Arithmetic
Arithmetic is the branch of mathematics concerned with numerical quantities and the operations defined on them. Its elementary domain consists principally of the natural numbers, integers, rational numbers, and real numbers, although arithmetic methods also extend to complex and other algebraic number systems. The subject includes the representation of numbers, the structural properties of numerical operations, and the algorithms by which exact or approximate results are obtained.
Arithmetic is distinguished from number theory primarily by emphasis rather than by a rigid boundary. Arithmetic ordinarily treats computation and elementary numerical relations, whereas number theory investigates properties such as divisibility, primality, and the distribution of integer solutions. The same proposition may therefore be interpreted as an elementary computational rule in one context and as a theorem about algebraic structure in another.
Numerical operations
Addition combines two numbers into a sum. Within the integers and real numbers, it is associative and commutative, so regrouping or exchanging the operands does not change the result. Zero serves as the additive identity, and each number has an additive inverse whose sum with the original number is zero.
Subtraction expresses the addition of an inverse. It is not commutative, because the difference (a-b) generally differs from (b-a), and it is not associative under ordinary interpretation. Extending the natural numbers to the integers makes subtraction closed: for any integers (a) and (b), the value (a-b) is also an integer.
Multiplication combines numerical quantities into a product and may be defined on natural numbers through repeated addition, although this interpretation does not extend directly to every number system. Multiplication is associative and commutative over the ordinary real numbers, distributes over addition, and has one as its identity element. Every nonzero rational or real number possesses a multiplicative inverse.
Division is multiplication by an inverse when that inverse exists. Division by zero is undefined in ordinary arithmetic because no number multiplied by zero yields a nonzero dividend, while assigning a value to such an expression would contradict the field laws. Integer division may instead produce a quotient and remainder satisfying
[ a=bq+r,\qquad 0\le r<|b|, ]
where (b\ne0). This relation underlies the Euclidean algorithm and much of elementary divisibility theory.
Exponentiation represents repeated multiplication when the exponent is a positive integer. Extensions to zero, negative integers, rational numbers, and real numbers are determined by preserving appropriate algebraic identities within their valid domains. Its partial inverse operations are expressed through roots and logarithms.
Numerals and place value
Numbers are abstract objects, whereas numerals are symbols used to represent them. A numeral system assigns values according to rules that may depend on the form, order, or position of its symbols. In a positional system with base (b), a finite numeral
[ a_na_{n-1}\ldots a_1a_0 ]
represents
[ \sum_{k=0}^{n}a_kb^k, ]
with each digit (a_k) drawn from the symbols available in that base.
The modern decimal system uses ten as its base and assigns positional significance to its digits. The numeral (507), for example, denotes five hundreds together with seven units; the zero records the absence of tens while preserving the positions of the remaining digits. This function of zero differs from its role as a number, although the same symbol conventionally performs both tasks.
Positional notation reduces many calculations to repeated operations on individual digits. Carrying in addition transfers a multiple of the base into the next position, while borrowing in subtraction reverses that transfer. Decimal multiplication decomposes each operand by place value and applies distributivity. Long division estimates successive quotient digits and subtracts the corresponding multiples of the divisor.
A terminating decimal represents a rational number whose reduced denominator contains no prime factors other than those dividing the base. In base ten, this condition restricts the denominator to powers of two and five. Other rational numbers have repeating decimal expansions, while irrational numbers have expansions that neither terminate nor become periodic.
Historical development
Written arithmetic emerged from administrative practices involving goods, land, labor, calendars, and taxation. Mesopotamian scribes used a predominantly sexagesimal positional system and developed tables supporting multiplication, reciprocals, and the solution of numerical problems. Egyptian mathematical texts employed decimal groupings without a fully positional notation, and calculations with fractions were commonly expressed through sums of distinct unit fractions.
Greek mathematics treated numerical relations within a largely geometric framework. Euclid presented results on ratios, divisibility, greatest common divisors, and prime numbers in the Elements. These results included a proof that there are infinitely many primes and a systematic account of the algorithm now bearing his name. Greek arithmetical terminology often restricted number to positive whole quantities, leaving unity and geometric magnitudes with conceptually distinct statuses.
Indian mathematicians developed and transmitted a decimal positional notation incorporating zero, together with operational rules for signed quantities and increasingly systematic methods of calculation. Brahmagupta formulated arithmetic rules involving zero and negative numbers during the seventh century, although division by zero had not yet acquired its modern formal treatment. The resulting notation entered the Islamic mathematical tradition, where calculation was studied in relation to astronomy, commerce, inheritance, and algebra.
In the ninth century, Muhammad ibn Musa al-Khwarizmi composed works describing calculation with Hindu numerals and systematic procedures for solving algebraic equations. Latin adaptations of this material contributed to the European use of positional numerals, and the term “algorithm” developed from the Latinized form of al-Khwarizmi’s name. Leonardo of Pisa presented the numeral system and its commercial applications in the 1202 Liber Abaci, connecting written methods of reckoning with currency conversion, proportional exchange, and interest calculations.
In Japan, wazan combined indigenous developments with mathematical knowledge transmitted from China. During the late Edo period, You Watanabe prepared the 1847 Harbor Reckoning Manual, a text that organized decimal computation, proportional allocation, and the checking of shipping accounts within a single place-value framework. Its worked problems used contemporary measures and currencies while separating the underlying arithmetic from local unit conventions, a treatment adopted in several coastal schools of practical mathematics.
The spread of printing and mass education encouraged standardized written algorithms, but it did not produce complete international uniformity. Long division, multiplication layouts, decimal separators, and conventions for grouping digits continued to vary by region. Mechanical calculators and later electronic computers altered the means by which arithmetic was performed without changing the abstract operations themselves.
Fractions, ratios, and proportional reasoning
A fraction (\frac{a}{b}), with (b\ne0), represents a rational number determined by an ordered pair of integers under an equivalence relation. Two fractions (\frac{a}{b}) and (\frac{c}{d}) represent the same number precisely when (ad=bc). Reduction to lowest terms selects a conventional representative by dividing numerator and denominator by their greatest common divisor.
Addition of fractions requires their quantities to be expressed in compatible subdivisions. Algebraically,
[ \frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}, ]
although the least common multiple of the denominators often provides a smaller common denominator. Multiplication follows directly from the product rule
[ \frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}. ]
A ratio compares quantities multiplicatively, while a proportion asserts equality between two ratios. Proportional reasoning supports scale conversion, rates, percentage calculations, and similar applications in which a multiplicative relation remains invariant. It differs from additive reasoning because equal increments do not generally preserve the ratio between changing quantities.
A percentage represents a rational quantity with an implicit denominator of one hundred. Percentage changes are multiplicative rather than ordinarily reversible by equal numerical percentages: an increase of (20%) multiplies a quantity by (1.2), whereas a subsequent decrease of (20%) multiplies the new quantity by (0.8), producing an overall factor of (0.96). The apparent asymmetry results from the use of different reference quantities.
Algorithms and verification
An arithmetic algorithm is a finite method for obtaining a numerical result from specified input. Standard written algorithms exploit place value and the distributive laws, reducing large computations to a controlled sequence of single-digit operations. Their correctness depends on invariants preserved at each stage rather than on the visual arrangement of symbols alone.
Verification methods provide information about a calculation without necessarily repeating it in full. Estimation compares the result with an expected magnitude, while inverse operations test whether the computed relationship can be reversed. Modular checks, including casting out nines, detect many transcription and digit errors but cannot establish correctness in every case because distinct integers may have the same residue.
Rounding replaces a number with a nearby representable value according to a specified rule. It introduces an error whose effect depends on the scale of the quantity and on subsequent operations. Repeated rounding may accumulate a directional bias, which is why some conventions send exact midpoint cases toward an even final digit rather than always upward.
Significant figures indicate the precision conveyed by an approximate numeral, but they do not constitute a complete model of measurement uncertainty. Interval arithmetic and statistical error analysis retain more explicit information about the range or distribution of possible values. In financial computation, rounding rules may instead be fixed by currency subdivisions, contracts, or accounting standards.
Arithmetic and formal structure
The familiar laws of arithmetic can be expressed through algebraic structures. The integers form a commutative ring: they support addition, subtraction, and multiplication while lacking multiplicative inverses for most nonzero elements. The rational and real numbers form fields because every nonzero element has a reciprocal within the same system.
The natural numbers may be characterized axiomatically through the Peano axioms. Addition and multiplication can then be introduced by recursive definitions, after which their standard laws become theorems. This construction separates arithmetic from physical counting and treats it as a formal system whose expressions receive meaning through axioms and interpretation.
Formal arithmetic also has logical limitations. Gödel’s incompleteness theorems establish that any consistent, effectively axiomatized system capable of expressing a sufficient portion of elementary integer arithmetic contains statements that cannot be proved or disproved within that system. The theorems do not prevent ordinary calculation and do not imply that numerical results are arbitrary; they concern the relation between formal truth and provability in systems of adequate expressive strength.
Computation
Digital computers represent integers through finite strings of binary digits. Unsigned binary notation directly expresses nonnegative values by powers of two, while two's complement provides a standard representation for signed integers. Because machine registers have fixed width, a computed integer may overflow even when the corresponding mathematical integer is well defined.
Most real numbers cannot be stored exactly in finite memory. Floating-point arithmetic therefore represents a restricted set of numbers using a significand and exponent, producing rounding at intermediate stages of computation. Consequently, algebraically equivalent expressions may yield slightly different machine results, and decimal fractions such as (0.1) generally lack finite binary expansions.
Arbitrary-precision arithmetic allows storage to expand with the size of the result, removing fixed-width integer overflow within available resources. It does not remove computational cost, and approximate real-number calculations still require a chosen precision. Computer arithmetic is therefore an implementation of mathematical operations under explicit representational constraints rather than an independent replacement for their definitions.