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Parentheses, Braces, and Brackets in Math
These symbols help determine the order of operations
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Deb Russell
You’ll come across many symbols in mathematics and arithmetic. In fact, the language of math is written in symbols, with some text inserted as needed for clarification. Three important—and related—symbols you’ll see often in math are parentheses, brackets, and braces. You will encounter parentheses, brackets, and braces frequently in prealgebra and algebra , so it’s important to understand the specific uses of these symbols as you move into higher math.
Using Parentheses ( )
Parentheses are used to group numbers or variables, or both. When you see a math problem containing parentheses, you need to use the order of operations to solve it. Take as an example the problem: 9 – 5 ÷ (8 – 3) x 2 + 6
You must calculate the operation within the parentheses first, even if it is an operation that would normally come after the other operations in the problem. In this problem, the times and division operations would normally come before subtraction (minus), but since 8 – 3 falls within the parentheses, you would work this part of the problem first. Once you’ve taken care of the calculation that falls within the parentheses, you would remove them. In this case (8 – 3) becomes 5, so you would solve the problem as follows:
9 – 5 ÷ (8 – 3) x 2 + 6
= 9 – 5 ÷ 5 x 2 + 6
= 9 – 1 x 2 + 6
= 9 – 2 + 6
= 7 + 6
= 13
Note that per the order of operations, you would work what’s in the parentheses first, then calculate numbers with exponents, then multiply and/or divide, then add or subtract. Multiplication and division, as well as addition and subtraction, hold an equal place in the order of operations, so you work these from left to right.
In the problem above, after taking care of the subtraction in the parentheses, you need to divide 5 by 5 first, yielding 1; then multiply 1 by 2, yielding 2; then subtract 2 from 9, yielding 7; and then add 7 and 6, yielding a final answer of 13.
Parentheses Can Also Mean Multiplication
In the problem 3(2 + 5), the parentheses tell you to multiply. However, you won’t multiply until you complete the operation inside the parentheses, 2 + 5, so you would solve the problem as follows:
3(2 + 5)
= 3(7)
= 21
Examples of Brackets [ ]
Brackets are used after the parentheses to group numbers and variables as well. Typically, you would use the parentheses first, then brackets, followed by braces. Here is an example of a problem using brackets:
4 – 3[4 – 2(6 – 3)] ÷ 3
= 4 – 3[4 – 2(3)] ÷ 3 (Do the operation in the parentheses first; leave the parentheses.)
= 4 – 3[4 – 6] ÷ 3 (Do the operation in the brackets.)
= 4 – 3[2] ÷ 3 (The bracket informs you to multiply the number within, which is 3 x 2.)
= 4 + 6 ÷ 3
= 4 + 2
= 6
Examples of Braces
Braces are also used to group numbers and variables. This example problem uses parentheses, brackets, and braces. Parentheses inside other parentheses (or brackets and braces) are also referred to as ” nested parentheses .” Remember, when you have parentheses inside brackets and braces, or nested parentheses, always work from the inside out:
21 + [4(2 + 1) + 3]
= 21 + [4(3) + 3]
= 21 + [12 + 3]
= 21 + [15]
= 216
= 32
Notes About Parentheses, Brackets, and Braces
Parentheses, brackets, and braces are sometimes referred to as round, square, and curly brackets, respectively. Braces are also used in sets, as in:
2, 3, 6, 8, 10…
When working with nested parentheses, the order will always be parentheses, brackets, braces, as follows:
[( )]
Examples of How Parentheses Are Used in Writing
The Associative and Commutative Properties
Learn How You Use BEDMAS in Math
Practice the Order of Operations With These Free Math Worksheets
What’s the Difference Between Brackets and Parentheses?
IEP Math Goals for Operations in the Primary Grades
Learn About the Associative Property in Math
Look Up Math Definitions With This Handy Glossary
What Is Area in Math?
9 Mental Math Tricks and Games for Students and Teachers
Realistic Math Problems Help 6thgraders Solve RealLife Questions
Discalculia
What Are the Probabilities for Rolling Three Dice?
ACT Math Practice Questions
Here’s How to Comment on a Math Report Card
What You Need to Know About Consecutive Numbers
›
Math
Parentheses, Braces, and Brackets in Math
These symbols help determine the order of operations
Share
Flipboard
Email
Math

Tutorials & Courses  Basics
 Arithmetic
 Geometry
 Pre Algebra & Algebra
 Statistics
 Exponential Decay
 Functions
 Worksheets by Grade
 Math Formulas & Math Tables
 Resources
View More
Deb Russell
You’ll come across many symbols in mathematics and arithmetic. In fact, the language of math is written in symbols, with some text inserted as needed for clarification. Three important—and related—symbols you’ll see often in math are parentheses, brackets, and braces. You will encounter parentheses, brackets, and braces frequently in prealgebra and algebra , so it’s important to understand the specific uses of these symbols as you move into higher math.
Using Parentheses ( )
Parentheses are used to group numbers or variables, or both. When you see a math problem containing parentheses, you need to use the order of operations to solve it. Take as an example the problem: 9 – 5 ÷ (8 – 3) x 2 + 6
You must calculate the operation within the parentheses first, even if it is an operation that would normally come after the other operations in the problem. In this problem, the times and division operations would normally come before subtraction (minus), but since 8 – 3 falls within the parentheses, you would work this part of the problem first. Once you’ve taken care of the calculation that falls within the parentheses, you would remove them. In this case (8 – 3) becomes 5, so you would solve the problem as follows:
9 – 5 ÷ (8 – 3) x 2 + 6
= 9 – 5 ÷ 5 x 2 + 6
= 9 – 1 x 2 + 6
= 9 – 2 + 6
= 7 + 6
= 13
Note that per the order of operations, you would work what’s in the parentheses first, then calculate numbers with exponents, then multiply and/or divide, then add or subtract. Multiplication and division, as well as addition and subtraction, hold an equal place in the order of operations, so you work these from left to right.
In the problem above, after taking care of the subtraction in the parentheses, you need to divide 5 by 5 first, yielding 1; then multiply 1 by 2, yielding 2; then subtract 2 from 9, yielding 7; and then add 7 and 6, yielding a final answer of 13.
Parentheses Can Also Mean Multiplication
In the problem 3(2 + 5), the parentheses tell you to multiply. However, you won’t multiply until you complete the operation inside the parentheses, 2 + 5, so you would solve the problem as follows:
3(2 + 5)
= 3(7)
= 21
Examples of Brackets [ ]
Brackets are used after the parentheses to group numbers and variables as well. Typically, you would use the parentheses first, then brackets, followed by braces. Here is an example of a problem using brackets:
4 – 3[4 – 2(6 – 3)] ÷ 3
= 4 – 3[4 – 2(3)] ÷ 3 (Do the operation in the parentheses first; leave the parentheses.)
= 4 – 3[4 – 6] ÷ 3 (Do the operation in the brackets.)
= 4 – 3[2] ÷ 3 (The bracket informs you to multiply the number within, which is 3 x 2.)
= 4 + 6 ÷ 3
= 4 + 2
= 6
Examples of Braces
Braces are also used to group numbers and variables. This example problem uses parentheses, brackets, and braces. Parentheses inside other parentheses (or brackets and braces) are also referred to as ” nested parentheses .” Remember, when you have parentheses inside brackets and braces, or nested parentheses, always work from the inside out:
21 + [4(2 + 1) + 3]
= 21 + [4(3) + 3]
= 21 + [12 + 3]
= 21 + [15]
= 216
= 32
Notes About Parentheses, Brackets, and Braces
Parentheses, brackets, and braces are sometimes referred to as round, square, and curly brackets, respectively. Braces are also used in sets, as in:
2, 3, 6, 8, 10…
When working with nested parentheses, the order will always be parentheses, brackets, braces, as follows:
[( )]
Examples of How Parentheses Are Used in Writing
The Associative and Commutative Properties
Learn How You Use BEDMAS in Math
Practice the Order of Operations With These Free Math Worksheets
What’s the Difference Between Brackets and Parentheses?
IEP Math Goals for Operations in the Primary Grades
Learn About the Associative Property in Math
Look Up Math Definitions With This Handy Glossary
What Is Area in Math?
9 Mental Math Tricks and Games for Students and Teachers
Realistic Math Problems Help 6thgraders Solve RealLife Questions
Discalculia
What Are the Probabilities for Rolling Three Dice?
ACT Math Practice Questions
Here’s How to Comment on a Math Report Card
What You Need to Know About Consecutive Numbers
Bracket (mathematics)
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In mathematics , various typographical forms of brackets are frequently used in mathematical notation such as parentheses ( ), square brackets [ ], braces , and angle brackets ⟨ ⟩. Generally such bracketing denotes some form of grouping: in evaluating an expression containing a bracketed subexpression, the operators in the subexpression take precedence over those surrounding it. Additionally, there are several uses and meanings for the various brackets.
Historically, other notations, such as the vinculum , were similarly used for grouping; in presentday use, these notations all have specific meanings. The earliest use of brackets to indicate aggregation (i.e. grouping) was suggested in 1608 by Christopher Clavius and in 1629 by Albert Girard .^{ [1] }
Contents
 1 Symbols for representing angle brackets
 2 Algebra
 3 Functions
 4 Coordinates and vectors
 5 Intervals
 6 Sets and groups
 7 Matrices
 8 Derivatives
 9 Falling and rising factorial
 10 Quantum mechanics
 11 Polynomial rings
 12 Lie bracket and commutator
 13 Floor/ceiling functions and fractional part
 14 See also
 15 Notes
Symbols for representing angle brackets[ edit ]
A variety of different symbols are used to represent angle brackets. In email and other ASCII text it is common to use the lessthan (<
) and greaterthan (>
) signs to represent angle brackets, because ASCII does not include angle brackets.^{ [2] } Unicode has three pairs of dedicated characters:
U+27E8 ⟨
MATHEMATICAL LEFT ANGLE BRACKET and U+27E9 ⟩ MATHEMATICAL RIGHT ANGLE BRACKET
 U+3008 〈 LEFT ANGLE BRACKET and U+3009 〉 RIGHT ANGLE BRACKET, used in Chinese punctuation
 U+2329 〈 LEFTPOINTING ANGLE BRACKET and U+232A 〉 RIGHTPOINTING ANGLE BRACKET, which are deprecated^{ [3] }
In LaTeX the markup is \langle
and \rangle
:
\displaystyle \langle \ \rangle
.
Algebra[ edit ]
In elementary algebra parentheses, ( ), are used to specify the order of operations . Terms inside the bracket are evaluated first; hence 2×(3 + 4) is 14 and 10 ÷ 5(1 + 0) is 2 and 8 ÷ 4(2 + 0) is 1 and (2×3) + 4 is 10. This notation is extended to cover more general algebra involving variables: for example
$$\displaystyle (x+y)\times (xy)
. Square brackets are also often used in place of a second set of parentheses when they are nested, to provide a visual distinction.
Also in mathematical expressions in general, parentheses are used to indicate grouping (that is, which parts belong together) when necessary to avoid ambiguities, or for the sake of clarity. For example, in the formula (εη)_{X} = ε_{Cod ηX}η_{X}, used in the definition of composition of two natural transformations , the parentheses around εη serve to indicate that the indexing by X is applied to the composition εη, and not just its last component η.
Functions[ edit ]
The arguments to a function are frequently surrounded by brackets:
$$\displaystyle f(x)
. It is common to omit the parentheses around the argument when there is little chance of ambiguity, thus:
$$\displaystyle \sin x
.
Coordinates and vectors[ edit ]
In the cartesian coordinate system brackets are used to specify the coordinates of a point: (2,3) denotes the point with xcoordinate 2 and ycoordinate 3.
The inner product of two vectors is commonly written as
$$\displaystyle \langle a,b\rangle
, but the notation (a, b) is also used.
Intervals[ edit ]
Both parentheses, ( ), and square brackets, [ ], can also be used to denote an interval . The notation
$$\displaystyle [a,c)
is used to indicate an interval from a to c that is inclusive of
$$\displaystyle a
but exclusive of
$$\displaystyle c
. That is,
$$\displaystyle [5,12)
would be the set of all real numbers between 5 and 12, including 5 but not 12. The numbers may come as close as they like to 12, including 11.999 and so forth (with any finite number of 9s), but 12.0 is not included. In some European countries, the notation
$$\displaystyle [5,12[
is also used for this.
The endpoint adjoining the square bracket is known as closed, while the endpoint adjoining the parenthesis is known as open. If both types of brackets are the same, the entire interval may be referred to as closed or open as appropriate. Whenever infinity or negative infinity is used as an endpoint in the case of intervals on the real number line, it is always considered open and adjoined to a parenthesis. The endpoint can be closed when considering intervals on the extended real number line .
Sets and groups[ edit ]
Braces are used to identify the elements of a set : a,b,c denotes a set of three elements.
Angle brackets are used in group theory to write group presentations , and to denote the subgroup generated by a collection of elements.
Matrices[ edit ]
An explicitly given matrix is commonly written between large round or square brackets:
Derivatives[ edit ]
The notation
stands for the nth derivative of function f, applied to argument x. So, for example, if
$$\displaystyle f(x)=\exp(\lambda x)
, then
$$\displaystyle f^(n)(x)=\lambda ^n\exp(\lambda x)
. This is to be contrasted with
$$\displaystyle f^n(x)=f(f(\ldots (f(x))\ldots ))
, the nfold application of f to argument x.
Falling and rising factorial[ edit ]
The notation (x)_{n} is used to denote the falling factorial , an nth degree polynomial defined by
Confusingly, the same notation may be encountered as representing the rising factorial, also called ” Pochhammer symbol “. Another notation for the same is x^{(n)}. It can be defined by
Quantum mechanics[ edit ]
In quantum mechanics , angle brackets are also used as part of Dirac ‘s formalism, bra–ket notation , to note vectors from the dual spaces of the bra
$$\displaystyle \left\langle A\right
and the ket
$$\displaystyle \left
.
In statistical mechanics, angle brackets denote ensemble or time average.
Polynomial rings[ edit ]
Square brackets are used to denote the variable in polynomial rings . For example,
$$\displaystyle \mathbb R [x]
is the polynomial ring with the
$$\displaystyle x
variable and real number coefficients.^{ [4] }
Lie bracket and commutator[ edit ]
In group theory and ring theory , square brackets are used to denote the commutator . In group theory, the commutator [g,h] is commonly defined as g^{−1}h^{−1}gh. In ring theory, the commutator [a,b] is defined as ab − ba. Furthermore, in theory, braces are used to denote the anticommutator where a,b is defined as ab + ba.
The Lie bracket of a Lie algebra is a binary operation denoted by
$$\displaystyle [\cdot ,\cdot ]:\mathfrak g\times \mathfrak g\to \mathfrak g
. By using the commutator as a Lie bracket, every associative algebra can be turned into a Lie algebra. There are many different forms of Lie bracket, in particular the Lie derivative and the Jacobi–Lie bracket .
Floor/ceiling functions and fractional part[ edit ]
Square brackets, as in [ π ] = 3, are sometimes used to denote the floor function , which rounds a real number down to the next integer. However the floor and ceiling functions are usually typeset with left and right square brackets where only the lower (for floor function) or upper (for ceiling function) horizontal bars are displayed, as in ⌊π⌋ = 3 or ⌈π⌉ = 4.
Braces, as in π < ^{1}/_{7}, may denote the fractional part of a real number.
See also[ edit ]
 Iverson bracket
 Binomial coefficient
 Poisson bracket
 Bracket polynomial
 Pochhammer symbol
 Frölicher–Nijenhuis bracket
 Nijenhuis–Richardson bracket , also known as algebraic bracket.
 Schouten–Nijenhuis bracket
 Dyck language
Notes[ edit ]
 ^ Cajori , Florian 1980. A history of mathematics. New York: Chelsea Publishing, p. 158
 ^ Raymond, Eric S. (1996), The New Hacker’s Dictionary , MIT Press, p. 41, ISBN 9780262680929
.
 ^ “Miscellaneous Technical” (PDF). unicode.org.
 ^ Stewart, Ian (1995). Concepts of Modern Mathematics . Dover Publications. p. 90.
 Arithmetic
 Mathematical notation
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