Cracking the Code: Unveiling the World of Mathematical Symbols.
Symbol  Symbol Name  Meaning / definition  Example 

=  equals sign  equality  5 = 2+3 5 is equal to 2+3 
≠  not equal sign  inequality  5 ≠ 4 5 is not equal to 4 
≈  approximately equal  approximation 
sin(0.01) ≈ 0.01, x ≈ y means x is approximately equal to y 
>  strict inequality  greater than  5 > 4 5 is greater than 4 
<  strict 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 
≤  inequality  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 
±  minus  plus  both minus and plus operations  3 ∓ 5 = 2 or 8 
*  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 }  power  exponent  2 ^{ 3 }= 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 } √ a = a  ^{ 3 }√ 8 = 2 
^{ 4 }√ a  fourth root  ^{ 4 }√ a ⋅ ^{ 4 } √ a ⋅ ^{ 4 } √ a ⋅ ^{ 4 } √ a = a  ^{ 4 }√ 16 = ±2 
^{ n }√ a  nth root (radical)  for n=3, ^{ n }√ 8 = 2  
%  percent  1% = 1/100  10% × 30 = 3 
‰  permille  1‰ = 1/1000 = 0.1%  10‰ × 30 = 0.3 
ppm  permillion  1ppm = 1/1000000  10ppm × 30 = 0.0003 
ppb  perbillion  1ppb = 1/1000000000  10ppb × 30 = 3×10 ^{7} 
ppt  pertrillion  1ppt = 10 ^{12}  10ppt × 30 = 3×10 ^{10} 
Symbol  Symbol Name  Meaning / definition  Example 

∠  angle  formed by two rays  ∠ABC = 30° 
measured angle  ABC = 30°  
spherical angle  AOB = 30°  
∟  right angle  = 90°  α = 90° 
°  degree  1 turn = 360°  α = 60° 
deg  degree  1 turn = 360deg  α = 60deg 
′  prime  arcminute, 1° = 60′  α = 60°59′ 
″  double prime  arcsecond, 1′ = 60″  α = 60°59′59″ 
line  infinite line  
AB  line segment  line from point A to point B  
ray  line that start from point A  
arc  arc from point A to point B  = 60°  
⊥  perpendicular  perpendicular lines (90° angle)  AC ⊥ BC 
∥  parallel  parallel lines  AB ∥ CD 
≅  congruent to  equivalence of geometric shapes and size  ∆ABC≅ ∆XYZ 
~  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... is the ratio between the circumference and diameter of a circle 
c = π⋅ d = 2⋅ π⋅ r 
rad  radians  radians angle unit  360° = 2π rad 
^{c}  radians  radians angle unit  360° = 2π ^{c} 
grad  gradians / gons  grads angle unit  360° = 400 grad 
^{g}  gradians / gons  grads angle unit  360° = 400 ^{g} 
Symbol  Symbol Name  Meaning / definition  Example 

x  x variable  unknown value to find  when 2 x = 4, then x = 2 
≡  equivalence  identical 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 
y ∝ x when y = kx, k constant 
∞  lemniscate  infinity symbol  
≪  much less than  much less than  1 ≪ 1000000 
≫  much greater than  much greater than  1000000 ≫ 1 
( )  parentheses  calculate expression inside first  2 * (3+5) = 16 
[ ]  brackets  calculate expression inside first  [(1+2)*(1+5)] = 18 
{ }  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! = 1*2*3*4 = 24 
 x   vertical bars  absolute value   5  = 5 
f ( x)  function of x  maps values of x to f(x)  f ( x) = 3 x+5 
( f ∘ g)  function composition  ( f ∘ g) ( x) = f ( g( x))  f ( x)=3 x, g( x)= x1 ⇒( f ∘ g)( x)=3( x1) 
( 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 = t _{ 1 } t _{ 0 } 
∆  discriminant  Δ = b ^{2}  4 ac  
∑  sigma  summation  sum of all values in range of series  ∑ x _{i}= x _{1} +x _{2} +...+x _{ n } 
∑∑  sigma  double summation  
∏  capital pi  product  product of all values in range of series  ∏ x _{i}=x _{1} ∙x _{2} ∙...∙x _{ n } 
e  e constant / Euler's number  e = 2.718281828...  e = lim (1+1/ x) ^{x} , x→∞ 
γ  EulerMascheroni constant  γ = 0.5772156649...  
φ  golden ratio  golden ratio constant  
π  pi constant 
π = 3.141592654... is the ratio between the circumference and diameter of a circle 
c = π⋅ d = 2⋅ π⋅ r 
Symbol  Symbol Name  Meaning / definition  Example 

·  dot  scalar product  a · b 
×  cross  vector product  a × b 
A⊗ B  tensor product  tensor product of A and B  A ⊗ B 
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 vertical bars  norm  
A ^{ T }  transpose  matrix transpose  ( A ^{T}) _{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 A ^{1} = I  
rank( A)  matrix rank  rank of matrix A  rank( A) = 3 
dim( U)  dimension  dimension of matrix A  dim( U) = 3 
Symbol  Symbol Name  Meaning / definition  Example 

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 density function (pdf)  P( a ≤ x ≤ b) = ∫ f ( x) dx  
F( x)  cumulative distribution function (cdf)  F( x) = P( X≤ x)  
μ  population mean  mean of population values  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 Y  E( X  Y=2) = 5 
var( X)  variance  variance of random variable X  var( X) = 4 
σ ^{2}  variance  variance of population values  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 
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 
ρ _{ X , Y }  correlation  correlation of random variables X and Y  ρ _{ X , Y } = 0.6 
∑  summation  summation  sum of all values in range of series  
∑∑  double summation  double summation  
Mo  mode  value that occurs most frequently in population  
MR  midrange  MR = ( x _{max} + x _{min} )/2  
Md  sample median  half the population is below this value  
Q _{1}  lower / first quartile  25% of population are below this value  
Q _{2}  median / second quartile  50% of population are below this value = median of samples  
Q _{3}  upper / third quartile  75% of population are below this value  
x  sample mean  average / arithmetic mean  x = (2+5+9) / 3 = 5.333 
s _{} ^{ 2 }  sample variance  population samples variance estimator  s ^{} ^{2} = 4 
s  sample standard deviation  population samples standard deviation estimator  s = 2 
z _{x}  standard score  z _{x} = ( x x) / s _{x}  
X ~  distribution of X  distribution of random variable X  X ~ N(0,3) 
N( μ, σ ^{ 2 })  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  
gamma( c, λ)  gamma distribution  f ( x) = λ c x ^{c1} e ^{ λx } / Γ( c), x≥0  
χ ^{ 2 }( k)  chisquare distribution  f ( x) = x ^{k} ^{/21} e ^{ x/2 } / ( 2 ^{k/2 }Γ( k/2) )  
F ( k _{ 1 } , k _{ 2 })  F distribution  
Bin( n, p)  binomial distribution  f ( k) = _{n}C _{k} p ^{k} (1 p) ^{nk}  
Poisson(λ)  Poisson distribution  f ( k) = λ ^{k}e ^{ λ } / k!  
Geom( p)  geometric distribution  f ( k) = p(1 p) ^{ k}  
HG( N, K, n)  hypergeometric distribution  
Bern( p)  Bernoulli distribution 
Symbol  Symbol Name  Meaning / definition  Example 

n!  factorial  n! = 1⋅2⋅3⋅...⋅ n  5! = 1⋅2⋅3⋅4⋅5 = 120 
_{n}P _{k}  permutation  _{5} P _{3} = 5! / (53)! = 60  
_{n}C _{k}

combination  _{5} C _{3} = 5!/[3!(53)!]=10 
Symbol  Symbol Name  Meaning / definition  Example 

{ }  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} 
2 ^{ A }  power set  all subsets of A  
power set  all subsets of A  
A = B  equality  both sets have the same members  A={3,9,14}, B={3,9,14}, A=B 
A ^{c}  complement  all the objects that do not belong to set A  
A \ B  relative complement  objects that belong to A and not to B  A = {3,9,14}, B = {1,2,3}, AB = {9,14} 
A  B  relative complement  objects that belong to A and not to B  A = {3,9,14}, B = {1,2,3}, AB = {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} 
a∈A  element of, belongs to 
set membership  A={3,9,14}, 3 ∈ A 
x∉A  not element of  no set membership  A={3,9,14}, 1 ∉ A 
( a, b)  ordered pair  collection of 2 elements  
A×B  cartesian product  set of all ordered pairs from A and B  A×B = {( a, b) a∈A , b∈B} 
A  cardinality  the number of elements of set A  A={3,9,14}, A=3 
#A  cardinality  the number of elements of set A  A={3,9,14}, #A=3 
  vertical bar  such that  A={x3<x<14} 
alephnull  infinite cardinality of natural numbers set  
alephone  cardinality of countable ordinal numbers set  
Ø  empty set  Ø = { }  C = {Ø} 
universal set  set of all possible values  
_{0}  natural numbers / whole numbers set (with zero)  _{0} = {0,1,2,3,4,...}  0 ∈ _{0} 
_{1}  natural numbers / whole numbers set (without zero)  _{1} = {1,2,3,4,5,...}  6 ∈ _{1} 
integer numbers set  = {...3,2,1,0,1,2,3,...}  6 ∈  
rational numbers set  = { x  x= a/ b, a, b∈ }  2/6 ∈  
real numbers set  = { x  ∞ < x <∞}  6.343434∈  
complex numbers set  = { z  z=a+ bi, ∞< a<∞, ∞< b<∞}  6+2 i ∈ 
Symbol  Symbol Name  Meaning / definition  Example 

⋅  and  and  x ⋅ y 
^  caret / circumflex  and  x ^ y 
&  ampersand  and  x & y 
+  plus  or  x + y 
∨  reversed caret  or  x ∨ y 
  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 ⊕ y 
~  tilde  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 
Symbol  Symbol Name  Meaning / definition  Example 

limit  limit value of a function  
ε  epsilon  represents a very small number, near zero  ε → 0 
e  e constant / Euler's number  e = 2.718281828...  e = lim (1+1/ x) ^{x} , x→∞ 
y '  derivative  derivative  Lagrange's notation  (3 x ^{3})' = 9 x ^{2} 
y ''  second derivative  derivative of derivative  (3 x ^{3})'' = 18 x 
y ^{ ( n) }  nth derivative  n times derivation  (3 x ^{3}) ^{(3)} = 18 
derivative  derivative  Leibniz's notation  d(3 x ^{3})/ dx = 9 x ^{2}  
second derivative  derivative of derivative  d ^{2}(3 x ^{3})/ dx ^{2} = 18 x  
nth derivative  n times derivation  
time derivative  derivative by time  Newton's notation  
time second derivative  derivative of derivative  
D _{x }y  derivative  derivative  Euler's notation  
D _{x} ^{ 2 } y  second derivative  derivative of derivative  
partial derivative  ∂( x ^{2}+ y ^{2})/∂ x = 2 x  
∫  integral  opposite to derivation  ∫ f(x)dx 
∫∫  double integral  integration of function of 2 variables  ∫∫ f(x,y)dxdy 
∫∫∫  triple integral  integration of function of 3 variables  ∫∫∫ f(x,y,z)dxdydz 
∮  closed contour / line integral  
∯  closed surface integral  
∰  closed volume integral  
[ a, b]  closed interval  [ a, b] = { x  a ≤ x ≤ b}  
( a, b)  open interval  ( a, b) = { x  a < x < b}  
i  imaginary unit  i ≡ √ 1  z = 3 + 2 i 
z*  complex conjugate  z = a+ bi → z*= a bi  z* = 3  2 i 
z  complex conjugate  z = a+ bi → z = a bi  z = 3  2 i 
Re( z)  real part of a complex number  z = a+ bi → Re( z)= a  Re(3  2 i) = 3 
Im( z)  imaginary part of a complex number  z = a+ bi → Im( z)= b  Im(3  2 i) = 2 
 z   absolute value/magnitude of a complex number   z =  a+ bi = √( a ^{2}+ b ^{2})  3  2 i = √13 
arg( z)  argument of a complex number  The angle of the radius in the complex plane  arg(3 + 2 i) = 33.7° 
∇  nabla / del  gradient / divergence operator  ∇ f ( x, y, z) 
vector  
unit vector  
x * y  convolution  y( t) = x( t) * h( t)  
Laplace transform  F( s) = { f ( t)}  
Fourier transform  X( ω) = { f ( t)}  
δ  delta function  
∞  lemniscate  infinity symbol 
Name  Western Arabic  Roman  Eastern Arabic  Hebrew 

zero  0  ٠  
one  1  I  ١  א 
two  2  II  ٢  ב 
three  3  III  ٣  ג 
four  4  IV  ٤  ד 
five  5  V  ٥  ה 
six  6  VI  ٦  ו 
seven  7  VII  ٧  ז 
eight  8  VIII  ٨  ח 
nine  9  IX  ٩  ט 
ten  10  X  ١٠  י 
eleven  11  XI  ١١  יא 
twelve  12  XII  ١٢  יב 
thirteen  13  XIII  ١٣  יג 
fourteen  14  XIV  ١٤  יד 
fifteen  15  XV  ١٥  טו 
sixteen  16  XVI  ١٦  טז 
seventeen  17  XVII  ١٧  יז 
eighteen  18  XVIII  ١٨  יח 
nineteen  19  XIX  ١٩  יט 
twenty  20  XX  ٢٠  כ 
thirty  30  XXX  ٣٠  ל 
forty  40  XL  ٤٠  מ 
fifty  50  L  ٥٠  נ 
sixty  60  LX  ٦٠  ס 
seventy  70  LXX  ٧٠  ע 
eighty  80  LXXX  ٨٠  פ 
ninety  90  XC  ٩٠  צ 
one hundred  100  C  ١٠٠  ק 
Upper Case Letter  Lower Case Letter  Greek Letter Name  English Equivalent  Letter Name Pronounce 

Α  α  Alpha  a  alfa 
Β  β  Beta  b  beta 
Γ  γ  Gamma  g  gama 
Δ  δ  Delta  d  delta 
Ε  ε  Epsilon  e  epsilon 
Ζ  ζ  Zeta  z  zeta 
Η  η  Eta  h  ehta 
Θ  θ  Theta  th  teta 
Ι  ι  Iota  i  iota 
Κ  κ  Kappa  k  kapa 
Λ  λ  Lambda  l  lamda 
Μ  μ  Mu  m  myoo 
Ν  ν  Nu  n  noo 
Ξ  ξ  Xi  x  xee 
Ο  ο  Omicron  o  omeecron 
Π  π  Pi  p  payee 
Ρ  ρ  Rho  r  row 
Σ  σ  Sigma  s  sigma 
Τ  τ  Tau  t  taoo 
Υ  υ  Upsilon  u  oopsilon 
Φ  φ  Phi  ph  fee 
Χ  χ  Chi  ch  khee 
Ψ  ψ  Psi  ps  psee 
Ω  ω  Omega  o  omega 
Number  Roman numeral 

0  not defined 
1  I 
2  II 
3  III 
4  IV 
5  V 
6  VI 
7  VII 
8  VIII 
9  IX 
10  X 
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 
90  XC 
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 
The symbol 'âˆš' denotes the square root of a number. It indicates the value that, when multiplied by itself, results in the original number. For example, 'âˆš9' equals 3 because 3 multiplied by 3 is 9.