# Notation

Pronunciation: /noʊˈteɪ.ʃən/ Explain

Notation is a way to write something down using special signs or symbols. 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+4icomplex 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
aba is less than or equal to b
a = ba is equal to b
aba is not equal to b
aba is greater than or equal to b
a > ba is greater than b
x | yx divides y
xy mod. qx is congruent with y modulo q
z = a + bicomplex number
a + bi = 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
absubtraction; 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 n
cube root of n, fourth root of n, etc.
fraction, division
a ÷ ba divided by b
a / ba divided by b
a : bratio of a to b, divided by
a * ban arbitrary operator
az, AZvariables
a1, a2, a3, …indexed variables
aba is identical to, is equivalent to b
c = a mod. bc is congruent to a modulo b.
approaches, implies
implies
aba varies as b, a is proportional to b.
infinity
fg(x)composition of functions
( )parenthesis, grouping of operations
[ ]brackets, grouping of operations
nºn 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
xceiling function of x
xfloor function of x
the 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
ABlength of line segment AB
line AB
ABray AB
αangle alpha
m∠αthe measure of angle alpha
ΔABCtriangle ABC
lml is parallel to m
lml is not parallel to m
lml is perpendicular to m
minor arc with endpoints J and K
major arc containing point B
P, Qpropositions
¬P, ~Pnegation, NOT P
PQ, P + Qdisjunction
PQ, P · Q conjunction, P AND Q
PQexclusive disjunction, P xor Q
PQP implies Q
PQP implies Q
PQequivalence, biconditional
PQequivalence
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
aAa is a member of A
aAa is not a member of A
ABA is a subset of B
ABA is a subset of or equal to B
BAB is a superset of A
ABA is not a subset of B
ABA is neither a subset of or equal to B
AB, A + BA union B
AB, A · BA intersection B
ABdifference 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
fg(x)composite function
f(X)image of set X
one to oneone to one correspondence
|X|cardinality of set X
0denumerable infinity
1, 1, 2nondenumerable 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

### References

1. McAdams, David E.. All Math Words Dictionary, notation. 2nd Classroom edition 20150108-4799968. pg 126. Life is a Story Problem LLC. January 8, 2015. Buy the book

McAdams, David E. Notation. 4/26/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/n/notation.html.

### Revision History

4/26/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)