it) was conceiving of zero as a
number comparable to the other basic digits.

Algorism comprises
all of the rules of performing arithmetic computations using a
decimal system for representing numbers in which numbers written
using ten symbols having the values 0 through 9 are combined using
a place-value system (positional notation), where each symbol has
ten times the weight of the one to its right. This notation allows
the addition of arbitrary numbers by adding the digits in each
place, which is accomplished with a 10 x 10 addition table. (A sum
of digits which exceeds 9 must have its 10-digit carried to the
next place leftward.) One can make a similar algorithm for
multiplying arbitrary numbers because the set of denominations is
closed under multiplication. Subtraction and division are achieved
by similar, though more complicated algorithms.

## Arithmetic operations

The traditional arithmetic operations are addition, subtraction, multiplication and division, although more advanced operations (such as manipulations of percentages, square root, exponentiation, and logarithmic functions) are also sometimes included in this subject. Arithmetic is performed according to an order of operations. Any set of objects upon which all four operations of arithmetic can be performed (except division by zero), and wherein these four operations obey the usual laws, is called a field.### Addition (+)

Addition is the basic operation of arithmetic. In its simplest form, addition combines two numbers, the addends or terms, into a single number, the sum.Adding more than two numbers
can be viewed as repeated addition; this procedure is known as
summation and includes
ways to add infinitely many numbers in an infinite
series; repeated addition of the number one is the
most basic form of counting.

Addition is commutative and associative so the order in
which the terms are added does not matter. The identity
element of addition (the additive
identity) is 0, that is, adding zero to any number will yield
that same number. Also, the inverse
element of addition (the additive
inverse) is the opposite of any number, that is, adding the
opposite of any number to the number itself will yield the additive
identity, 0. For example, the opposite of 7 is (-7), so 7 + (-7) =
0. Addition can be given geometrically as follows.

If a and b are the lengths of
two sticks, then if we place the sticks one after the other, the
length of the stick thus formed will be a+b

### Subtraction (−)

Subtraction is essentially the opposite of addition. Subtraction finds the difference between two numbers, the minuend minus the subtrahend. If the minuend is larger than the subtrahend, the difference will be positive; if the minuend is smaller than the subtrahend, the difference will be negative; and if they are equal, the difference will be zero.Subtraction is neither
commutative nor associative. For that reason, it is often helpful
to look at subtraction as addition of the minuend and the opposite
of the subtrahend, that is
a − b = a + (−b).
When written as a sum, all the properties of addition
hold.

### Multiplication (×, ·, or *)

Multiplication is in essence repeated addition, or the sum of a list of identical numbers. Multiplication finds the product of two numbers, the multiplier and the multiplicand, sometimes both simply called factors.Multiplication, as it is
really repeated addition, is commutative and associative; further
it is distributive
over addition and subtraction. The multiplicative
identity is 1, that is, multiplying any number by 1 will yield
that same number. Also, the multiplicative
inverse is the reciprocal
of any number, that is, multiplying the reciprocal of any number by
the number itself will yield the multiplicative
identity.

### Division (÷ or /)

Division is essentially the opposite of multiplication. Division finds the quotient of two numbers, the dividend divided by the divisor. Any dividend divided by zero is undefined. For positive numbers, if the dividend is larger than the divisor, the quotient will be greater than one, otherwise it will be less than one (a similar rule applies for negative numbers). The quotient multiplied by the divisor always yields the dividend.Division is neither
commutative nor associative. As it is helpful to look at
subtraction as addition, it is helpful to look at division as
multiplication of the dividend times the reciprocal
of the divisor, that is
a ÷ b = a × 1⁄b.
When written as a product, it will obey all the properties of
multiplication.

### Examples

#### Multiplication table

## Number theory

The term arithmetic is also used to refer to number theory. This includes the properties of integers related to primality, divisibility, and the solution of equations by integers, as well as modern research which is an outgrowth of this study. It is in this context that one runs across the fundamental theorem of arithmetic and arithmetic functions. A Course in Arithmetic by Serre reflects this usage, as do such phrases as first order arithmetic or arithmetical algebraic geometry. Number theory is also referred to as 'the higher arithmetic', as in the title of H. Davenport's book on the subject.## Arithmetic in education

Primary education in mathematics often places a strong focus on algorithms for the arithmetic of natural numbers, integers, rational numbers (vulgar fractions), and real numbers (using the decimal place-value system). This study is sometimes known as algorism.The difficulty and unmotivated
appearance of these algorithms has long led educators to question
this curriculum, advocating the early teaching of more central and
intuitive mathematical ideas. One notable movement in this
direction was the New Math of the
1960s and '70s, which attempted to teach arithmetic in the spirit
of axiomatic development from set theory, an echo of the prevailing
trend in higher mathematics.

Since the introduction of the
electronic calculator, which can perform
the algorithms far more efficiently than humans, an influential
school of educators has argued that mechanical mastery of the
standard arithmetic algorithms is no longer necessary. In their
view, the first years of school mathematics could be more
profitably spent on understanding higher-level ideas about what
numbers are used for and relationships among number, quantity,
measurement, and so on. However, most research mathematicians still
consider mastery of the manual algorithms to be a necessary
foundation for the study of algebra and computer science. This
controversy was central to the "Math Wars" over
California's primary school curriculum in the 1990s, and continues
today.

Many mathematics texts for
K-12 instruction were developed, funded by grants from the United
States
National Science Foundation based on standards created by the
NCTM and given
high ratings by United States Department of Education, though
condemned by many mathematicians. Some widely adopted texts such as
TERC were
based on the spirit of research papers which found that instruction
of basic arithmetic was harmful to mathematical understanding.
Rather than teaching any traditional method of arithmetic, teachers
are instructed to instead guide students to invent their own (some
critics claim inefficient) methods, instead using such techniques
as skip
counting, and the heavy use of manipulatives, scissors and
paste, and even singing rather than multiplication tables or long
division. Although such texts were designed to be a complete
curricula, in the face of intense protest and criticism, many
districts have chosen to circumvent the intent of such radical
approaches by supplementing with traditional texts. Other districts
have since adopted traditional
mathematics texts and discarded such reform-based approaches as
misguided failures.

### Related topics

## Footnotes

## References

- Cunnington, Susan. The story of arithmetic, a short history of its origin and development. Swan Sonnenschein, London, 1904.
- Dickson, Leonard Eugene. History of the theory of numbers. Three volumes. Reprints: Carnegie Institute of Washington, Washington, 1932. Chelsea, New York, 1952, 1966.
- Leonhard Euler, Elements of Algebra Tarquin Press, 2007
- Fine, Henry Burchard (1858-1928). The number system of algebra treated theoretically and historically. Leach, Shewell & Sanborn, Boston, 1891.
- Karpinski, Louis Charles (1878-1956). The history of arithmetic. Rand McNally, Chicago, 1925. Reprint: Russell & Russell, New York, 1965.
- Ore, Øystein. Number theory and its history. McGraw-Hill, New York, 1948.
- Weil, Andre. Number theory: an approach through history. Birkhauser, Boston, 1984. Reviewed: Math. Rev. 85c:01004.

## External links

- What is arithmetic?
- MathWorld article about arithmetic
- Interactive Arithmetic Lessons and Practice
- Talking Math Game for kids
- The New Student's Reference Work/Arithmetic (historical)
- Arithmetic Game
- Math Games for kids and adults
- Maximus Planudes' the Great Calculation an early western work on arithmetic at Convergence

arithmetic in Arabic:
حساب

arithmetic in Bengali:
পাটীগণিত

arithmetic in Belarusian:
Арыфметыка

arithmetic in Belarusian
(Tarashkevitsa): Арытмэтыка

arithmetic in Breton:
Aritmetik

arithmetic in Bulgarian:
Аритметика

arithmetic in Catalan:
Aritmètica

arithmetic in Czech:
Aritmetika

arithmetic in Danish:
Aritmetik

arithmetic in German:
Arithmetik

arithmetic in Estonian:
Aritmeetika

arithmetic in Spanish:
Aritmética

arithmetic in Esperanto:
Aritmetiko

arithmetic in Basque:
Aritmetika

arithmetic in Persian:
حساب

arithmetic in French:
Arithmétique

arithmetic in Scottish
Gaelic: Àireamhachd

arithmetic in Galician:
Aritmética

arithmetic in Korean:
산술

arithmetic in Hindi:
अंकगणित

arithmetic in Croatian:
Aritmetika

arithmetic in Ido:
Aritmetiko

arithmetic in Indonesian:
Aritmatika

arithmetic in Interlingua
(International Auxiliary Language Association):
Arithmetica

arithmetic in Icelandic:
Talnareikningur

arithmetic in Italian:
Aritmetica

arithmetic in Hebrew:
אריתמטיקה

arithmetic in Javanese: Ilmu
hitung

arithmetic in Georgian:
არითმეტიკა

arithmetic in Swahili
(macrolanguage): Hesabu

arithmetic in Latin:
Arithmetica

arithmetic in Lithuanian:
Aritmetika

arithmetic in Lojban:
sapme'ocmaci

arithmetic in Macedonian:
Аритметика

arithmetic in Marathi:
अंकगणित

arithmetic in Malay
(macrolanguage): Aritmetik

arithmetic in Dutch:
Rekenen

arithmetic in Japanese:
算術

arithmetic in Norwegian:
Aritmetikk

arithmetic in Norwegian
Nynorsk: Aritmetikk

arithmetic in Novial:
Aritmetike

arithmetic in Polish:
Arytmetyka

arithmetic in Portuguese:
Aritmética

arithmetic in Romanian:
Aritmetică

arithmetic in Quechua: Yupa
hap'ichiy

arithmetic in Russian:
Арифметика

arithmetic in Sardinian:
Aritmètica

arithmetic in Simple English:
Arithmetic

arithmetic in Slovak:
Aritmetika

arithmetic in Slovenian:
Aritmetika

arithmetic in Serbian:
Аритметика

arithmetic in Finnish:
Aritmetiikka

arithmetic in Swedish:
Aritmetik

arithmetic in Tagalog:
Aritmetika

arithmetic in Tamil:
எண்கணிதம்

arithmetic in Thai:
เลขคณิต

arithmetic in Turkish:
Aritmetik

arithmetic in Ukrainian:
Арифметика

arithmetic in Urdu:
حساب

arithmetic in Võro:
Arvokunst

arithmetic in Yiddish:
חשבון

arithmetic in Chinese:
算术

# Synonyms, Antonyms and Related Words

Boolean algebra, Euclidean
geometry, Fourier analysis, Lagrangian function, algebra, algebraic geometry,
analysis, analytic
geometry, associative algebra, binary arithmetic, calculation, calculus, ciphering, circle geometry,
descriptive geometry, differential calculus, division algebra,
equivalent algebras, estimation, figuring, game theory, geodesy, geometry, graphic algebra,
group theory, higher algebra, higher arithmetic, hyperbolic
geometry, infinitesimal calculus, integral calculus, intuitional
geometry, invariant subalgebra, inverse geometry, line geometry,
linear algebra, mathematical physics, matrix algebra, metageometry, modular
arithmetic, n-tuple linear algebra, natural geometry, nilpotent
algebra, number theory, plane trigonometry, political arithmetic,
projective geometry, proper subalgebra, quaternian algebra,
reckoning, reducible
algebra, set theory, simple algebra, solid geometry, speculative
geometry, spherical trigonometry, statistics, subalgebra, systems analysis,
topology, trig, trigonometry, universal
algebra, universal geometry, vector algebra, zero
algebra