Math Notation Flashcards
≠
not equal sign
Inequality
5 ≠ 4
5 is not equal to 4
=
equals sign
Equality
5 = 2+3
5 is equal to 2+3
≈
approximately equal
approximation
x ≈ y means x is approximately equal to y
>
inequality
greater than
5 > 4
5 is greater than 4
inequality
less than
4 < 5
4 is less than 5
≥
inequality
greater than or equal to
5 ≥ 4,
x ≥ y means x is greater than or equal to y
≤
less than or equal to
4 ≤ 5,
x ≤ y means x is less than or equal to y
( )
parentheses
calculate expression inside first
2 × (3+5) = 16
[ ]
Brackets
calculate expression inside first
[(1+2)×(1+5)] = 18
+
plus sign
Addition
1 + 1 = 2
−
minus sign
Subtraction
2 − 1 = 1
±
plus - minus
both plus and minus operations
3 ± 5 = 8 or -2
*
Asterisk
Multiplication
2 * 3 = 6
×
times sign
Multiplication
2 × 3 = 6
⋅
multiplication dot
Multiplication
2 ⋅ 3 = 6
÷
division sign / obelus
Division
6 ÷ 2 = 3
/
division slash
Division
6 / 2 = 3
—
horizontal line
division / fraction
mod
Modulo
remainder calculation
7 mod 2 = 1
.
Period
decimal point, decimal separator
2.56 = 2+56/100
a^b
Caret
Exponent
2 ^ 3 = 8
√a
square root
√a ⋅ √a = a
√9 = ±3
3√a (3 superindex)
cube root
3√a ⋅ 3√a ⋅ 3√a = a
3√8 = 2
n√a (n superindex)
“n-th root (radical)
for n=3, n√8 = 2”
%
Percent
1% = 1/100
10% × 30 = 3
‰
Per-mille
1‰ = 1/1000 = 0.1%
10‰ × 30 = 0.3
ppm
Per-million
1ppm = 1/1000000
10ppm × 30 = 0.0003
ppb
Per-billion
1ppb = 1/1000000000
10ppb × 30 = 3×10-7
∠
Angle
formed by two rays
∠ABC = 30°
°
Degree
1 turn = 360°
α = 60°
deg
Degree
1 turn = 360deg
α = 60deg
≅
congruent to
equivalence of geometric shapes and size
∆ABC≅ ∆XYZ
~ (geometry)
Similarity
same shapes, not same size
∆ABC~ ∆XYZ
Δ
Triangle
triangle shape
ΔABC≅ ΔBCD
|x-y|
Distance
distance between points x and y
x-y | = 5
π
pi constant
π = 3.141592654…
c = π⋅d = 2⋅π⋅r
rad
Radians
radians angle unit
360° = 2π rad
c
Radians
radians angle unit
360° = 2π c
x
x variable
unknown value to find
when 2x = 4, then x = 2
≡
Equivalence
identical to
≜
equal by definition
:=
equal by definition
~
approximately equal
weak approximation
11 ~ 10
≈
approximately equal
Approximation
sin(0.01) ≈ 0.01
∝
proportional to
y ∝ x when y = kx, k constant
∞
Lemniscate
infinity symbol
≪
much less than
1 ≪ 1000000
≫
much greater than
1000000 ≫ 1
{ }
Braces
set
⌊x⌋
floor brackets
rounds number to lower integer
⌊4.3⌋ = 4
⌈x⌉
ceiling brackets
rounds number to upper integer
⌈4.3⌉ = 5
x!
exclamation mark
Factorial
4! = 123*4 = 24
x |
vertical bars
absolute value
-5 | = 5
f (x)
function of x maps values of x to f(x)
f (x) = 3x+5
(f ∘ g)
function composition (f ∘ g) (x) = f (g(x))
f (x)=3x,g(x)=x-1 ⇒(f ∘ g)(x)=3(x-1)
(a,b)
open interval
(a,b) = {x | a < x < b}
x∈ (2,6)
[a,b]
closed interval
[a,b] = {x | a ≤ x ≤ b}
x ∈ [2,6]
∆
Delta
change / difference
∆t = t1 - t0
∑
Sigma
summation - sum of all values in range of series
∑ xi= x1+x2+…+xn
∑∑
Sigma
double summation
∑+∑
∏
capital pi
product - product of all values in range of series
∏ xi=x1∙x2∙…∙xn
e
Euler’s number
e = 2.718281828…
e = lim (1+1/x)x , x→∞
γ
Euler-Mascheroni constant
γ = 0.5772156649…
φ
golden ratio constant
·
Dot
scalar product
a · b
×
Cross
vector product
a × b
⊗
tensor product
tensor product of A and B
A ⊗ B
[ ]
Brackets
matrix of numbers
|| x ||
double vertical bars
norm
AT
Transpose
matrix transpose
(AT)ij = (A)ji
A -1
inverse matrix
dim(U)
Dimension
dimension of matrix A
dim(U) = 3
P(A)
probability function
probability of event A
P(A) = 0.5
P(A ⋂ B)
probability of events intersection
probability that of events A and B
P(A⋂B) = 0.5
P(A ⋃ B)
probability of events union
probability that of events A or B
P(A⋃B) = 0.5
P(A | B)
conditional probability function
probability of event A given event B occured
P(A | B) = 0.3
f (x) （probability)
probability density function (pdf)
P(a ≤ x ≤ b) = ∫ f (x) dx
F(x)
cumulative distribution function (cdf)
F(x) = P(X≤ x)
μ
Mu
population mean
μ = 10
E(X)a
expectation value
expected value of random variable X
E(X) = 10
E(X | Y)
conditional expectation
expected value of random variable X given Y
E(X | Y=2) = 5
var(X)
Variance
variance of random variable X
var(X) = 4
σ2
Sigma (variation)
Variance
variance of random variable X
σ2 = 4
std(X)
standard deviation
standard deviation of random variable X
std(X) = 2
σX
standard deviation
standard deviation value of random variable X
σX = 2
x̃
Median
middle value of random variable x
cov(X,Y)
Covariance
covariance of random variables X and Y
cov(X,Y) = 4
corr(X,Y)
Correlation
correlation of random variables X and Y
corr(X,Y) = 0.6
ρX,Y
Correlation
correlation of random variables X and Y
ρX,Y = 0.6
MR
Mid-range
MR = (xmax+xmin)/2
Md
sample median
half the population is below this value
Q1
lower / first quartile
25% of population are below this value
Q2
median / second quartile
50% of population are below this value = median of samples
Q3
upper / third quartile
75% of population are below this value
s 2
sample variance
population samples variance estimator
s 2 = 4
x-bar
sample mean
average / arithmetic mean
x-bar = (2+5+9) / 3 = 5.333
zx
standard score
zx = (x-xbar) / sx
s
sample standard deviation
population samples standard deviation estimator
s = 2
X ~
distribution of X
distribution of random variable X
X ~ N(0,3)
N(μ,σ2)
normal distribution
gaussian distribution
X ~ N(0,3)
U(a,b)
uniform distribution
equal probability in range a,b
X ~ U(0,3)
exp(λ)
exponential distribution
f (x) = λe-λx , x≥0
λ
Lambda
gamma(c, λ)
gamma distribution
f (x) = λ c xc-1e-λx / Γ(c), x≥0
χ 2(k)
chi-square distribution
f (x) = xk/2-1e-x/2 / ( 2k/2 Γ(k/2) )
F (k1, k2)
F distribution
Bin(n,p)
binomial distribution
f (k) = nCk pk(1-p)n-k
Poisson(λ)
Poisson distribution
f (k) = λke-λ / k!
Geom(p)
geometric distribution
f (k) = p(1-p) k
Bern(p)
Bernoulli distribution
n!
Factorial
n! = 1⋅2⋅3⋅…⋅n
5! = 1⋅2⋅3⋅4⋅5 = 120
nPk
Permutation
P(n,r)=n! (n−r)!
5P3 = 5! / (5-3)! = 60
nCk
Combination
r! (n−r)!
{ }
Set
a collection of elements
A = {3,7,9,14},
B = {9,14,28}
A ∩ B
Intersection
objects that belong to set A and set B
A ∩ B = {9,14}
A ∪ B
Union
objects that belong to set A or set B
A ∪ B = {3,7,9,14,28}
A ⊆ B
Subset
A is a subset of B. set A is included in set B.
{9,14,28} ⊆ {9,14,28}
A ⊂ B
proper subset / strict subset
A is a subset of B, but A is not equal to B.
{9,14} ⊂ {9,14,28}
A ⊄ B
not subset
set A is not a subset of set B
{9,66} ⊄ {9,14,28}
A ⊇ B
Superset
A is a superset of B. set A includes set B
{9,14,28} ⊇ {9,14,28}
A ⊃ B
proper superset / strict superset
A is a superset of B, but B is not equal to A.
{9,14,28} ⊃ {9,14}
A ⊅ B
not superset
set A is not a superset of set B
{9,14,28} ⊅ {9,66}
A ⊅ B
not superset
set A is not a superset of set B
{9,14,28} ⊅ {9,66}
A - B
relative complement
objects that belong to A and not to B
A = {3,9,14},
B = {1,2,3},
A-B = {9,14}
A ∆ B
symmetric difference
objects that belong to A or B but not to their intersection
A = {3,9,14},
B = {1,2,3},
A ∆ B = {1,2,9,14}
A ⊖ B
symmetric difference
objects that belong to A or B but not to their intersection
A = {3,9,14},
B = {1,2,3},
A ⊖ B = {1,2,9,14}
∈
element of,
belongs to
set membership
A={3,9,14}, 3 ∈ A
∉
not element of
no set membership
A={3,9,14}, 1 ∉ A
(a,b) (Set theory symbols)
ordered pair
collection of 2 elements
A×B
cartesian product
set of all ordered pairs from A and B
|A| #A
Cardinality
the number of elements of set A
A={3,9,14}, |A|=3
|
Set theory symbols
vertical bar
such that
A={x|3
Ø
empty set
Ø = { }
C = {Ø}
∀
For All
∀x>1, x2>x
∴
Therefore
a=b ∴ b=a
⋅ (Logic)
And
x ⋅ y
&
Ampersand
and
x & y
(logic)+
Or
x + y
(logic)
vertical line
Or
x | y
⇒
implies
¬
not - negation
¬ x
⇔
Equivalent
if and only if (iff)
∵
because / since
y ‘
Derivative
derivative - Lagrange’s notation
(3x3)’ = 9x2
y ‘’
second derivative
derivative of derivative
(3x3)’’ = 18x
y(n)
nth derivative
(3x3)(3) = 18
∫
Integral
opposite to derivation
∫ f(x)dx
∫∫
double integral
integration of function of 2 variables
∫∫ f(x,y)dxdya
i
imaginary unit
i ≡ √-1
z = 3 + 2i
z*
complex conjugate
z = a+bi → z*=a-bi
z* = 3 - 2i
Re(z)
real part of a complex number
z = a+bi → Re(z)=a
Re(3 - 2i) = 3
Im(z)
imaginary part of a complex number
z = a+bi → Im(z)=b
Im(3 - 2i) = -2
z |
absolute value/magnitude of a complex number
|z| = |a+bi| = √(a2+b2)
|3 - 2i| = √13
Α α
Alpha
Β β
Beta
Γ γ
Gamma
Δ δ
Delta
Ε ε
Epsilon
Ζ ζ
Zeta
Η η
Eta
Θ θ
Theta
Ι ι
Iota
Κ κ
Kappa
Λ λ
Lambda
Μ μ
Mu
Ν ν
Nu
Ξ ξ
Xi
Π π
Pi
Ρ ρ
Rho
Σ σ
Sigma
Τ τ
Tau
Υ υ
Upsilon
Φ φ
Phi
Χ χ
Chi
Ψ ψ
Psi
Ω ω
Omega
