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.
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 |
a + bi = a - bi | 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. |
a^{b} | 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 |
AB | length of line segment AB |
line AB | |
AB | ray AB |
∠α | angle alpha |
m∠α | the measure of angle alpha |
ΔABC | 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 | |
P, Q | propositions |
¬P, ~P | negation, NOT P |
P ∨ Q, P + Q | disjunction |
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 + B | A union B |
A ∩ 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( e_{1}, e_{2} ) | conditional probability of e_{1} given e_{2}. |
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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