Mathematical Symbols

List of all mathematical symbols and signs - meaning and examples.

 Basic math symbols
 Geometry symbols
 Algebra symbols
 Probability & statistics symbols
 Set theory symbols
 Logic symbols
 Calculus & analysis symbols
 Number symbols
 Greek symbols
 Roman numerals

Basic math symbols

Symbo Symbol Name Meaning / definition Example
l

= 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 approximation sin(0.01) ≈ 0.01,

equal x ≈ y means x is
approximately equal to y
Symbo Symbol Name Meaning / definition Example
l
Symbo Symbol Name Meaning / definition Example
l
Symbo Symbol Name Meaning / definition Example
l

> strict greater than 5>4

inequality 5 is greater than 4

< strict less than 4<5

inequality 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 2 × (3+5) = 16

inside first

[] brackets calculate expression [(1+2)×(1+5)] = 18

inside first

+ plus sign addition 1+1=2

− minus sign subtraction 2−1=1

± plus - minus both plus and minus 3 ± 5 = 8 or -2

operations

± minus - plus both minus and plus 3 ∓ 5 = -2 or 8

operations

* asterisk multiplication 2*3=6

× times sign multiplication 2×3=6

⋅ multiplication multiplication 2⋅3=6

dot
Symbo Symbol Name Meaning / definition Example
l

÷ division sign / division 6÷2=3

obelus

/ division slash division 6/2=3

— horizontal line division / fraction

mod modulo remainder calculation 7 mod 2 = 1

. period decimal point, decimal 2.56 = 2+56/100

separator
b 3

a power exponent 2  = 8

a^b caret exponent 2 ^ 3 = 8

√a square root √a  ⋅ √a   = a √9 = ±3

√a  ⋅  √a   ⋅  √a   = a

3 3 3 3 3

√a cube root √8 = 2

4 4

√a √a ⋅  √a  ⋅  √a  ⋅  √a = a √16 = ±2

4 4 4 4
fourth root
n
n for n=3,  √8 = 2
√a n-th root  
(radical)

% 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
ppt per-trillion 1ppt = 10-12 10ppt × 30 = 3×10-10

Geometry symbols
Symbo Symbol Name Meaning / definition Example
l

∠ angle formed by two rays ∠ABC = 30°

measured angle   ABC = 30°

spherical   AOB = 30°
angle

∟ right angle = 90° α = 90°

° degree 1 turn = 360° α = 60°

Symbo Symbol Name Meaning / definition Example
l
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 ∆ABC≅ ∆XYZ

and size

~ 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

is the ratio between the circumference and diameter of a circle

rad radians radians angle unit 360° = 2π rad

c
radians radians angle unit 360° = 2π c
grad gradians / grads angle unit 360° = 400 grad
gons
g
gradians / grads angle unit 360° = 400 g
gons

Algebra symbols
Symbol Symbol Name Meaning / Example
definition

x x variable unknown value to when 2x = 4, then x = 2

find

≡ equivalence identical to  
Symbol Symbol Name Meaning / Example
definition

≜ equal by equal by  
definition definition

:= equal by equal by  
definition definition

~ approximately weak approximation 11 ~ 10

equal

≈ approximately approximation sin(0.01) ≈ 0.01

equal

∝ proportional to proportional to y ∝ x when y = kx, kconstant

∞ lemniscate infinity symbol  

≪ much less than much less than 1 ≪ 1000000

≫ much greater than much greater than 1000000 ≫ 1

() parentheses calculate 2 * (3+5) = 16

expression inside
first

[] brackets calculate [(1+2)*(1+5)] = 18

expression inside
first

{} braces set  

⌊x⌋ floor brackets rounds number to ⌊4.3⌋ = 4

lower integer

⌈x⌉ ceiling brackets rounds number to ⌈4.3⌉ = 5

upper integer

x! exclamation mark factorial 4! = 1*2*3*4 = 24

| x | single vertical absolute value | -5 | = 5

bar

f (x) function of x maps values of x f (x) = 3x+5

to f(x)
(f ∘ g) function (f ∘ g) (x) = f (g(x)) f (x)=3x,g(x)=x-1 ⇒(f ∘ g)
composition (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]

Symbol Symbol Name Meaning / Example
definition

∆ delta change / ∆t = t - t

1  0

difference

∆ discriminant Δ = b2 - 4ac  

∑ sigma summation - sum of ∑ xi= x1+x2+...+xn

all values in
range of series

∑∑ sigma double summation

∏ capital pi product - product ∏ xi=x1∙x2∙...∙xn

of all values in
range of series

e e constant / e = 2.718281828... e = lim (1+1/x)x , x→∞

Euler's number

γ Euler-Mascheroni γ = 0.5772156649...  
constant

φ golden ratio golden ratio  

constant

π pi constant π = 3.141592654... c = π⋅d = 2⋅π⋅r

is the ratio between the
circumference and diameter of a
circle

Linear Algebra Symbols

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  
Symbol Symbol Name Meaning / definition Example
T

A 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

-1

A  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

Probability and statistics symbols

Symbol Symbol Name Meaning / Example
definition

P(A) probability probability of P(A) = 0.5

function event A

P(A ⋂ B) probability of probability that P(A⋂B) = 0.5

events of events A and B
intersection

P(A ⋃ B) probability of probability that P(A⋃B) = 0.5

events union of events A or B

P(A | B) conditional probability of P(A | B) = 0.3

probability event A given
function event B occured

f (x) probability P(a ≤ x ≤ b) = ∫  

density f  (x)  dx
function (pdf)

F(x) cumulative F(x) = P(X≤ x)  

distribution
function (cdf)

μ population mean mean of μ = 10

population values

E(X) expectation expected value of E(X) = 10

value random variable X

E(X | Y) conditional expected value of E(X | Y=2) = 5

expectation random variable X
given Y

var(X) variance variance of var(X) = 4

random variable X
2 2 

σ variance variance of σ =4
 Symbol Symbol Name Meaning / Example
definition
population values

std(X) standard standard std(X) = 2

deviation deviation of
random variable X

σ X
standard standard σ     =  2
X

deviation deviation value

of random
variable X
median middle value of
random variable x

cov(X,Y) covariance covariance of cov(X,Y) = 4

random variables
X and Y

corr(X,Y) correlation correlation of corr(X,Y) = 0.6

random variables
X and Y

ρX,Y
correlation correlation of ρ  = 0.6
X,Y

random variables
X and Y

∑ summation summation - sum

of all values in
range of series

∑∑ double double summation

summation

Mo mode value that occurs  

most frequently
in population

MR mid-range MR = (xmax+xmin)/2  

Md sample median half the  

population is
below this value

Q1 lower / first 25% of population  

quartile are below this
value

Q2 median / second 50% of population  

quartile are below this
value = median of
samples
 Symbol Symbol Name Meaning / Example
definition

Q3 upper / third 75% of population  

quartile are below this
value

x sample mean average / x = (2+5+9) / 3 = 5.333

arithmetic mean
2

s  sample variance population s  2 = 4

samples variance
estimator

s sample standard population s = 2

deviation samples standard
deviation
estimator

zx standard score zx = (x-x) / sx  

X ~ distribution o distribution of X ~ N(0,3)

f X random variable X
2
N(μ,σ ) normal gaussian X ~ N(0,3)
distribution distribution
U(a,b) uniform equal probability X ~ U(0,3)
distribution in range a,b 
exp(λ) exponential f  (x)  = λe-λx , x≥0  
distribution
gamma(c, λ) gamma f  (x)  = λ c xc-1e-λx /  
distribution Γ(c), x≥0
 2
χ (k) chi-square f  (x)  = xk/2-1e-x/2 /  
distribution ( 2k/2 Γ(k/2) )
F  (k , k )
1 2
F distribution    
Bin(n,p) binomial f  (k)  =  nCk pk(1-p)n-k  
distribution
Poisson(λ) Poisson f  (k)  = λke-λ / k!  
distribution
Geom(p) geometric f  (k)  =  p(1-p)  k  
distribution
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

n Pk permutation P3  =  5! / (5-3)! = 60

5

nCk combination C3  =  5!/[3!(5-3)!]=10

5

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 intersection objects that belong to set A A ∩ B = {9,14}
and set B
A ∪ B union objects that belong to set A A ∪ B =
or set B {3,7,9,14,28}
A ⊆ B subset A is a subset of B. set A is {9,14,28} ⊆
included in set B. {9,14,28}
A ⊂ B proper subset / strict A is a subset of B, but A is {9,14} ⊂
subset not equal to B. {9,14,28}
A ⊄ B not subset set A is not a subset of set {9,66} ⊄
B {9,14,28}
A ⊇ B superset A is a superset of B. set A {9,14,28} ⊇
includes set B {9,14,28}
A ⊃ B proper superset / A is a superset of B, but B {9,14,28} ⊃
strict superset is not equal to A. {9,14}
A ⊅ B not superset set A is not a superset of {9,14,28} ⊅
set B {9,66}
A
2 power set all subsets of A  
power set all subsets of A  

A = B equality both sets have the same A={3,9,14},

members B={3,9,14},
A=B
Symbol Symbol Name Meaning / definition Example
Ac complement all the objects that do not  
belong to set A
A \ B relative complement objects that belong to A and A = {3,9,14},
not to B B = {1,2,3},
A-B = {9,14}
A - B relative complement objects that belong to A and A = {3,9,14},
not to B B = {1,2,3},
A-B = {9,14}
A ∆ B symmetric difference objects that belong to A or A = {3,9,14},
B but not to their B = {1,2,3},
intersection A∆B=
{1,2,9,14}
A ⊖ B symmetric difference objects that belong to A or A = {3,9,14},
B but not to their B = {1,2,3},
intersection A⊖B=
{1,2,9,14}

a∈A element of, set membership A={3,9,14}, 3

belongs to ∈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| cardinality the number of elements of A={3,9,14}, |A|

set A =3

#A cardinality the number of elements of A={3,9,14},

set A #A=3
aleph-null infinite cardinality of  
natural numbers set
aleph-one cardinality of countable  
ordinal numbers set

Ø empty set Ø={} C = {Ø}

universal set set of all possible values  

0 natural numbers /  = {0,1,2,3,4,...}
0 0 ∈  0
whole numbers  set
(with zero)
1 natural numbers /  = {1,2,3,4,5,...}
1 6 ∈  1
Symbol Symbol Name Meaning / definition Example
whole numbers  set
(without zero)
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<∞,      6+2i ∈ 
-∞<b<∞}

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

∨ 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    
Symbol Symbol Name Meaning / definition Example

∄ there does not exists    

∴ therefore    

∵ because / since    

Calculus & analysis symbols

Symbol Symbol Name Meaning / definition Example
limit limit value of a function  

ε epsilon represents a very small ε  → 0

number, near zero

e e constant / e = 2.718281828... e = lim

Euler's number (1+1/x)x, x→∞

y ' derivative derivative - Lagrange's (3x3)' = 9x2

notation

y '' second derivative derivative of derivative (3x3)'' = 18x

(n)

y nth derivative n times derivation (3x3)(3) = 18

derivative derivative - Leibniz's d(3x3)/dx = 9x2

notation

second derivative derivative of derivative d2(3x3)/dx2 = 18x

nth derivative n times derivation  

time derivative derivative by time -  

Newton's notation
time second derivative of derivative  
derivative

Dx  y derivative derivative - Euler's  

notation

Dx2y second derivative derivative of derivative  

partial derivative   ∂(x2+y2)/∂x = 2x

∫ integral opposite to derivation ∫ f(x)dx

∫∫ double integral integration of function of ∫∫ f(x,y)dxdy

 Symbol Symbol Name Meaning / definition Example
2 variables

∫∫∫ triple integral integration of function of ∫∫∫ f(x,y,z)dxdydz

3 variables

∮ 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 + 2i

z* complex conjugate z = a+bi → z*=a-bi z* = 3 - 2i

z complex conjugate z = a+bi → z  =  a-bi z = 3 - 2i

∇ nabla / del gradient / divergence ∇f (x,y,z)

operator
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  

Numeral symbols
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
 Name Western Arabic Roman Eastern Arabic Hebrew
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
 

Greek alphabet letters

Upper Case Lower Case Greek Letter English Letter Name
Letter Letter Name Equivalent Pronounce

Α α Alpha a al-fa
 Β β Beta b be-ta

Γ γ Gamma g ga-ma

Δ δ Delta d del-ta

Ε ε Epsilon e ep-si-lon

Ζ ζ Zeta z ze-ta

Η η Eta h eh-ta

Θ θ Theta th te-ta

Ι ι Iota i io-ta

Κ κ Kappa k ka-pa

Λ λ Lambda l lam-da

Μ μ Mu m m-yoo

Ν ν Nu n noo

Ξ ξ Xi x x-ee

Ο ο Omicron o o-mee-c-ron

Π π Pi p pa-yee

Ρ ρ Rho r row

Σ σ Sigma s sig-ma

Τ τ Tau t ta-oo

Υ υ Upsilon u oo-psi-lon

Φ φ Phi ph f-ee

Χ χ Chi ch kh-ee

Ψ ψ Psi ps p-see

Ω ω Omega o o-me-ga

15 XV
Mathematics is the language in which God has written the universe”
― Galileo Galilei

People sometimes have trouble understanding mathematical ideas:

not necessarily because the ideas are difficult,
but because they are being presented in a foreign language—the language of
mathematics.

The language of mathematics makes it easy to express the kinds of thoughts that
mathematicians like to express.
It is:

 precise (able to make very fine distinctions)

 concise (able to say things briefly)

 powerful (able to express complex thoughts with relative ease)

Every language has its vocabulary (the words)

and its rules for combining these words into complete thoughts (the sentences).
Mathematics is no exception.

As a first step in studying the mathematical language,

we will make a very broad classification between the ‘nouns’ of mathematics (used to
name mathematical objects of interest)
and the ‘sentences’ of mathematics (which state complete mathematical thoughts).

DEFINITION expression
An expression is the mathematical analogue of an English noun; it
is a correct arrangement of mathematical symbols used to represent a
mathematical object of interest.

An expression does not state a complete thought;

it does not make sense to ask if an expression is true or false.
The most common expression types are numbers, sets, and functions.

Numbers have lots of different names: for example, the expressions

55 2+32+3 102102 (6−2)+1(6−2)+1 1+1+1+1+11+1+1+1+1

all look different, but are all just different names for the same number.

This simple idea—that numbers have lots of different names—is extremely important
in mathematics!

DEFINITION sentence
A mathematical sentence is the analogue of an English sentence; it
is a correct arrangement of mathematical symbols that states a
complete thought.

Sentences have verbs.

In the mathematical sentence  ‘3+4=73+4=7’ , the verb is ‘==’.

A sentence can be (always) true, (always) false, or sometimes true/sometimes false.

For example, the sentence  ‘1+2=31+2=3’  is true.
The sentence  ‘1+2=41+2=4’  is false.
The sentence  ‘x=2x=2’  is sometimes true/sometimes false: it is true when xx is 22,
and false otherwise.
The sentence  ‘x+3=3+xx+3=3+x’  is (always) true, no matter what number is chosen
for xx.

Click on t

EXAMPLES:

22 is an expression
1+11+1 is an expression
x+1x+1 is an expression
 1+1=21+1=2 is a (true) sentence
1+1=31+1=3 is a (false) sentence
x+1=3x+1=3 is a (sometimes true/sometimes false) sentence

So, xx is to mathematics as  cat  is to English:

hence the title of the book,
        One Mathematical Cat, Please!

Master the ideas from this section

by practicing both exercises at the bottom of this page.

When you're done practicing, move on to:

Basic Addition Practice

 
 
Bilateral symmetry – Geometry
Home/Biology, Math/Bilateral symmetry – Geometry

A butterfly has bilateral symmetry. Thanks to Wikimedia Commons

What is bilateral symmetry?

A shape has bilateral symmetry when it is the same on both sides of a line drawn
down the middle.

Are humans symmetrical?

Venus of Lespugue, ca. 25000 BC, now in the Musee du Quai Branly, Paris

You are (mostly) bilaterally symmetrical. Suppose I drew a line down through your
nose perpendicular to the ground. You’d have one eye, one arm, and one leg on
each side of the line. So you’d be symmetrical. But you wouldn’t be entirely the same
on both sides of the line. You only have one heart, on the left side. And you only
have one liver and one appendix, on the right side.

This butterfly has bilateral symmetry too. It has one wing and one antenna on each
side of the red line.

T’ang Dynasty landscape painting and poem

Because symmetry makes it clear that both sides of you are healthy and undamaged,
many animals, including humans, tend to find partners more attractive when they are
the same on both sides. That’s why people usually part their hair in the middle, and
wear symmetrical clothing. But sometimes we like to mess with that a little bit, as a
sort of tease, or to show we’re not boring. We might part our hair on the side, or wear
an asymmetrical skirt.
How to figure out the area of an isosceles triangle.

Chinese painters, for example, created asymmetrical but carefully balanced

paintings, so they would look interesting.

Geometry and bilateral symmetry

Some geometric shapes have bilateral symmetry, while others don’t. For example,
a square has bilateral symmetry, and so does a circle. So does a rectangle.
An isosceles triangle also has bilateral symmetry.

This is a rhombus, divided so you can figure out the area.

But most parallelograms do not have bilateral symmetry. A rhombus does not have

bilateral symmetry vertically, but it does if you draw a diagonal line connecting two
opposite corners. (Thanks to Jeffrey Paules for pointing this out!)

Three-dimensional solids can have bilateral symmetry, too. A sphere and

a cube both have bilateral symmetry.

Learn by doing: painting bilateral symmetry

Perpendicular
Rectangles
Squares
Circles
Triangles
More about Geometry
 

Bibliography and further reading about geometry:

   

More Geometry
More about Math
Spiral, plane curve that, in general, winds around a point while moving ever
farther from the point. Many kinds of spiral are known, the first dating from the
days of ancient Greece. The curves are observed in nature, and human beings
have used them in machines and in ornament, notably architectural—for
example, the whorl in an Ionic capital. The two most famous spirals are
described below.

Although Greek mathematician Archimedes did not discover the spiral that bears

his name (see figure), he did employ it in his On Spirals (c. 225 BC) to square the
circle and trisect an angle. The equation of the spiral of Archimedes is r = aθ, in
which a is a constant, r is the length of the radius from the centre, or beginning,
of the spiral, and θ is the angular position (amount of rotation) of the radius. Like
the grooves in a phonograph record, the distance between successive turns of the
spiral is a constant—2πa, if θ is measured in radians.

The equiangular, or logarithmic, spiral (see figure) was discovered by the French

scientist René Descartes in 1638. In 1692 the Swiss mathematician Jakob
Bernoulli named it spira mirabilis(“miracle spiral”) for its mathematical
properties; it is carved on his tomb. The general equation of the logarithmic
spiral is r = aeθ cot b, in which r is the radius of each turn of the spiral, a and bare
constants that depend on the particular spiral, θ is the angle of rotation as the
curve spirals, and e is the base of the natural logarithm. Whereas successive
turns of the spiral of Archimedes are equally spaced, the distance between
successive turns of the logarithmic spiral increases in a geometric progression
(such as 1, 2, 4, 8,…). Among its other interesting properties, every ray from its
centre intersects every turn of the spiral at a constant angle (equiangular),
represented in the equation by b. Also, for b = π/2 the radius reduces to the
constant a—in other words, to a circle of radius a. This approximate curve is
observed in spider webs and, to a greater degree of accuracy, in the chambered
mollusk, nautilus (seephotograph),
 Spiral
QUICK FACTS

View Media Page

Set
Set, In mathematics and logic, any collection of objects (elements), which may
be mathematical (e.g., numbers, functions) or not. The intuitive idea of a set is
probably even older than that of number. Members of a herd of animals, for
example, could be matched with stones in a sack without members of either set
actually being counted. The notion extends into the infinite. For example, the set
of integers from 1 to 100 is finite, whereas the set of all integers is infinite. A set
is commonly represented as a list of all its members enclosed in braces. A set
with no members is called an empty, or null, set, and is denoted ∅. Because an
infinite set cannot be listed, it is usually represented by a formula that generates
its elements when applied to the elements of the set of counting numbers. Thus,
{2x | x = 1,2,3,...} represents the set of positive even numbers (the vertica


set theory


… created a theory of abstract sets of entities and made it into a
mathematical discipline. This theory grew out of his investigations of some
concrete problems regarding certain types of infinite sets of real numbers. A
set, wrote Cantor, is a collection of definite, distinguishable objects of
perception or thought…





mathematics: Cantor


… on the concept of a set. Cantor had begun work in this area because of his
interest in Riemann’s theory of trigonometric series, but the problem of what
characterized the set of all real numbers came to occupy him more and more. He
began to discover unexpected properties of sets.…





history of logic: Set theory


…as a theory of infinite sets and of kinds of infinity but also as a universal
language in which mathematical theories can be formulated and discussed.
Because it covered much of the same ground as higher-order logic, however, set
theory was beset by the same paradoxes that had plagued higher-order…


A mathematical sentence, also called mathematical statement, statement, or proposal,
is a sentence that can be identified as either true or false.

For example, " 6 is a prime number " is a mathematical sentence or simply statement.

Of course, " 6 is a prime number " is a false statement!

More examples of mathematical sentences or
statements
6 + 8 = 2  × 7  ( This is a true statement )

9 + 1 = 0 + 11 ( This is a false statement )

If an integer n is odd, then 2n is an even number.( This is a true statement)

A stop sign is not in the shape of an octagon (This is a false statement)

What are Fractals?

A fractal is a never-ending pattern. Fractals are infinitely complex patterns

that are self-similar across different scales. They are created by repeating a
simple process over and over in an ongoing feedback loop. Driven by
recursion, fractals are images of dynamic systems – the pictures of Chaos.
Geometrically, they exist in between our familiar dimensions. Fractal patterns
are extremely familiar, since nature is full of fractals. For instance: trees,
rivers, coastlines, mountains, clouds, seashells, hurricanes, etc. Abstract
fractals – such as the Mandelbrot Set – can be generated by a computer
calculating a simple equation over and set