Operation (mathematics)

In mathematics, an operation is a function which takes one or more inputs (called operands) and produces an output. Some of the most common operations include addition, subtraction, multiplication and division, all of which take two inputs and produce an output. These are called binary operations, and are often used when solving math problems. Other types of operations are called unary operations, which take only one input and produce an output. 
There are more operations than these, including raising numbers to exponents, taking the root and applying the logarithm.
Below is a list of the most used operations.
List of mathematical operations.
Addition.
Addition is the first arithmetic operation and hyperoperation. It is the inverse operation of subtraction. The terms in an addition are called "addends", and the result of an addition is called a "sum".
The symbol for addition is +
Examples of additions:
formula_1
formula_2
formula_3
formula_4
formula_5
Any number plus zero is the same number (formula_6). This is called the "additive identity".
For example: formula_7
Changing the order of the addends in an addition does not change its sum. This is called the "commutative property of addition".
For example: formula_8
Changing how addends are grouped in an addition does not change its sum either. This is called the "associative property of addition".
For example: formula_9
Additive inverses (opposites).
The opposite of a number formula_10 is formula_11. A number formula_10 plus its opposite formula_11 is always equal to 0: formula_14
For example, the opposite of 5 is -5, because formula_15
The absolute value of two opposite numbers is always the same.
Subtraction.
Subtraction is the second arithmetic operation and the inverse operation of addition. The number that is being subtracted is the "subtrahend" and the number it is subtracted from is the "minuend". The result of a subtraction is called a "difference".
The symbol for subtraction is −
Examples of subtractions:
formula_16
formula_17
formula_18
formula_19
formula_20
formula_21
Because of the "additive identity", any number minus zero is the same number (formula_22).
In a subtraction of two terms, switching the minuend and the subtrahend changes the sign of the answer, meaning subtraction is anticommutative.
For example: formula_23 and formula_24
Multiplication.
Multiplication is the third arithmetic operation and the second hyperoperation. It is the inverse operation of division. The terms in a multiplication are called "factors", and the result of a multiplication is called a "product".
Multiplication is repeated addition.
The symbol for multiplication is × (• in algebra)
Examples of multiplications:
formula_25
formula_26
formula_27
formula_28
formula_29
Any number times one is the same number (formula_30). This is called the "multiplicative identity".
For example: formula_31
Changing the order of the factors in a multiplication does not change its product. This is called the "commutative property of multiplication".
For example: formula_32
Changing how factors are grouped in a multiplication does not change its product either. This is called the "associative property of multiplication".
For example: formula_33
Multiplicative inverses (reciprocals).
The reciprocal of a number formula_34 is formula_35. A number formula_34 times its reciprocal formula_35 is always equal to 1: formula_38
For example, the reciprocal of 3 is 1/3, because formula_39
To get the reciprocal of a fraction, switch the numerator and the denominator: the reciprocal of formula_40 is formula_41
Division.
Division is the fourth arithmetic operation and the inverse operation of multiplication. The number that is being divided is the "dividend" and the number it is divided by is the "divisor". The number on top of a fraction is called the "numerator" and the number at the bottom is called the "denominator". The result of a division is called a "quotient".
Division is repeated subtraction.
The symbol for division is / or a fraction.
Examples of divisions:
formula_42
formula_43
formula_44
formula_45
formula_46
Because of the "multiplicative identity", any number divided by one is the same number (formula_47).
Division by zero is undefined (formula_48).
In a fraction, switching the numerator and the denominator gives the reciprocal of the fraction.
For example: formula_49
Exponentiation.
Exponentiation is the fifth arithmetic operation and the third hyperoperation. It is one of the inverse operations of roots and logarithms. The number that is being multiplied is the "base" and the number of times it is multiplied is the "exponent". The result of an exponentiation is called a "power".
Exponentiation is repeated multiplication or division.
The symbol for exponentiation is the superscript (xy) or the caret (^).
Examples of exponentiations:
formula_50
formula_51
formula_52
formula_53
formula_54
formula_55
Because of the "multiplicative identity", the first power of any number is the same number, and the zeroth power of any number is one (formula_56 and formula_57).
Roots.
Roots are the sixth arithmetic operation and one of the inverse operations of exponentiation and logarithms. The first term is called the "index", and the second term is called the "radicand". The result of a root is called a "base". When there is no index, this means it is a square (2) root.
The symbol for roots is the radical (formula_58).
Examples of roots:
formula_59
formula_60
formula_61
formula_62
formula_63
The first root of any number is the same number (formula_64).
Logarithms.
Logarithms are the seventh arithmetic operation and one of the inverse operations of exponentiation and roots. The first term is called the "base", and the second term is called the "power". The result of a logarithm is called an "exponent". When there is no base, this means it is a base 10 logarithm.
The symbol for logarithm is logb(a)
Examples of logarithms:
formula_65
formula_66
formula_67
formula_68
formula_69
The logarithm of 1 (formula_70) is 0 in every base. This is because formula_71
The logarithm base e, or natural logarithm, is written as formula_72.
Modulation.
Modulation is the eighth arithmetic operation. It gives the remainder of a division. The first term is called the "modulend" and the second term is called the "modulator". The result of a modulation is called a "modulus".
The symbol for modulation is \
Examples of modulations:
formula_73\formula_74
formula_75\formula_76
formula_77\formula_78
formula_79\formula_80
formula_81\formula_82
formula_77\formula_34 is always equal to zero, because zero can be divided by any number (formula_85).
Factorial.
Factorial is a function which gives the number of ways to arrange formula_10 objects. The term is called the "index". The result of a factorial is also called a factorial.
The symbol for factorial is !
The first factorials are:
formula_87
formula_88
formula_89
formula_90
formula_91
formula_92
formula_93 is equal to one because there is exactly one way of arranging 0 objects. Factorials are undefined for negative numbers. Factorials of fractional numbers can be calculated using the Gamma function.
Absolute value.
Absolute value is a function which gives the distance from zero (or magnitude) of a number.
The symbol for absolute value is formula_94
Examples of absolute values:
formula_95
formula_96
formula_97
formula_98
formula_99
The absolute value of formula_100 is the same as the absolute value of formula_101 (formula_102). This is because subtraction is anticommutative.