In mathematics, an operation is a function which takes one or more inputs (named operands) and produces an output. Common operations are addition, subtraction, multiplication and division, all of which take two inputs and produce an output. These are named binary operations, and are frequently used when solving math problems. Other kinds of operations are named unary operations, which take only one input and produce an output.
Below is a list of the most useful operations.
List of mathematical operations change
The symbol for addition is +
Any number plus zero is the same number ( ). This is named the additive identity.
Changing the order of the addends in an addition does not change its sum. This is named the commutative property of addition.
Changing how addends are grouped in an addition does not change its sum either. This is named the associative property of addition.
Additive inverses (opposites) change
The opposite of a number is . A number plus its opposite is always equal to 0:
For example, the opposite of 5 is -5, because
The absolute value of two opposite numbers is always the same.
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 named a difference.
The symbol for subtraction is −
Because of the additive identity, any number minus zero is the same number ( ).
In a subtraction of two terms, switching the minuend and the subtrahend changes the sign of the answer, meaning subtraction is anticommutative.
For example: and
Multiplication is the third arithmetic operation and the second hyperoperation. It is the inverse operation of division. The terms in a multiplication are named factors, and the result of a multiplication is named a product.
Multiplication is repeated addition.
Any number times one is the same number ( ). This is named the multiplicative identity.
Changing the order of the factors in a multiplication does not change its product. This is named the commutative property of multiplication.
Changing how factors are grouped in a multiplication does not change its product either. This is named the associative property of multiplication.
Multiplication can also be implied. For example, means , and means .
With the Hindu-Arabic numerals, putting two digits next to each other could be misunderstood (e.g. 235 is read as "two hundred and thirty-five" and not ). Instead, one of the numbers (normally the second) is put in brackets.
Multiplicative inverses (reciprocals) change
The reciprocal of a number is . A number times its reciprocal is always equal to 1:
For example, the reciprocal of 3 is 1/3, because
To get the reciprocal of a fraction, switch the numerator and the denominator: the reciprocal of is
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 named the numerator and the number at the bottom is named the denominator. The result of a division is named a quotient.
Division is repeated subtraction.
The symbol for division is / or a fraction.
Because of the multiplicative identity, any number divided by one is the same number ( ).
Division by zero is undefined ( ).
In a fraction, switching the numerator and the denominator gives the reciprocal of the fraction.
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 named a power.
Exponentiation is repeated multiplication.
The symbol for exponentiation is the superscript ( ) or the caret (^).
Because of the multiplicative identity, the first power of any number is the same number, and the zeroth power of any number is one ( and ).
Roots are the sixth arithmetic operation and one of the inverse operations of exponentiation and logarithms. The first term is named the index, and the second term is named the radicand. The result of a root is named a base. When there is no index, this means it is a square (index 2) root.
The symbol for roots is the radical ( ).
The first root of any number is the same number ( ).
Logarithms are the seventh arithmetic operation and one of the inverse operations of exponentiation and roots. The first term is named the base, and the second term is named the power. The result of a logarithm is named an exponent. When there is no base, this means it is a decimal (base 10) logarithm.
The symbol for logarithm is
The logarithm of 1 ( ) is 0 in every base. This is because
The logarithm base , or natural logarithm, is written as .
Modulation is the eighth arithmetic operation. It gives the remainder of a division. The first term is named the modulend and the second term is named the modulator. The result of a modulation is named a modulus.
The symbol for modulation is \
is always equal to zero, because zero can be divided by any number ( ).
The symbol for factorial is !
The first factorials are:
is equal to one because there is exactly one way of arranging 0 objects. Factorials are undefined for negative integers. Factorials of fractional numbers can be calculated using the Gamma function.
Absolute value change
The symbol for absolute value is
The absolute value of is the same as the absolute value of ( ). This is because subtraction is anticommutative.
Related pages change
- "Definition of Operation (Illustrated Mathematics Dictionary)". mathisfun.com. Retrieved 2021-10-21.
- "Order of Operations". mathisfun.com. Retrieved 2021-11-21.
- Weisstein, Eric W. "Binary Operation". mathworld.wolfram.com. Retrieved 2020-08-26.
- "Definition of Binary Operation (Illustrated Mathematics Dictionary)". mathisfun.com. Retrieved 2021-11-21.
- "Definition of Unary Operation (Illustrated Mathematics Dictionary)". mathisfun.com. Retrieved 2021-11-21.