Arithmetic

The simplest non-positional system is the unary numeral system. It relies on one symbol for the number 1. All higher numbers are written by repeating this symbol. For example, the number 7 can be represented by repeating the symbol for 1 seven times. This system makes it cumbersome to write large numbers, which is why many non-positional systems include additional symbols to directly represent larger numbers.^[30] Variations of the unary numeral systems are employed in tally sticks using dents and in tally marks.^[31]

A numeral system is positional if the position of a basic numeral in a compound expression determines its value. Positional numeral systems have a radix that acts as a multiplicand of the different positions. For each subsequent position, the radix is raised to a higher power. In the common decimal system, also called the Hindu–Arabic numeral system, the radix is 10. This means that the first digit is multiplied by ${\displaystyle 10^{0}}$, the next digit is multiplied by ${\displaystyle 10^{1}}$, and so on. For example, the decimal numeral 532 stands for ${\displaystyle 5\cdot 10^{2}+3\cdot 10^{1}+2\cdot 10^{0}}$. Because of the effect of the digits' positions, the numeral 532 differs from the numerals 325 and 253 even though they have the same digits.^[34]

Another positional numeral system used extensively in computer arithmetic is the binary system, which has a radix of 2. This means that the first digit is multiplied by ${\displaystyle 2^{0}}$, the next digit by ${\displaystyle 2^{1}}$, and so on. For example, the number 13 is written as 1101 in the binary notation, which stands for ${\displaystyle 1\cdot 2^{3}+1\cdot 2^{2}+0\cdot 2^{1}+1\cdot 2^{0}}$. In computing, each digit in the binary notation corresponds to one

Operations

Arithmetic operations are ways of combining, transforming, or manipulating numbers. They are

Two important concepts in relation to arithmetic operations are identity elements and inverse elements. The identity element or neutral element of an operation does not cause any change if it is applied to another element. For example, the identity element of addition is 0 since any sum of a number and 0 results in the same number. The inverse element is the element that results in the identity element when combined with another element. For instance, the additive inverse of the number 6 is -6 since their sum is 0.^[41]

There are not only inverse elements but also inverse operations. In an informal sense, one operation is the inverse of another operation if it undoes the first operation. For example, subtraction is the inverse of addition since a number returns to its original value if a second number is first added and subsequently subtracted, as in ${\displaystyle 13+4-4=13}$. Defined more formally, the operation "${\displaystyle \star }$" is an inverse of the operation "${\displaystyle \circ }$" if it fulfills the following condition: ${\displaystyle t\star s=r}$ if and only if ${\displaystyle r\circ s=t}$.^[42]

are laws governing the order in which some arithmetic operations can be carried out. An operation is commutative if the order of the arguments can be changed without affecting the results. This is the case for addition, for instance, ${\displaystyle 7+9}$ is the same as ${\displaystyle 9+7}$. Associativity is a rule that affects the order in which a series of operations can be carried out. An operation is associative if, in a series of two operations, it does not matter which operation is carried out first. This is the case for multiplication, for example, since ${\displaystyle (5\times 4)\times 2}$ is the same as ${\displaystyle 5\times (4\times 2)}$.^[43]

Addition is an arithmetic operation in which two numbers, called the addends, are combined into a single number, called the sum. The symbol of addition is ${\displaystyle +}$. Examples are ${\displaystyle 2+2=4}$ and ${\displaystyle 6.3+1.26=7.56}$.^[44] The term summation is used if several additions are performed in a row.^[45] Counting is a type of repeated addition in which the number 1 is continuously added.^[46]

Subtraction is the inverse of addition. In it, one number, known as the subtrahend, is taken away from another, known as the minuend. The result of this operation is called the difference. The symbol of subtraction is ${\displaystyle -}$.^[47] Examples are ${\displaystyle 14-8=6}$ and ${\displaystyle 45-1.7=43.3}$. Subtraction is often treated as a special case of addition: instead of subtracting a positive number, it is also possible to add a negative number. For instance ${\displaystyle 14-8=14+(-8)}$. This helps to simplify mathematical computations by reducing the number of basic arithmetic operations needed to perform calculations.^[48]

The additive identity element is 0 and the additive inverse of a number is the negative of that number. For instance, ${\displaystyle 13+0=13}$ and ${\displaystyle 13+(-13)=0}$. Addition is both commutative and associative.^[49]

Multiplication is an arithmetic operation in which two numbers, called the multiplier and the multiplicand, are combined into a single number called the product.^[50][d] The symbols of multiplication are ${\displaystyle \times }$, ${\displaystyle \cdot }$, and *. Examples are ${\displaystyle 2\times 3=6}$ and ${\displaystyle 0.3\cdot 5=1.5}$. If the multiplicand is a natural number then multiplication is the same as repeated addition, as in ${\displaystyle 2\times 3=2+2+2}$.^[52]

Division is the inverse of multiplication. In it, one number, known as the dividend, is split into several equal parts by another number, known as the divisor. The result of this operation is called the quotient. The symbols of division are ${\displaystyle \div }$ and ${\displaystyle /}$. Examples are ${\displaystyle 48\div 8=6}$ and ${\displaystyle 29.4/1.4=21}$.^[53] Division is often treated as a special case of multiplication: instead of dividing by a number, it is also possible to multiply by its reciprocal. The reciprocal of a number is 1 divided by that number. For instance, ${\displaystyle 48\div 8=48\times {\tfrac {1}{8}}}$.^[54]

The

element is 1 and the multiplicative inverse of a number is the reciprocal of that number. For example, ${\displaystyle 13\times 1=13}$ and ${\displaystyle 13\times {\tfrac {1}{13}}=1}$. Multiplication is both commutative and associative.^[55]

Exponentiation is an arithmetic operation in which a number, known as the base, is raised to the power of another number, known as the exponent. The result of this operation is called the power. Exponentiation is sometimes expressed using the symbol ^ but the more common way is to write the exponent in

right after the base. Examples are ${\displaystyle 2^{4}=16}$ and ${\displaystyle 3}$^${\displaystyle 3=27}$. If the exponent is a natural number then exponentiation is the same as repeated multiplication, as in ${\displaystyle 2^{4}=2\times 2\times 2\times 2}$.^[56][e]

Roots are a special type of exponentiation using a fractional exponent. For example, the square root of a number is the same as raising the number to the power of ${\displaystyle {\tfrac {1}{2}}}$ and the cube root of a number is the same as raising the number to the power of ${\displaystyle {\tfrac {1}{3}}}$. Examples are ${\displaystyle {\sqrt {4}}=4^{\frac {1}{2}}=2}$ and ${\displaystyle {\sqrt[{3}]{27}}=27^{\frac {1}{3}}=3}$.^[58]

Logarithm is the inverse of exponentiation. The logarithm of a number ${\displaystyle x}$ to the base ${\displaystyle b}$ is the

to which ${\displaystyle b}$ must be raised to produce ${\displaystyle x}$. For instance, since ${\displaystyle 1000=10^{3}}$, the logarithm base 10 of 1000 is 3. The logarithm of ${\displaystyle x}$ to base ${\displaystyle b}$ is denoted as ${\displaystyle \log _{b}(x)}$, or without parentheses, ${\displaystyle \log _{b}x}$, or even without the explicit base, ${\displaystyle \log x}$, when the base can be understood from context. So, the previous example can be written ${\displaystyle \log _{10}1000=3}$.^[59]

Exponentiation and logarithm do not have general identity elements and inverse elements like addition and multiplication. The neutral element of exponentiation in relation to the exponent is 1, as in ${\displaystyle 14^{1}=14}$. However, exponentiation does not have a general identity element since 1 is not the neutral element for the base.^[60] Exponentiation and logarithm are neither commutative nor associative.^[61]

Different types of arithmetic systems are discussed in the academic literature. They differ from each other based on what type of number they operate on, what numeral system they use to represent them, and whether they operate on mathematical objects other than numbers.^[62]

Integer arithmetic is the branch of arithmetic that deals with the manipulation of positive and negative whole numbers.

Addition table

+	0	1	2	3	4	...
0	0	1	2	3	4	...
1	1	2	3	4	5	...
2	2	3	4	5	6	...
3	3	4	5	6	7	...
4	4	5	6	7	8	...
...	...	...	...	...	...	...

Multiplication table

×	0	1	2	3	4	...
0	0	0	0	0	0	...
1	0	1	2	3	4	...
2	0	2	4	6	8	...
3	0	3	6	9	12	...
4	0	4	8	12	16	...
...	...	...	...	...	...	...

For operations on numbers with more than one digit, different techniques can be employed to calculate the result by using several one-digit operations in a row. For example, in the method addition with carries, the two numbers are written one above the other. Starting from the rightmost digit, each pair of digits is added together. The rightmost digit of the sum is written below them. If the sum is a two-digit number then the leftmost digit, called the "carry", is added to the next pair of digits to the left. This process is repeated until all digits have been added.^[65] Other methods used for integer additions are the number line method, the partial sum method, and the compensation method.^[66] A similar technique is utilized for subtraction: it also starts with the rightmost digit and uses a "borrow" or a negative carry for the column on the left if the result of the one-digit subtraction is negative.^[67]

A basic technique of integer multiplication employs repeated addition. For example, the product of ${\displaystyle 3\times 4}$ can be calculated as ${\displaystyle 3+3+3+3}$.

Integer arithmetic is not closed under division. This means that when dividing one integer by another integer, the result is not always an integer. For instance, 7 divided by 2 is not a whole number but 3.5.[73] One way to ensure that the result is an integer is to round the result to a whole number. However, this method leads to inaccuracies as the original value is altered.^[74] Another method is to perform the division only partially and retain the remainder. For example, 7 divided by 2 is 3 with a remainder of 1. These difficulties are avoided by rational number arithmetic, which allows for the exact representation of fractions.^[75]

A simple method to calculate exponentiation is by repeated multiplication. For instance, the exponentiation of ${\displaystyle 3^{4}}$ can be calculated as ${\displaystyle 3\times 3\times 3\times 3}$.^[76] A more efficient technique used for large exponents is exponentiation by squaring. It breaks down the calculation into a number of squaring operations. For example, the exponentiation ${\displaystyle 3^{65}}$ can be written as ${\displaystyle (((((3^{2})^{2})^{2})^{2})^{2})^{2}\times 3}$. By taking advantage of repeated squaring operations, only 7 individual operations are needed rather than the 64 operations required for regular repeated multiplication.^[77] Methods to calculate logarithms include the Taylor series and continued fractions.^[78] Integer arithmetic is not closed under logarithm and under exponentiation with negative exponents, meaning that the result of these operations is not always an integer.^[79]

Number theory studies the structure and properties of integers as well as the relations and laws between them.

Influential theorems in number theory include the

number 18

is not a prime number and can be represented as ${\displaystyle 2\times 3\times 3}$, all of which are prime numbers. The Fermat's Last Theorem is the statement that no positive integer values exist for ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$ that solve the equation ${\displaystyle a^{n}+b^{n}=c^{n}}$ if ${\displaystyle n}$ is greater than ${\displaystyle 2}$.[92]

Rational number arithmetic is the branch of arithmetic that deals with the manipulation of numbers that can be expressed as a ratio of two integers.^[93] Most arithmetic operations on rational numbers can be calculated by performing a series of integer arithmetic operations on the numerators and the denominators of the involved numbers. If two rational numbers have the same denominator then they can be added by adding their numerators and keeping the common denominator. For example, ${\displaystyle {\tfrac {2}{7}}+{\tfrac {3}{7}}={\tfrac {5}{7}}}$. A similar procedure is used for subtraction. If the two numbers do not have the same denominator then they must be transformed to find a common denominator. This can be achieved by scaling the first number with the denominator of the second number while scaling the second number with the denominator of the first number. For instance, ${\displaystyle {\tfrac {1}{3}}+{\tfrac {1}{2}}={\tfrac {1\cdot 2}{3\cdot 2}}+{\tfrac {1\cdot 3}{2\cdot 3}}={\tfrac {2}{6}}+{\tfrac {3}{6}}={\tfrac {5}{6}}}$.^[94]

Two rational numbers are multiplied by multiplying their numerators and their denominators respectively, as in ${\displaystyle {\tfrac {2}{3}}\cdot {\tfrac {2}{5}}={\tfrac {2\cdot 2}{3\cdot 5}}={\tfrac {4}{15}}}$. Dividing one rational number by another can be achieved by multiplying the first number with the reciprocal of the second number. This means that the numerator and the denominator of the second number change position. For example, ${\displaystyle {\tfrac {3}{5}}:{\tfrac {2}{7}}={\tfrac {3}{5}}\cdot {\tfrac {7}{2}}={\tfrac {21}{10}}}$.^[95] Unlike integer arithmetic, rational number arithmetic is closed under division as long as the divisor is not 0.^[96]

Both integer arithmetic and rational number arithmetic are not closed under exponentiation and logarithm.^[97] One way to calculate exponentiation with a fractional exponent is to perform two separate calculations: one exponentiation using the numerator of the exponent followed by drawing the nth root of the result based on the denominator of the exponent. For example, ${\displaystyle 5^{\frac {2}{3}}={\sqrt[{3}]{5^{2}}}}$. The first operation can be completed using methods like repeated multiplication or exponentiation by squaring. One way to get an approximate result for the second operation is to employ Newton's method, which uses a series of steps to gradually refine an initial guess until it reaches the desired level of accuracy.^[98] The Taylor series or the continued fraction method can be utilized to calculate logarithms.^[99]

The

notation is a special way of representing rational numbers whose denominator is a power of 10. For instance, the rational numbers ${\displaystyle {\tfrac {1}{10}}}$, ${\displaystyle {\tfrac {371}{100}}}$, and ${\displaystyle {\tfrac {44}{10000}}}$ are written as 0.1, 3.71, and 0.0044 in the decimal fraction notation.^[100] Modified versions of integer calculation methods like addition with carry and long multiplication can be applied to calculations with decimal fractions.^[101] Not all rational numbers have a finite representation in the decimal notation. For example, the rational number ${\displaystyle {\tfrac {1}{3}}}$ corresponds to 0.333... with an infinite number of 3s. The shortened notation for this type of repeating decimal is 0.3.^[102] Every repeating decimal expresses a rational number.^[103]

Real number arithmetic is the branch of arithmetic that deals with the manipulation of both rational and irrational numbers. Irrational numbers are numbers that cannot be expressed through fractions or repeated decimals, like the root of 2 and π.^[104] Unlike rational number arithmetic, real number arithmetic is closed under exponentiation as long as it uses a positive number as its base. The same is true for the logarithm of positive real numbers as long as the logarithm base is positive and not 1.^[105]

Irrational numbers involve an infinite non-repeating series of decimal digits. Because of this, there is often no simple and accurate way to express the results of arithmetic operations like ${\displaystyle {\sqrt {2}}+\pi }$ or ${\displaystyle e\cdot {\sqrt {3}}}$.^[106] In cases where absolute precision is not required, the problem of calculating arithmetic operations on real numbers is usually addressed by truncation or rounding. For truncation, a certain number of leftmost digits are kept and remaining digits are discarded or replaced by zeros. For example, the number π has an infinite number of digits starting with 3.14159.... If this number is truncated to 4 decimal places, the result is 3.141. Rounding is a similar process in which the last preserved digit is increased by one if the next digit is 5 or greater but remains the same if the next digit is less than 5, so that the rounded number is the best approximation of a given precision for the original number. For instance, if the number π is rounded to 4 decimal places, the result is 3.142 because the following digit is a 5, so 3.142 is closer to π than 3.141.^[107] These methods allow computers to efficiently perform approximate calculations on real numbers.^[108]

In science and engineering, numbers represent estimates of physical quantities derived from measurement or modeling. Unlike mathematically exact numbers such as π or ⁠${\displaystyle {\sqrt {2}}}$⁠, scientifically relevant numerical data are inherently inexact, involving some

In performing calculations with uncertain quantities, the

.)

More sophisticated methods of dealing with uncertain values include

measurement errors

. Interval arithmetic includes operations like addition and multiplication on intervals, as in ${\displaystyle [1,2]+[3,4]=[4,6]}$ and ${\displaystyle [1,2]\times [3,4]=[3,8]}$.[115] It is closely related to affine arithmetic, which aims to give more precise results by performing calculations on affine forms rather than intervals. An affine form is a number together with error terms that describe how the number may deviate from the actual magnitude.^[116]

The precision of numerical quantities can be expressed uniformly using normalized scientific notation, which is also convenient for concisely representing numbers which are much larger or smaller than 1. Using scientific notation, a number is decomposed into the product of a number between 1 and 10, called the significand, and 10 raised to some integer power, called the exponent. The significand consists of the significant digits of the number, and is written as a leading digit 1–9 followed by a decimal point and a sequence of digits 0–9. For example, the normalized scientific notation of the number 8276000 is ${\displaystyle 8.276\times 10^{6}}$ with significand 8.276 and exponent 6, and the normalized scientific notation of the number 0.00735 is ${\displaystyle 7.35\times 10^{-3}}$ with significand 7.35 and exponent −3.^[117] Unlike ordinary decimal notation, where trailing zeros of large numbers are implicitly considered to be non-significant, in scientific notation every digit in the significand is considered significant, and adding trailing zeros indicates higher precision. For example, while the number 1200 implicitly has only 2 significant digits, the number ⁠${\displaystyle 1.20\times 10^{3}}$⁠ explicitly has 3.^[118]

A common method employed by computers to approximate real number arithmetic is called

A consequence of this behavior is that certain laws of arithmetic are violated by floating-point arithmetic. For example, floating-point addition is not associative since the rounding errors introduced can depend on the order of the additions. This means that the result of ${\displaystyle (a+b)+c}$ is sometimes different from the result of ${\displaystyle a+(b+c)}$.[121] The most common technical standard used for floating-point arithmetic is called IEEE 754. Among other things, it determines how numbers are represented, how arithmetic operations and rounding are performed, and how errors and exceptions are handled.^[122] In cases where computation speed is not a limiting factor, it is possible to use arbitrary-precision arithmetic, for which the precision of calculations is only restricted by the computer's memory.^[123]

mental arithmetic

are done exclusively in the mind without relying on external aids.

Forms of arithmetic can also be distinguished by the

One such technique is the compensation method, which consists in altering the numbers to make the calculation easier and then adjusting the result afterward. For example, instead of calculating ${\displaystyle 85-47}$, one calculates ${\displaystyle 85-50}$ which is easier because it uses a round number. In the next step, one adds ${\displaystyle 3}$ to the result to compensate for the earlier adjustment.[125] Mental arithmetic is often taught in primary education to train the numerical abilities of the students.^[126]

The human body can also be employed as an arithmetic tool. The use of hands in

Tally marks are a simple system based on external tools other than the body. This system relies on mark making, such as strokes drawn on a surface or notches carved into a wooden stick, to keep track of quantities. Some forms of tally marks arrange the strokes in groups of five to make them easier to read.^[129]

The abacus is a more advanced tool to represent numbers and perform calculations. An abacus usually consists of a series of rods, each holding several beads. Each bead represents a quantity, which is counted if the bead is moved from one end of a rod to the other. Calculations happen by manipulating the positions of beads until the final bead pattern reveals the result.^[130] Related aids include counting boards, which use tokens whose value depends on the area on the board in which they are placed,^[131] and counting rods, which are arranged in horizontal and vertical patterns to represent different numbers.^[132][f]

Sectors and slide rules are more refined calculating instruments that rely on geometric relationships between different scales to perform both basic and advanced arithmetic operations.^[134][g] Printed tables were particularly relevant as an aid to look up the results of operations like logarithm and trigonometric functions.^[136]

Others

There are many other types of arithmetic. Modular arithmetic operates on a finite set of numbers. If an operation would result in a number outside this finite set then the number is adjusted back into the set, similar to how the hands of clocks start at the beginning again after having completed one cycle. The number at which this adjustment happens is called the modulus. For example, a regular clock has a modulus of 12. In the case of adding 4 to 9, this means that the result is not 13 but 1. The same principle applies also to other operations, such as subtraction, multiplication, and division.^[139]

Some forms of arithmetic deal with operations performed on mathematical objects other than numbers. Interval arithmetic describes operations on intervals.

Arithmetic systems can be classified based on the numeral system they rely on. For instance, decimal arithmetic describes arithmetic operations in the decimal system. Other examples are binary arithmetic, octal arithmetic, and hexadecimal arithmetic.^[142]

Compound unit arithmetic describes arithmetic operations performed on magnitudes with compound units. It involves additional operations to govern the transformation between single unit and compound unit quantities. For example, the operation of reduction is used to transform the compound quantity 1 h 90 min into the single unit quantity 150 min.^[143]

Non-Diophantine arithmetics are arithmetic systems that violate traditional arithmetic intuitions and include equations like ${\displaystyle 1+1=1}$ and ${\displaystyle 2+2=5}$.^[144] They can be employed to represent some real-world situations in modern physics and everyday life. For instance, the equation ${\displaystyle 1+1=1}$ can be used to describe the observation that if one raindrop is added to another raindrop then they do not remain two separate entities but become one.^[145]

The Dedekind–Peano axioms provide an axiomatization of the arithmetic of natural numbers. Their basic principles were first formulated by Richard Dedekind and later refined by Giuseppe Peano. They rely only on a small number of primitive mathematical concepts, such as 0, natural number, and successor.^[h] The Peano axioms determine how these concepts are related to each other. All other arithmetic concepts can then be defined in terms of these primitive concepts.^[147]

• 0 is a natural number.
• For every natural number, there is a successor, which is also a natural number.
• The successors of two different natural numbers are never identical.
• 0 is not the successor of a natural number.
• If a set contains 0 and every successor then it contains every natural number.^[148][i]

Numbers greater than 0 are expressed by repeated application of the successor function ${\displaystyle s}$. For example, ${\displaystyle 1}$ is ${\displaystyle s(0)}$ and ${\displaystyle 3}$ is ${\displaystyle s(s(s(0)))}$. Arithmetic operations can be defined as mechanisms that affect how the successor function is applied. For instance, to add ${\displaystyle 2}$ to any number is the same as applying the successor function two times to this number.^[150]

Various axiomatizations of arithmetic rely on set theory. They cover natural numbers but can also be extended to integers, rational numbers, and real numbers. Each natural number is represented by a unique set. 0 is usually defined as the empty set ${\displaystyle \varnothing }$. Each subsequent number can be defined as the union of the previous number with the set containing the previous number. For example, ${\displaystyle 1=0\cup \{0\}=\{0\}}$, ${\displaystyle 2=1\cup \{1\}=\{0,1\}}$, and ${\displaystyle 3=2\cup \{2\}=\{0,1,2\}}$.^[151] Integers can be defined as ordered pairs of natural numbers where the second number is subtracted from the first one. For instance, the pair (9, 0) represents the number 9 while the pair (0, 9) represents the number −9.^[152] Rational numbers are defined as pairs of integers where the first number represents the numerator and the second number represents the denominator. For example, the pair (3, 7) represents the rational number ${\displaystyle {\tfrac {3}{7}}}$.^[153] One way to construct the real numbers relies on the concept of Dedekind cuts. According to this approach, each real number is represented by a partition of all rational numbers into two sets, one for all numbers below the represented real number and the other for the rest.^[154] Arithmetic operations are defined as functions that perform various set-theoretic transformations on the sets representing the input numbers to arrive at the set representing the result.^[155]

The earliest forms of arithmetic are sometimes traced back to counting and tally marks used to keep track of quantities. 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.^[156] However, a basic sense of numbers may predate these findings and might even have existed before the development of language.^[157]

It was not until the emergence of

Early civilizations primarily used numbers for concrete practical purposes, like commercial activities and tax records, but lacked an abstract concept of number itself.

The ancient Indians were the first to develop the concept of

Indian mathematicians also developed the

The first

In the 18th and 19th centuries, mathematicians such as

In various fields

Education

Arithmetic education forms part of

Later stages focus on a more abstract understanding and introduce the students to different types of numbers, such as negative numbers, fractions, real numbers, and complex numbers. They further cover more advanced numerical operations, like exponentiation, extraction of roots, and logarithm.[187] They also show how arithmetic operations are employed in other branches of mathematics, such as their application to describe geometrical shapes and the use of variables in algebra. Another aspect is to teach the students the use of algorithms and calculators to solve complex arithmetic problems.^[188]

The

One of the concepts studied by psychology is

Philosophy

The philosophy of arithmetic studies the fundamental concepts and principles underlying numbers and arithmetic operations. It explores the nature and ontological status of numbers, the relation of arithmetic to language and logic, and how it is possible to acquire arithmetic knowledge.^[195]

According to

The traditionally dominant view in the

Others

Arithmetic is relevant to many fields. In

Arithmetic is intimately connected to many branches of mathematics that depend on numerical operations.

See also

• Algorism
• Expression (mathematics)
• Finite field arithmetic
• Outline of arithmetic
• Plant arithmetic

External links