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.
Notation | Example | Meaning |
---|---|---|
Decimal Numbers | 34.7 | Decimal numbers are a notation for expressing numbers using base 10. |
Scientific Notation | 3.753×10^{23} | 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 = 3x^{2} + 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. |
Notation | Description | |
---|---|---|
Numbers | ||
14 | integer | |
1.5 | real number | |
3+4i | complex number | |
2.2i | imaginary number | |
7/15 | rational number, fraction | |
∞ | infinity | |
– | negative | |
+ | positive | |
% | percent | |
3.77×10^{4} | scientific notation | |
2.64E05 | E notation | |
a<b | a is less than b | |
a≤b | a is less than or equal to b | |
a=b | a is equal to b | |
a≠b | a is not equal to b | |
a≥b | a is greater than or equal to b | |
a>b | a is greater than b | |
x|y | x divides y | |
x≡y mod. q | x is congruent with y modulo q | |
z = a + bi | complex number | |
complex conjugate | ||
ℑ, Im | imaginary part of a complex number | |
ℜ, Re | real 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+b | addition, add a to b | |
a–b | subtraction; subtract b from a | |
-b | negation; negative b | |
a±b | a plus or minus b | |
a×b | multiply a by b | |
a·b | multiply a by b | |
ab | multiply a by b | |
a*b | multiply a by b in some computer languages. | |
ab | exponentiation: a raised to the b power; a multiplied times itself b times. | |
a^b | exponentiation in some computer languages | |
a**b | exponentiation in some computer languages. | |
square root of n | ||
cube root of n, fourth root of n, etc. | ||
fraction, division | ||
a÷b | a divided by b | |
a/b | a divided by b | |
a:b | ratio of a to b, divided by | |
a*b | an arbitrary operator | |
a…z, A…Z | variables | |
a_{1},a_{2},a_{3},… | indexed variables | |
a≡b | a is identical to, is equivalent to b | |
c=a mod. b | c is congruent to a modulo b. | |
→ | approaches, implies | |
⇒ | implies | |
a∝b | a 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 | |
nº | n degrees | |
n' | n minutes (1/60th degree) | |
n' | n feet. | |
n" | n seconds (1/60th minute) | |
Δx | change 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 | |
the limit of f(x) as x approaches a is equal to b. | ||
Geometry | ||
≅ | is congruent with | |
≇ | is not congruent with | |
~ | is similar to | |
AB | line segment AB | |
length of line segment AB | ||
line AB | ||
AB | ray AB | |
∠α | angle alpha | |
m∠α | the measure of angle alpha | |
triangle ABC | ||
l∥;m | l is parallel to m | |
l∦m | l is not parallel to m | |
l⊥m | l is perpendicular to m | |
minor arc with endpoints J and K | ||
major arc containing point B Logic | ||
P, Q | propositions | |
¬P, ~P | negation, NOT P | |
P∨Q, P+Q | disjunction | P OR Q |
P∧Q, P·Q | conjunction, P AND Q | |
P⊕Q | exclusive disjunction, P xor Q | |
P→Q | P implies Q | |
P⇒Q | P implies Q | |
P↔Q | equivalence, biconditional | |
P≡Q | equivalence | |
P=Q | equivalence | |
≡ | identity | |
0, F | false | |
1, T | true | |
∴ | therefore, in conclusion | |
Q.E.D., | ¦ End of proof | |
Set Theory | ||
A,B,C,... | set | |
a,b,c,... | member of a set | |
a∈A | a is a member of A | |
a∉A | a is not a member of A | |
A⊂B | A is a subset of B | |
A⊆B | A is a subset of or equal to B | |
B⊃A | B is a superset of A | |
A⊄B | A is not a subset of B | |
A⊈B | A is neither a subset of or equal to B | |
A∪B,+ | A union B | |
A∩B,· | A intersection B | |
A–B | difference of A and B. | |
∅, { } | empty set, null set | |
A' | complement of set A | |
A/S | complement 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×B | Cartesian product A cross B | |
f∘g(x) | composite function | |
f(X) | image of set X | |
one to one | one to one correspondence | |
|X| | cardinality of set X | |
ℵ_{0} | denumerable infinity | |
ℵ_{1}, ℵ_{1}, ℵ_{2} | nondenumerable 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 |
# | A | B | C | D |
E | F | G | H | I |
J | K | L | M | N |
O | P | Q | R | S |
T | U | V | W | X |
Y | Z |
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