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Practicenglish 417
MUSIC VIDEO
useful information
symbols and mathematical terms
Basic math symbols
Symbol Symbol Name Meaning / definition Example
Equality equals sign = 5 = 2 + 3
≠ not equal sign ≠ inequality 5 4
> Strict inequality Greater than 5> 4
<Strict inequality less than 4 <5
Inequality ≥ Greater than or equal to 5 ≥ 4
≤ inequality less than or equal to 4 ≤ 5
() Parentheses first calculate expression inside 2x (3 + 5) = 16
[] Brackets inside first calculate expression [(1 + 2) * (1 + 5)] = 18
Addition plus sign + 1 + 1 = 2
- Minus sign subtraction 2-1 = 1
± plus - minus Both plus and minus operations ± 5 3 = 8 and -2
∓
minus - plus and minus Both operations plus three
∓
5 = -2 and 8
* Asterisk multiplication 2 * 3 = 6
Times multiplication sign × 2 × 3 = 6
Multiplication multiplication dot ∙ 2 ∙ 3
= 6
÷ division sign / obelus division 6 ÷ 2 = 3
/ Division division slash 6/2 = 3
- Horizontal line division / fraction \ frac {6} {2} = 3
remainder calculation module mod 7 mod 2 = 1
. period decimal point, decimal separator 2.56 = 2 + 56/100
ab power exponent 23 = 8
a ^ b caret exponent 2 ^ 3 = 8
√a square root
√a · √a = a
√9 = ± 3
3√a cube root 3√a · · 3√a 3√8 3√a = a = 2
4√a fourth root 4√a 4√a · · · 4√a 4√16 4√a = a = ± 2
n√a n-th root (radical) for n = 3, n√8 = 2
Percent% 1% 10% = 1/100 × 30 = 3
Per-mille ‰ 1 ‰ = 1/1000 = 0.1% 10 × 30 = 0.3 ‰
ppm per-million millionth 10ppm 1ppm = 0.0003 × 30 =
per-billion ppb 1 ppb = 1 billionths 10ppb × 30 = 3 × 10-7
ppt per-trillion 10ppt 1ppt = 10-12 × 30 = 3 × 10-10
Geometry symbols
Symbol Symbol Name Meaning / definition Example
∠
FORMED by two rays angle = 30 °
∠
ABC
Measured angle ABC = 30 °
spherical angle AOB = 30 °
∟ right angle = 90 °
α
= 90 °
Th degree 1 turn = 360 °
α
= 60 °
'60' arcminute 1 =
α
= 60º59'
'' Arcsecond = 1'60''
α
= 60º59'59 ''
line infinite line
Line segment AB line from point A to point B
That ray line start from point A
arc arc from point A to point B = 60
| Perpendicular lines perpendicular (90 ° angle) AC | BC
Parallel parallel lines AB || CD ||
≅
congruent to equivalence of geometric shapes and size
Δ
ABC
≅
Δ
XYZ
~ Similarity same shapes, same size
Δ
ABC ~ not
Δ
XYZ
Δ
ABC triangle shape triangle
Δ
≅
Δ
BCD
| X | distance Distance between points x and y | x | = 5
constant pi
π
π
= 3.141592654 ...
is the ratio Between the circumference and diameter of a circle
c =
π
· d = 2 ·
π
· r
radians radians rad angle 2
π
rad = 360 unit
grad grads grads unit 360 ° angle = 400 deg
Algebra symbols
Symbol Symbol Name Meaning / definition Example
x x Variable unknown value to find When 2x = 4, then a x = 2
≡ identical equivalence to
≜
equal by definition equal by definition
: = Equal by definition equal by definition
Approximately equal weak approximation ~ 11 ~ 10
≈ approximately equal approximation sin (0.01) ≈ 0.01
proportional to proportional to
α
f (x)
α
g (x)
Lemniscate infinity symbol ∞
"Much much less than less than 1« 1000000
"Much much Greater Than Greater Than 1000000» 1
() Parentheses calculate expression inside first 2 * (3 + 5) = 16
[] Brackets inside first calculate expression [(1 + 2) * (1 + 5)] = 18
{} Braces September
⌊
x
⌋
brackets rounds floor to lower integer number
⌊
4.3
⌋
= 4
⌈
x
⌉
rounds ceiling brackets to upper integer number
⌈
4.3
⌉
= 5
x! exclamation mark factorial 4! = 1 * 2 * 3 * 4 = 24
| X | vertical single bar absolute value | -5 | = 5
f (x) function of x values
of x maps to f (x) f (x) = 3x + 5
(F
∘
g) function composition
(F
∘
g) (x) = f (g (x))
f (x) = 3x, g (x) = x-1
⇒
(
∘
g f) (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]
Δ
delta change / difference t = t1 - t0
Discriminant
Δ
Δ
= b2 - 4ac
Sigma summation
Σ
- sum of all values
in range of series
Σ
xi = x1 + x2 + ... + xn
Double sigma summation
ΣΣ
Capital pi
Π
product - product of all values
in range of series
Π
xi = x1 ∙ ∙ ... ∙ x2 xn
e and constant / Euler's number e = 2.718281828 ... e = lim (1 + 1 / x) x, x → ∞
Euler-Mascheroni constant
γ
γ
= 0.527721566 ...
φ
golden ratio golden ratio constant
constant pi
π
π
= 3.141592654 ...
is the ratio Between the circumference and diameter of a circle
c =
π
· d = 2 ·
π
· r
Symbols Linear Algebra
Symbol Symbol Name Meaning / definition Example
∙ a ∙ dot product scalar b
× cross vector product a × b
A
⊗
B tensor product tensor product of A and B A
⊗
B
\ Langle x, y \ rangle inner product
[] Brackets matrix of numbers
() Parentheses matrix of numbers
| A | determinant determinant of matrix A
det (A) determinant determinant of matrix A
|| X || double upright bars norm
A T transpose matrix transpose
(AT) ij = (A) ji
† A Hermitian matrix matrix conjugate transpose
(A †) ij = (A) ji
A * Hermitian matrix matrix conjugate transpose
(A *) ij = (A) ji
A -1 inverse matrix A-1 A = I
rank (A) matrix rank rank of matrix A
rank (A) = 3
dim (U) dimension dimension of matrix A
rank (U) = 3
Probability and statistics symbols
Symbol Symbol Name Meaning / definition Example
P (A) probability function probability of event A P (A) = 0.5
P (A ∩ B) probability of events That intersection probability of events A and B P (A∩B) = 0.5
P (A
∪
B) probability of events That Union probability of events A or B P (A
∪
B) = 0.5
P (A | B) probability function conditional probability of event A event B occured Given P (A | B) = 0.3
f (x) probability density function (PDF) P (a ≤ x ≤ b) = ∫ f (x) dx
F (x) cumulative distribution function (cdf) F (x) = P (X ≤ x)
μ
population mean population mean values
of
μ
= 10
E (X) expectation value expected value of random variable X E (X) = 10
E (X | Y) conditional expectation expected value of random variable X Given and E (X | Y = 2) = 5
var (X) variance of random variable X variance var (X) = 4
σ
2 variance values
population variance of
σ
2 = 4
std (X) standard deviation standard deviation of random variable X std (X) = 2
σ
X standard deviation standard deviation value of random variable X
σ
X = 2
middle median 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
ρ
X, correlation and correlation of random variables X and Y
ρ
X, Y = 0.6
Summation summation
Σ
- sum of all values
in range of series
Double double summation summation
ΣΣ
That mode Mo OCCURS value Most Frequently in population
MR mid-range
MR = (xmax + xmin) / 2
Md sample median population is below half the value esta
Lower Q1 / first quartile 25% of population are below esta value
Q2 mediate / second quartile 50% of population are below esta value = median of samples
Q3 upper / third quartile 75% of population are below esta value
x average sample mean / arithmetic mean x = (2 + 5 + 9) / 3 = 5,333
s two sample variance s population variance estimator samples 2 = 4
population standard deviation s sample standard deviation estimator samples s = 2
zx standard score
zx = (x-x) / sx
X X ~ distribution of distribution of random variable X X ~ N (0,3)
N (
μ
,
σ
2) the normal distribution Gaussian distribution X ~ N (0,3)
U (a, b) uniform probability distribution in range equal to b X ~ U (0,3)
exp (
λ
) exponential distribution f (x) =
λ
e-
λ
x, x≥0
gamma (c,
λ
) gamma distribution
f (x) =
λ
c xc-1e-
λ
x /
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) = pk NCK (1-p) n-k
Poisson (
λ
) Poisson distribution
f (k) =
λ
ke-
λ
/ k!
Geom (p) geometric distribution
f (k) = p (1-p) k
HG (N, K, n) hyper-geometric distribution
Bern (p) Bernoulli distribution
Combinatorics Symbols
Symbol Symbol Name Meaning / definition Example
n! factorial n! = 1 · 2 · 3 · ... · n 5! = 1 · 2 · 3 · 4 · 5 = 120
NPK permutation _ P_ {n} {k} = \ frac {n!} {(nk)!} 5P3 = 5! / (5-3)! = 60
combination _ {n} C {k} = \ binom {n} {k} = \ frac {n!} {k! (nk)!} 5C3 = 5! / [3! (5-3)!] = 10
Set Theory symbols
Symbol Symbol Name Meaning / definition Example
{} Set a collection of elements A = {3,7,9,14},
B = {9,14,28}
A ∩ B That intersection objects belong to September and September A B A ∩ B = {9.14}
A
∪
B Union belong That objects or September to September A B A
∪
B = {3,7,9,14,28}
A subset B
⊆
subset has fewer elements or equal to the set {9,14,28}
⊆
{9,14,28}
A proper subset
⊂
B / strict subset subset has fewer elements than the set {9.14}
⊂
{9,14,28}
A subset
⊄
B not left in September not a subset of right
⊄
set {9.66} {9,14,28}
A set superset A
⊇
B elements have more or equal to the set B
⊇
{9,14,28} {9,14,28}
A proper superset
⊃
B / strict superset September A has more elements than September B
⊃
{9,14,28} {9.14}
A superset not
⊅
B SET A is not a superset of set B {9,14,28} {9.66}
⊅
2A power set all subsets of A
\ Mathcal {P} (A) power set all subsets of A
Both equality A = B Have the same members sets A = {3,9,14},
B = {3,9,14},
A = B
Ac That complement all the objects do not belong to Set A
A \ B relative complement That objects belong to A and not to B A = {3,9,14},
B = {1,2,3},
A-B = {9.14}
A - B relative complement That objects belong to A and not to B A = {3,9,14},
B = {1,2,3},
A-B = {9.14}
A symmetric difference
Δ
B That objects belong to A or B but not to Their intersection A = {3,9,14},
B = {1,2,3},
A
Δ
B = {1,2,9,14}
A
⊖
B symmetric difference That objects 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
∈
A membership element of set A = {3,9,14}, 3
∈
A
x
∉
A not no element of membership in September A = {3,9,14} 1
∉
A
(A, b) ordered pair of 2 elements collection
A × B Cartesian product set of all ordered pairs from A and B
| A | cardinality The number of elements of set A = {3,9,14}, | A | = 3
#A Cardinality The number of elements of set A = {3,9,14}, # A = 3
cardinality aleph-null infinite nature of numbers in September
aleph-one cardinality of countable ordinal numbers in September
Ø empty set or = {} C = {o}
\ Mathbb {U} universal set set of all possible values
\ Mathbb {N} 0 Natural numbers / whole numbers in September (with zero) \ mathbb {N} 0 = {0,1,2,3,4, ...}
∈
0 \ mathbb {N} 0
\ Mathbb {N} 1 Natural numbers / whole numbers set (without zero) \ mathbb {N} 1 = {1,2,3,4,5, ...} 6
∈
\ mathbb {N} 1
\ Mathbb {Z} integer numbers set \ mathbb {Z} = {...- 3, -2, -1,0,1,2,3 ...} -6
∈
\ mathbb {Z}
\ Mathbb {Q} rational numbers set \ mathbb {Q} = {x | x = a / b, a, b
∈
\ mathbb {Z}} 2/6
∈
\ mathbb {Q}
\ Mathbb {R} actual numbers set \ mathbb {R} = {x | -∞ <x <∞
∈
6.343434} \ mathbb {R}
\ Mathbb {C} complex numbers set \ mathbb {C} = {z | z = a + bi, -∞ <a <∞, -∞ <b <∞} 6 + 2i
∈
\ mathbb {C}
Logic symbols
Symbol Symbol Name Meaning / definition Example
· And and x · y
^ Caret / circumflex and x ^ y
& Ampersand and x & y
+ Plus or x + y
Caret or reversed
∨
x
∨
and
| Vertical line or x | y
x 'single quote not - negation x'
x bar not - negation x
¬ not not - negation ¬ x
! exclamation mark not - negation! x
⊕
circled plus / oplus exclusive or - xor x
⊕
and
Tittle negation ~ x ~
⇒
IMPLIES
⇔
equivalent if and only if (iff)
↔ equivalent if and only if (iff)
∀
for all
∃
there exists
∄
there does not exists
THEREFORE
∴
∵
Because / since
Calculus & Analysis symbols
Symbol Symbol Name Meaning / definition Example
\ Lim_ {x \ to x0} f (x) limit limit value of a function
ε
epsilon Represents a very small number, near zero
ε
→ 0
e and constant / Euler's number e = 2.718281828 ... e = lim (1 + 1 / x) x, x → ∞
and 'derivative derivative - Lagrange's notation (3x3)' = 9x2
and '' second derivative derivative of derivative (3x3) '' = 18x
and (n) nth derivative derivation n times (3x3) (3) = 18
\ Frac {d} {dx} derivative derivative - Leibniz's notation d (3x3) / dx = 9x2
\ Frac {d} {dx ^ 2y ^ 2} second derivative derivative derivative of d2 (3x3) / dx2 = 18x
\ Frac {d ^ ny} {dx ^ n} n times nth derivative derivation
\ Dot {y} derivative time derivative by time - Newton's notation
second derivative time derivative of derivative
Derivative and derivative Dx - Euler's notation
Dx2 derivative and second derivative of derivative
\ Frac {\ partial f (x, y)} {\ partial x} partial derivative ∂ (x2 + y2) / ∂x = 2x
∫ complete opposite to derivation
∬
comprehensive double integration of function of two variables
∭
integration of triple integral function of 3 variables
∮
closed contour / comprehensive line
∯
comprehensive closed surface
∰
comprehensive volume closed
[A, b] closed interval [a, b] = {x | a ≤ x ≤ b}
(A, b) open interval (a, b) = {x | a <x <b}
imaginary unit i i ≡ √-1 z = 3 + 2i
complex conjugate z * z = a + bi → z * = a-bi z * = 3 + 2i
complex conjugate z = a + bi z → z = a-bi z = 3 + 2i
∇
nabla / the gradient / divergence operator
∇
f (x, y, z)
vector
vector unit
and convolution x * y (t) = x (t) * h (t)
Laplace transform F (s) = {f (t)}
Fourier transform X (
ω
) = {f (t)}
delta function
δ
Lemniscate infinity symbol ∞
Numeral symbols
Name European Roman Hindu Arabic Hebrew
zero 0 0
One 1 I 1
א
two 2 II 2
ב
three 3 III 3
ג
four 4 IV 4
ד
five 5 V 5
ה
Six 6 VI 6
ו
Seven 7 VII 7
ז
eight 8 VIII 8
ח
Nine 9 IX 9
ט
ten 10 X 10
י
Eleven 11 XI 11
יא
twelve 12 XII 12
יב
thirteen 13 XIII 13
יג
fourteen 14 XIV 14
יד
fifteen 15 XV 15
טו
sixteen 16 16
טז
XVI
17 17 seventeen seventeenth
יז
XVIII eighteen 18 18
יח
nineteen 19 XIX 19
יט
twenty XX 20 20
כ
thirty 30 XXX 30
ל
fourty 40 XL 40
מ
50 L 50
נ
fifty
sixty 60 LX 60
ס
70 seventy LXX 70
ע
80 eighty eightieth 80
פ
90 XC 90
צ
ninety
one hundred
ק
100 C 100
Greek alphabet letters
Greek Greek Letter Name Symbol Equivalent Inglés Pronunciation
Upper Case Lower Case
Α
α
Alpha al-fa
Β
β
Beta be-ta b
Γ
γ
Gamma g ga-ma
Δ
δ Delta d-ta
Ε
ε Epsilon e ep-si-lon
Ζ
ζ Zeta z ze-ta
Η
η Eta h er-ta
Theta Θ θ th te-ta
Ι
ι i io-ta Iota
Kappa k Κ κ ka-pa
Lambda Λ λ l lam-da
Mu Μ μ m m-yoo
Ν
ν Nu n noo
Xi Ξ ξ x x-ee
Ο
ο Omicron or o-mee-c-ron
Pi
Π
π
p pa-yee
Rho
Ρ
ρ
r row
Sigma
Σ
σ
s sig-ma
Τ
τ
Tau ta t-oo
Υ
υ
Upsilon or oo-psi-lon
Φ
φ
Phi ph f-ee
Chi
Χ
χ
kh ch-ee
Psi
Ψ
ψ
ps p-see
Omega Ω ω o-me-ga
Roman numerals
Number Roman Numeral
0 not defined
1 I
2 II
3 III
4 IV
5 V
6 VI
7 VII
8 VIII
9 IX
10X
11 XI
12 XII
13 XIII
14 XIV
15 XV
16 XVI
17 XVII
18 XVIII
19 XIX
20 XX
30 XXX
40 XL
50 L
60 LX
70 LXX
80 LXXX
XC 90
100 C
200 CC
300 CCC
400 CD
500 D
600 DC
700 DCC
800 DCCC
900 CM
1000 M
5000 V
10000 X
50000 L
100000 C
500000 D
1000000 M
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