Operand

Last updated October 01, 2023

In mathematics, an operand is the object of a mathematical operation, i.e., it is the object or quantity that is operated on.^[1]

Example

The following arithmetic expression shows an example of operators and operands:

3+6=9

In the above example, '+' is the symbol for the operation called addition.

The operand '3' is one of the inputs (quantities) followed by the addition operator, and the operand '6' is the other input necessary for the operation.

The result of the operation is 9. (The number '9' is also called the sum of the augend 3 and the addend 6.)

An operand, then, is also referred to as "one of the inputs (quantities) for an operation".

Notation

Expressions as operands

Operands may be nested, and may consist of expressions also made up of operators with operands.

(3+5)\times 2

In the above expression '(3 + 5)' is the first operand for the multiplication operator and '2' the second. The operand '(3 + 5)' is an expression in itself, which contains an addition operator, with the operands '3' and '5'.

Order of operations

Rules of precedence affect which values form operands for which operators:^[2]

3+5\times 2

In the above expression, the multiplication operator has the higher precedence than the addition operator, so the multiplication operator has operands of '5' and '2'. The addition operator has operands of '3' and '5 × 2'.

Positioning of operands

Depending on the mathematical notation being used the position of an operator in relation to its operand(s) may vary. In everyday usage infix notation is the most common,^[3] however other notations also exist, such as the prefix and postfix notations. These alternate notations are most common within computer science.

Below is a comparison of three different notations — all represent an addition of the numbers '1' and '2'

1+2

(infix notation)

+\;1\;2

(prefix notation)

1\;2\;+

(postfix notation)

Infix and the order of operation

In a mathematical expression, the order of operation is carried out from left to right. Start with the leftmost value and seek the first operation to be carried out in accordance with the order specified above (i.e., start with parentheses and end with the addition/subtraction group). For example, in the expression

4\times 2^{2}-(2+2^{2})

,

the first operation to be acted upon is any and all expressions found inside a parenthesis. So beginning at the left and moving to the right, find the first (and in this case, the only) parenthesis, that is, (2 + 2²). Within the parenthesis itself is found the expression 2². The reader is required to find the value of 2² before going any further. The value of 2² is 4. Having found this value, the remaining expression looks like this:

4\times 2^{2}-(2+4)

The next step is to calculate the value of expression inside the parenthesis itself, that is, (2 + 4) = 6. Our expression now looks like this:

4\times 2^{2}-6

Having calculated the parenthetical part of the expression, we start over again beginning with the left most value and move right. The next order of operation (according to the rules) is exponents. Start at the left most value, that is, 4, and scan your eyes to the right and search for the first exponent you come across. The first (and only) expression we come across that is expressed with an exponent is 2². We find the value of 2², which is 4. What we have left is the expression

4\times 4-6

.

The next order of operation is multiplication. 4 × 4 is 16. Now our expression looks like this:

16-6

The next order of operation according to the rules is division. However, there is no division operator sign (÷) in the expression, 16 − 6. So we move on to the next order of operation, i.e., addition and subtraction, which have the same precedence and are done left to right.

16-6=10

.

So the correct value for our original expression, 4 × 2² − (2 + 2²), is 10.

It is important to carry out the order of operation in accordance with rules set by convention. If the reader evaluates an expression but does not follow the correct order of operation, the reader will come forth with a different value. The different value will be the incorrect value because the order of operation was not followed. The reader will arrive at the correct value for the expression if and only if each operation is carried out in the proper order.

Arity

The number of operands of an operator is called its arity.^[4] Based on arity, operators are chiefly classified as nullary (no operands), unary (1 operand), binary (2 operands), ternary (3 operands). Higher arities are less frequently denominated through a specific terms, all the more when function composition or currying can be used to avoid them. Other terms include:

quaternary, tetranary (4)
quinary, quintenary, quinquennary (5)
hexanary, senary, sexenary (6)
septenary (7)
octonary (8)
nonary, novenary (9)
denary (10)
undenary (11)
duodenary (12)
tridecennary (13)
quindenary (15)
vigenary (20)
quadringenary (40)
quinquagenary (50)
sexagenary (60)
septuagenary (70)
octogenary (80)
nonagenary (90)
centenary (100)
sesquicentenary (150)
bicentenary (200)
tercentenary, tricentenary (300)
quadringentenary, quatercentenary (400)
quincentenary (500)
sexcentenary (600)
septcentenary (700)
octocentenary (800)

Computer science

In computer programming languages, the definitions of operator and operand are almost the same as in mathematics.

In computing, an operand is the part of a computer instruction which specifies what data is to be manipulated or operated on, while at the same time representing the data itself.^[5] A computer instruction describes an operation such as add or multiply X, while the operand (or operands, as there can be more than one) specify on which X to operate as well as the value of X.

Additionally, in assembly language, an operand is a value (an argument) on which the instruction, named by mnemonic, operates. The operand may be a processor register, a memory address, a literal constant, or a label. A simple example (in the x86 architecture) is

MOVDS,AX

where the value in register operand AX is to be moved ( MOV ) into register DS. Depending on the instruction, there may be zero, one, two, or more operands.

Related Research Articles

In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs.

In mathematics, a binary operation or dyadic operation is a rule for combining two elements to produce another element. More formally, a binary operation is an operation of arity two.

<span class="mw-page-title-main">Logical conjunction</span> Logical connective AND

In logic, mathematics and linguistics, and is the truth-functional operator of conjunction or logical conjunction. The logical connective of this operator is typically represented as $or or (prefix) or or in which is the most modern and widely used.$

Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division. The result of a multiplication operation is called a product.

Polish notation (PN), also known as normal Polish notation (NPN), Łukasiewicz notation, Warsaw notation, Polish prefix notation or simply prefix notation, is a mathematical notation in which operators precede their operands, in contrast to the more common infix notation, in which operators are placed between operands, as well as reverse Polish notation (RPN), in which operators follow their operands. It does not need any parentheses as long as each operator has a fixed number of operands. The description "Polish" refers to the nationality of logician Jan Łukasiewicz, who invented Polish notation in 1924.

Reverse Polish notation (RPN), also known as reverse Łukasiewicz notation, Polish postfix notation or simply postfix notation, is a mathematical notation in which operators follow their operands, in contrast to prefix or Polish notation (PN), in which operators precede their operands. The notation does not need any parentheses for as long as each operator has a fixed number of operands.

In logic, mathematics, and computer science, arity is the number of arguments or operands taken by a function, operation or relation. In mathematics, arity may also be called rank, but this word can have many other meanings. In logic and philosophy, arity may also be called adicity and degree. In linguistics, it is usually named valency.

In mathematics, a unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function $f : A \to A$ , where $A$ is a set. The function $f$ is a unary operation on $A$ .

In mathematics, function composition is an operation $\circ$ that takes two functions $f$ and $g$ , and produces a function $h = g \circ f$ such that $h (x) = g (f (x))$ . In this operation, the function $g$ is applied to the result of applying the function $f$ to $x$ . That is, the functions $f : X \to Y$ and $g : Y \to Z$ are composed to yield a function that maps $x$ in domain $X$ to $g (f (x))$ in codomain $Z$ . Intuitively, if $z$ is a function of $y$ , and $y$ is a function of $x$ , then $z$ is a function of $x$ . The resulting composite function is denoted $g \circ f : X \to Z$ , defined by $(g \circ f)(x) = g (f (x))$ for all $x$ in $X$ .

<span class="mw-page-title-main">Infix notation</span> Mathematics notation with operators between operands

Infix notation is the notation commonly used in arithmetical and logical formulae and statements. It is characterized by the placement of operators between operands—"infixed operators"—such as the plus sign in 2 + 2.

In mathematics and computer programming, the order of operations is a collection of rules that reflect conventions about which operations to perform first in order to evaluate a given mathematical expression.

In programming languages, scientific calculators and similar common operator notation or operator grammar is a way to define and analyse mathematical and other formal expressions. In this model a linear sequence of tokens are divided into two classes: operators and operands.

In programming language theory, the associativity of an operator is a property that determines how operators of the same precedence are grouped in the absence of parentheses. If an operand is both preceded and followed by operators, and those operators have equal precedence, then the operand may be used as input to two different operations. The choice of which operations to apply the operand to, is determined by the associativity of the operators. Operators may be associative, left-associative, right-associative or non-associative. The associativity and precedence of an operator is a part of the definition of the programming language; different programming languages may have different associativity and precedence for the same type of operator.

In computer programming, operators are constructs defined within programming languages which behave generally like functions, but which differ syntactically or semantically.

In computer science, an operator precedence parser is a bottom-up parser that interprets an operator-precedence grammar. For example, most calculators use operator precedence parsers to convert from the human-readable infix notation relying on order of operations to a format that is optimized for evaluation such as Reverse Polish notation (RPN).

There are various ways in which calculators interpret keystrokes. These can be categorized into two main types:

In computer science, the shunting yard algorithm is a method for parsing arithmetical or logical expressions, or a combination of both, specified in infix notation. It can produce either a postfix notation string, also known as Reverse Polish notation (RPN), or an abstract syntax tree (AST). The algorithm was invented by Edsger Dijkstra and named the "shunting yard" algorithm because its operation resembles that of a railroad shunting yard. Dijkstra first described the shunting yard algorithm in the Mathematisch Centrum report MR 34/61.

In mathematics, an operation is a function which takes zero or more input values to a well-defined output value. The number of operands is the arity of the operation.

A formula calculator is a software calculator that can perform a calculation in two steps:

Enter the calculation by typing it in from the keyboard.
Press a single button or key to see the final result.

<span class="mw-page-title-main">Computer algebra</span> Scientific area at the interface between computer science and mathematics

In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects. Although computer algebra could be considered a subfield of scientific computing, they are generally considered as distinct fields because scientific computing is usually based on numerical computation with approximate floating point numbers, while symbolic computation emphasizes exact computation with expressions containing variables that have no given value and are manipulated as symbols.

References

↑ American Heritage Dictionary
↑ "Physical Review Style and Notation Guide" (PDF). American Physical Society. Section IV–E–2–e. Retrieved 5 August 2012.
↑ "The Implementation and Power of Programming Languages" . Retrieved 30 August 2014.
↑ Michiel Hazewinkel (2001). Encyclopaedia of Mathematics, Supplement III. Springer. p. 3. ISBN 978-1-4020-0198-7.: "Each connective has associated with it a natural number, called its rank, or arity."
↑ Nell Dale and John Lewis (2012). Computer Science Illuminated, 5th Edition. Jones and Bartlett. ISBN 978-1449672843.

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[1] American Heritage Dictionary

[2] "Physical Review Style and Notation Guide" (PDF). American Physical Society. Section IV–E–2–e. Retrieved 5 August 2012.

[Infix,_Postfix_and_Prefix-3] "The Implementation and Power of Programming Languages" . Retrieved 30 August 2014.

[Hazewinkel2001-4] Michiel Hazewinkel (2001). Encyclopaedia of Mathematics, Supplement III. Springer. p. 3. ISBN 978-1-4020-0198-7.: "Each connective has associated with it a natural number, called its rank, or arity."

[5] Nell Dale and John Lewis (2012). Computer Science Illuminated, 5th Edition. Jones and Bartlett. ISBN 978-1449672843.

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