Notation

Pronunciation: /noʊˈteɪ ʃən/ Explain

Notation is a way to write something down using special signs or symbols.[1] In mathematics, notation is used a lot. Mathematical notation allows one to express a mathematical idea in a shorter and much more easily understood format than plain English. The examples in table 1 show why this is true.

NotationExampleMeaning
Decimal Numbers 34.7 Decimal numbers are a notation for expressing numbers using base 10.
Scientific Notation 3.753×1023 Scientific notation is used to express very large or very small numbers in base 10 without using a lot of zeros. The number expressed in scientific notation could be written in decimal notation as 375,300,000,000,000,000,000,000.
Algebraic notation y = 3x2 + 4 A sentence that could be used to mean the same thing as the equation is: "Let y be equal to three times the value of x squared plus 4." However, in this sentence it is not clear whether one multiplies three times x first or squares x first. The algebraic order of operations tells us what has to be done first in the equation.
Probability function P(A) = 0.5 A sentence that could be used to mean the same thing as the equation is: "The probability of event A happening is 0.5.
Table 1: Examples of Mathematical Notation.

Summary of Mathematical Notation

NotationDescription
Numbers
14integer
1.5real number
3+ 4i complex number
2.2iimaginary number
7/15rational number, fraction
infinity
negative
+positive
%percent
3.77×104scientific notation
2.64E05E notation
a<ba is less than b
a≤ba is less than or equal to b
a=ba is equal to b
a≠ba is not equal to b
a≥ba is greater than or equal to b
a>ba is greater than b
x|yx divides y
x=y mod. qx is congruent with y modulo q
z = a + bicomplex number
(a+bi) with a line over it = a-bicomplex conjugate
ℑ, Imimaginary part of a complex number
ℜ, Rereal part of a complex number
Sets of Numbers
[m,n]closed interval from m to n
(m,n), ]m,n[open interval from m to n
(m,n], ]m,n]half open interval on the left from m to n
[m,n), [m,n[half open interval on the right from m to n
(-8,8)interval of all real numbers
ellipsis
sup.supremum
inf.Infimum
Arithmetic
a+baddition, add a to b
a–bsubtraction; subtract b from a
-bnegation; negative b
a±ba plus or minus b
a×bmultiply a by b
a·bmultiply a by b
abmultiply a by b
a*bmultiply a by b in some computer languages.
abexponentiation: a raised to the b power; a multiplied times itself b times.
a^bexponentiation in some computer languages
a**bexponentiation in some computer languages.
square root of nsquare root of n
cube root of n, fourth root of ncube root of n, fourth root of n, etc.
fraction a over b.fraction, division
a÷ba divided by b
a/ba divided by b
a:bratio of a to b, divided by
a*ban arbitrary operator
a…z, A…Zvariables
a1,a2,a3,…indexed variables
a≡ba is identical to, is equivalent to b
c=a mod. bc is congruent to a modulo b.
approaches, implies
implies
a∝ba varies as b, a is proportional to b.
infinity
f∘g(x)composition of functions
( )parenthesis, grouping of operations
[ ]brackets, grouping of operations
{ }braces, grouping of operations, see also sets
n degrees
n'n minutes (1/60th degree)
n'n feet.
n"n seconds (1/60th minute)
Δxchange in x, delta x
sum of a sequence
f(x)function of x
n!n factorial
|x|absolute value of x, magnitude of x
⌈x⌉ceiling function of x
⌊x⌋floor function of x
Limit as x approaches a of f(x) equals bthe limit of f(x) as x approaches a is equal to b.
Geometry
is congruent with
is not congruent with
~is similar to
ABline segment AB
AB with a line segment above it.length of line segment AB
AB with a two headed arrow above it.line AB
ABray AB
∠αangle alpha
m∠αthe measure of angle alpha
A triangle followed by ABCtriangle ABC
l∥;ml is parallel to m
l∦ml is not parallel to m
l⊥ml is perpendicular to m
JK with an ark over it.minor arc with endpoints J and K
ABC with an ark over it.major arc containing point B Logic
P, Qpropositions
¬P, ~Pnegation, NOT P
P∨Q, P+QdisjunctionP OR Q
P∧Q, P·Q conjunction, P AND Q
P⊕Qexclusive disjunction, P xor Q
P→QP implies Q
P⇒QP implies Q
P↔Qequivalence, biconditional
P≡Qequivalence
P=Qequivalence
identity
0, Ffalse
1, Ttrue
therefore, in conclusion
Q.E.D.,¦ End of proof
Set Theory
A,B,C,...set
a,b,c,...member of a set
a∈Aa is a member of A
a∉Aa is not a member of A
A⊂BA is a subset of B
A⊆BA is a subset of or equal to B
B⊃AB is a superset of A
A⊄BA is not a subset of B
A⊈BA is neither a subset of or equal to B
A∪B,+A union B
A∩B,·A intersection B
A–Bdifference of A and B.
∅, { }empty set, null set
A'complement of set A
A/Scomplement of set A in S.
{x:P(x)}the set of all x with property P
{a,b,c,...}set
(a,b,c,...)ordered set
<a,b,c,...>ordered set
A×BCartesian product A cross B
f∘g(x)composite function
f(X)image of set X
one to oneone to one correspondence
|X|cardinality of set X
0denumerable infinity
1, ℵ1, ℵ2nondenumerable infinities
P(A)power set of A
Probability
P(e)probability of event e
P(e1,e2)conditional probability of e1 given e2.
E(X)expectation of X
E(X,c)conditional expectation of X. given condition c
e'complement of event e.
Table 2: Summary of mathematics notation

Cite this article as:

McAdams, David E. Notation. 8/7/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/n/notation.html.

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Revision History

8/7/2018: Changed vocabulary links to WORDLINK format. (McAdams, David E.)
12/21/2009: Change More Information to Reference. (McAdams, David E.)
4/14/2008: Initial version. (McAdams, David E.)

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