"What's
Special About This Number" Facts
eople
have always been fascinated by NUMBERS... Numbers
are actually basic elements of mathematics used for counting,
measuring, ranking, comparing quantities, and solving equations.
Numbers have unique properties: for some ones of us they are
merely concise symbols manipulated according to arbitrary rules,
for others numbers carry occult powers and mystic virtues.
Almost all numeration systems start as simple tally
marks, using single strokes to represent each additional unit.
The first known use of numbers dates back to around 30,000 BC when
tally marks were precisely used by stone age people.
To show that each number is unique and has its own beauty, we have
collected for you a huge amount of facts pertaining to the magical
world of numbers, covering a range of different topics including mathematics, history, philosophy, psychology, symbolism, etymology, language,
and/or ethnology...
The
number facts in this 'Numberopedia' are available as features for
print and electronic publishing.
A
free 197page eBook containing a huge collection of number
facts is available for
you to download.

If
you got a distinctive fact about any number listed here you think Archimedes'
Lab community might enjoy, why not post it here? 
Conoscete
un numero con delle proprietà originali? Contattateci! 
Connaissezvous
un nombre avec des propriétés étonnantes? Contacteznous! 
Number
list: lista
dei numeri (it), liste des nombres (fr), lista de números (es,
por), Liste besonderer Zahlen (ger), getallen en getalverzamelingen (du), seznam čísel (cz), 數表 (ch), 数の一覧 (jap), список чисел (ru), שמות מספרים (he).
06  712  1323  2469  70200  201786  5H0P
NaN 
NaN (Not a Number)
is, in computing, a value (or symbol) that is usually produced
as the result of an operation on invalid input operands, especially
in floatingpoint
calculations. NaNs are close to some undefined or inderterminate expressions
in mathematics. In short, NaN is not really a number but a symbol
that represents a numerical quantity whose magnitude cannot be
determined by the operating system.
= 1
= log (n) = ln (n)
= 0 / 0
= 0^{0}
= 1^{∞}
= ∞^{0}
= ∞ / ∞ = ∞ / ∞ = ∞ / ∞ =
∞ / ∞
= 0 x ∞ = 0 x ∞
= (∞) + ∞ = ∞ + (∞)
= ln 0 / ln ±∞
= e^{±∞} x ln 0
= (m / ±∞) x (n / 0) if m ±∞ and n 0
The infinite
series ∑ = 1  1 + 1  1 + 1  1 + … , called Grandi's
series, also written:
is a
divergent series, meaning that it lacks a sum in the usual sense...
In fact, if you treat this series like a telescoping series and/or
use different bracketing procedures to sum it, you may obtain contradictory
results...
∑ 
=
1  1 + 1  1 + 1  1 + 1  1 + ... 
= 
∑_{1} 
=
(11) + (11) + (11) + (11) + ... 
=
0 
∑_{2} 
=
1 + (1+1) + (1+1) + (1+1) + ... 
=
1 
∑_{3} 
=
1 + (11+11+11+ ...) = 1 + ∑ => ∑ 
=
1/2 
As shown
above, it appears to equal 0 and 1, yet in some sense 'sums' to 1/2,
producing a paradox... The error here is that the associative law
cannot be applied freely to an infinite sum unless the sum is absolutely
convergent. We can say that the sum of Grandi's series is NaN. 

= i,
is the imaginary
unit of any imaginary number. Discovered by the Italian mathematician Girolamo
Cardano.
An imaginary
number is a number of the form bi where
'b' is a real number, 'i' is the square root of 1, for
b 0.
Imaginary numbers (and complex numbers in general) are essential
for describing physical reality and have concrete applications
in: electromagnetism, signal processing, control theory, quantum
mechanics, cryptology, and cartography...
is the
result of the folowing equations:
x^{2} + 1 = 0 (for x i)
Square
roots of negative numbers other than 1 can be written under
the form:
n
= in
e^{i}^{/2} =
cos (/2)
+ i sin (/2) = i
i^{i} = e^{}^{/2} ≈ 0.207879576...
(cf. i to
the i is a Real Number)
the reciprocal of i is i:
i^{1} = 1/i = i/i^{2 }= i/1
= i
Powers
of i repeat in a definite pattern (i,
1, i, 1, ...):
i^{1} = i
i^{2} = 1
i^{3} = i^{2}i = (1)i =
i
i^{4} = i^{3}i = (i)i =
(i^{2}) = (1) = 1
i^{5} = i^{4}i = (1)i = i
...
Multiplicative
table with i

1 
1 
i 
i 
1 
1 
1 
i 
i 
1 
1 
1 
i 
i 
i 
i 
i 
1 
1 
i 
i 
i 
1 
1 
The
first roots of i are:
^{1}i = i
^{2}i = ±(1
+ i)/2
^{3}i =
(3
+ i)/2
^{4}i = ±(i(2
 2)
+ (2
+ 2))/2
^{5}i = i
A 'paradox'
(or a math
fallacy?) with i:
a) 1
= 1
b) (1/1)
= (1/1)
c) 1/1
= 1/1
d) (1)^{2} =
(1)^{2}
e) 1 = 1 and then 2 = 0 ??? Is this possible? Can you discover
what led to this poetic licenced conclusion?

1 
is
the first and largest negative integer.
is a Heegner
number.
In
number theory, "Wilson's theorem" states that a natural
number greater than 1 is a prime number if and only if:
(n  1)! == 1(mod n)
Multiplying
any number by 1 is equivalent to changing the sign on the number.
x^{1} =
1/x
1/(1)
= (1)/1
=
e^{i}
≈ sin2017(2)^{1/5}

0
⠼⠚

is
a separate and special entity called 'Identity
element'. 0 is actually the identity element under addition
for the real numbers, since if a is any real number, a +
0 = 0 + a = a. Mathematicians refers
to 0 as the additive identity (or better said, the reflexive
identity of addition).
is considered
to be a purely imaginary number: 0 is the only complex number
which is both real and purely imaginary.
identifies
the concept of "almost" impossible in probability.
More generally, the concept of almost nowhere in measure theory.
0 = n / ∞
0 =
log_{a}1
a^{0} = 1, only when a doesn't
equal 0.
By convention,
you cannot divide any number by zero.
In theory, zero multiplied by infinity is undetermined (as is zero
divided by zero).
It is
the only integer (actually, the only real number) that is neither
negative nor positive. The question whether 'zero' is odd or
even seems to be totally subjective!
Mathematical
equations with one or more unknown factors are solved by equalizing
them to zero.
is the
number of n x n magic squares for n =
2.
The
difference between 3, 30 and 300 is only some extra zeros, but
those little circles are actually one of the world's greatest
inventions! As early as 200 B.C., Hindu scholars were working
with nine oddly shaped symbols and
a dot that eventually would bring order out of a world of mathematical
chaos. The dot and nine symbols were the earliest known forerunners
of the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Comprised of only
ten symbols and based on multiples of ten, the Hindu numeral
system was easily learned and easily used. Who first thought
of using a dot (bindu, in sanskrit) as the tenth number
is not known. But it can be supposed that a Hindu, working on
his abacus, wanted to keep a written record of the answers on
his abacus. One day he used a symbol '.' which
he called shunya ()
to indicate a column on his counting board in which he had moved
no beads... Shunya, the dot, was originally not zero the number,
but merely a mark to indicate empty space.
The
word "zero" was coined by the Italian mathematician
Leonardo Pisano, said Fibonacci. He transformed the Arabic word
'صِفْر', sifr (from the
semitic root s.p.r., 'empty') into Italian equivalent zefiro,
shortened to zero afterwards. Many languages have adopted
the word "zero": english, catalan, french (zéro),
portuguese, romanian, spanish (cero), wallon (zérô),
albanian, polish, japanese...
Europe is divided into two regions: the 'zero region'
(see above) and the 'nullus region' (nullus,
'zero' in Latin). The 'nullus region' includes the Germanic, the
Skandinavian and some Slavonic countries. The following is a table
of the number 0 in a sample of the languages of the 'nullus region':
Dutch 
nul 
Czech 
nula 
German 
null 
Russian 
nol' 
Swedish 
noll 
Slovak 
nula 
The
Greek word for zero is μηδεν, read as
'meden', which means, etymologically, not even one (i.e. nothing).
The Oracle of Delphi in ancient Greece had a wise motto, like
this: "meden agan"  nothing too much (or nothing
in excess)... 
posted by George Pantazis
Love is
a score of 0 in tennis.
What
English word contains 0 vowels?
Answer: hymn, gypsyfy, myth, rhythm, sylph, syzygy, etc.
The
Czech phrase: Strc prst skrz krk meaning "thrust
finger through neck", contains 0 vowels
and semivowels!
In German,
the expression in Null Komma nichts (in
zero point nothing) means 'in a trice'
In Italian,
the expression a chilometri zero (in
zero kilometers from any location) means 'local'. For instance, un
gelato a chilometri zero translates as 'an icecream produced
with local products'.
There
are no letters assigned to the numbers 0 and 1 on
a phone dial. These numbers remain unassigned
because they are socalled 'flag' numbers, kept for special purposes
such as emergency or operator services.
"Wuji" (Number
0), in the Mystical Numbers of Taoism, represents the Null, the
Chaos, the Origin and the End.
Joke:
Chuck Norris can divide by zero. (More Chuck
Norris facts)
Zero
Star Hotel (Null Stern Hotel)
The "Null
Stern", or "Zero
Star" Hotel is a cross between a hostel and an art installation
by Swiss concept artists Frank and Patrik Riklin. This hotel is
actually a converted bomb shelter...

1/2 
=
sinus(30°)
= cosinus(60°)
= cosinus(30°)/3
= 1/3 + 1/6
Using
all digits from 1 to 9 once:
= 6 729 / 13 458
= 9 327 / 18 654 (there are 10 other possibilities
to write similar fractions by using all digits from 1 to 9 once)
= (123  45) / (67 + 89)
Using
the same number twice, but just swapping the place of ONE digit:
=
105263157894736842 / 210526315789473684
= 157894736842105263 / 315789473684210526
= 210526315789473684 / 421052631578947368
= 263157894736842105 / 526315789473684210
= 315789473684210526 / 631578947368421052
= 368421052631578947 / 736842105263157894
= 421052631578947368 / 842105263157894736
= 473684210526315789 / 947368421052631578
Numbers with such properties are called 'parasit
or parasitic numbers'.
≈ angular
magnitude of the Sun, and of the Moon.
In a
group of 23 people, at least two have the same birthday with
the probability greater than 1/2.
Another
'paradox'
(or math
fallacy?) involving 1/2:
Since (1/2)^{2} = 1/4
and (1/2)^{3} = 1/8
then (1/2)^{3} < (1/2)^{2
}using the logarithms we obtain:
3 log (1/2) < 2 log (1/2)
and after dividing by log (1/2):
3 < 2
How can that be?
The
population of the Roman Empire under Augustus was
about one hundred millions, of which more than one half were
slaves!
There
is a 1/2 percent probability you are related to Genghis Khan...
An international
group of geneticists studying Ychromosome data have found
that nearly 8 percent of the men living in the region of the former
Mongol empire carry ychromosomes that are nearly identical to
those of Genghis Khan, the fearsome Mongolian warrior of the 13th
century, whose adopted name means "Universal Ruler" in
Altaic, his native tongue. That translates to 0.5 percent of the
global population in the world (or roughly 16 million descendants
living today).
Did
you know that the Romans too could transcribe unit fractions?
e.g. to write 1/2 they
used the letter S (semis). Knowing that, what represents
the Roman numeral SIX? Obviously
not 6, but 8.5! (10  1  1/2)
In
Italy, "fojetta" (small leaf, in Roman
dialect) is a measure corresponding to half a liter of
wine.
A
typical 'fojetta' >


Tupper's
selfreferential formula
is an amazing formula concocted by Jeff Tupper that, when graphed
in 2 dimensions, can visually reproduce the formula itself:
If one graphs the set of points (x, y)
with 0 < x < 106 and k < y < (k +
17), such that they satisfy the inequality given above, the resulting
selfreferential 'plot' looks like this:

1
⠼⠁
I

is
a separate and special entity called 'Unity' or 'Identity
element'. 1 is actually the identity element under multiplication
for the real numbers, since a x 1 = 1 x a = a.
Mathematicians refers to 1 as the multiplicative identity (or
better said, the reflexive identity of multiplication).
is NOT prime!
Primes or prime
numbers can be poetically described as the 'atoms' of mathematics
 the building blocks of the world of numbers. But, mathematically
speaking: "a prime number is a positive integer with
exactly TWO positive divisors: 1 and itself". Modern
textbooks consider 1 neither prime nor composite, whereas older
texts generally asserted the contrary. In 1859, Henri
Lebesgue stated explicitly that 1 is prime in "Exercices
d'analyse numérique". It is also prime in "Primary
Elements of Algebra for Common Schools and Academies" (1866)
by Joseph Ray, and in "Standard Arithmetic" (1892)
by William J. Milne. A list of primes to 10,006,721 published
in 1914 by Derrick N. Lehmer includes 1 ("List of prime
numbers from 1 to 10,006,721", Carnegie Institution of Washington).
is the
solution of the equation: x^{3}  x = 0 (for x 1,
or 0)
[Note: x^{3}  x can be factorized as follows:
x(x + 1)(x  1)]
is the
only real solution of the equation x^{3} + 3x 
4 = 0
Benford's
law states that in a huge assortment of number sequences
 in listings, tables of statistics, random samples from a
day's stock quotations, a tournament's tennis scores, the populations
of towns, electricity bills in the Solomon Islands, and much
more, the digit 1 tends to occur with probability ∼30%,
much greater than the expected 11.1% (i.e., one digit out of
9). Dr. Nigrini gained recognition by applying a system he
devised based on Benford's Law to some fraud cases in Brooklyn.
The idea underlying his system is that if the numbers in a
set of data like a tax return more or less match the frequencies
and ratios predicted by Benford's Law, the data are probably
honest. But if a graph of such numbers is markedly different
from the one predicted by Benford's Law, he said, "I think
I'd call someone in for a detailed audit".
Mathematicians
define a 'sphere' as the surface of a sphere, not a solid ball,
so a sphere has 2 sides: the outside and the inside. However,
there are also 1sided
surfaces!
f(x) = e^{x} at the point x = 0 is exactly
1.
=0!
Why 0! = 1? Because 4! = 4x3x2x1 and 3! = 3x2x1. Therefore 4! =
4x3! In the same way 3! = 3x2! and 2! = 2x1! So it follows
that 1! = 1x0! Therefore 0! must be equal to 1 or 1! would
be 0... And so 2! would be zero and then 3! and so on.
Simple
math relations:
x^{1} = x and 1^{x} =
1
=
log_{a}a = log_{b}a /
log_{a}b
= a^{0} (for a0)
= a^{2}  (a + 1)(a  1)
= 35  3^{2} 
5^{2}
= 75  7^{2} 
5^{2}
= 1/2 + 1/3 + 1/6
= 1/2 + 1/4 + 1/6 + 1/12
= 1/2^{1} + 1/2^{2} + 1/2^{3} + 1/2^{4} +
1/2^{5 }+ ...
= (2 + 5)^{1/3} +
(2  5)^{1/3}
= (1 + i)(1  i)/2
= sin^{2} (a) + cos^{2} (a)
=  F_{n} x F_{n+3}  F_{n+1} x
F_{n+2}  (F = Fibonacci numbers)
= 1/(1x2)
+ 1/(2x3) + 1/(3x4) + 1/(4x5) + ... + 1/n(n+1)
= 0.9 =
0.9 + 0.09 + 0.009 + 0.0009 + 0.00009 + ...
n =
1
(1
+ 2 + 3 + ... + n)^{1/n} = 1
(sin x)/x =
1
(e^{x} 
1)/x = 1
ln(1
+ x)/x = 1
a^{p1} ≡ 1
(mod p) [p = prime number]
=
e^{2i}
=
35/70 + 148/296 (all digits 0 through 9 were used
once!)
= ^{3}(133/36 + 5/8)
 ^{3}(133/36  5/8)
Curious
multiplications using 1's:
1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
etc...
The
numbers in the series 1, 11, 111, 1111, 11111,
etc... are all triangular
numbers in base9.
During
any police lineup the suspects wear nos. 2 through 9 because
it is considered too suggestive to make anyone display the no. 1!
Symbolizes
the essence of all phenomena, which is a single unity, before
being divided. It represents also the contrast between essence
and existence; the enduring and the ephemeral; the unity in diversity
(one/many). According to Hopper, the first advance towards counting
is with the use of words for one and for many,
the differentiation from the self from the group. We still say
'numero uno' to speak of ourselves.
"Taiji" (also
termed as "Dayi" or "Taiyi"), in the Mystical
Numbers of Taoism, represents the ONE, the Ultimate, the Order.
The martial art known as "Taijiquan" based its movement's
philosophies upon the notion of Taiji.
In
the English language, there is a word with just ONE vowel which
occurs 6 times: indivisibility.
'Strengths'
is the longest word in the English language with just ONE vowel.
Impoverished
counting system: 1 + 1 = ?
When it comes to counting, a remote Amazonian tribespeople have
been found to be lost for words. In fact, researchers discovered
that Pirahã tribe
of Brazil, with a population of 200, have no words beyond ONE,
two and many.
The word for "one" can also mean "a few", while "two" can
also be used to refer to "not many"... (But is there
any case where not having words for something doesn't allow you
to think about it?) Source BBC
ONE in
different languages (© G.
Sarcone)
(Old English ān) 
Reconstructed
protolanguage:  *TIK 
Indoeuropean  *OIN, *OIW,
*OIK ,  *SEM 
Sanskrit
 EKA 
ekaḥ (m)
/ ekā (f) / ekam (n)
ProtoHellenic
 *HEMS (< sems) 
Greek, Attic  'EIΣ, MIA, 'EN HÊS (m), MIA (f), HÉN (n)

Latin  VNVS, A 
Archaic Latin  *ŒNVS, A ,  *OINOS,
A 

Italian,
Spanish uno; Romanian, French,
and Catalan un; Provençal uns;
Portuguese um; Romansh in;
Sardinian únu. 
Old
Celtic  OINO  
Breton unan;
Welsh un; Irish a
haon (cardinal), amháin (thing), duine (person). 
Old
Germanic  AINAZ  
Dutch een;
German eins; Danish and Norwegian en, et;
Swedish ett; Icelandic einn. 
Old
Slavic  JEDINU, A, 0  
Russian один odÕn;
Czech and Polish jeden; Slovenian êna. 
Proto
IndoIranian  *AIWAS  
Persian یک yek;
Hindi एक ek. 
Evolution
from 'seal
script' to modern sinograph 一 :
Old Chinese (pron.)  iêt  
Chinese 一 yī. 幺 yāo is
used as a replacement for yī in series of digits
such as phone numbers, room numbers, etc... to prevent confusion
between similar sounding words. 
ProtoSemitic
 *HAD ;  'IShT 
Semitic root  WHD  or  ?HD 
(? = glotal stop)
Ancient Egyptian [w'.] ua; Akkadian ishte'n;
Punic e'hd 
Arabic واحِد wa:hid;
Hebrew אחת 'aHat;
Maltese: wiehed; Amharic and. 
More
languages
Magyar egy.
Turkish bir.
Mayan hun.
Nahuatl cē.
Suomi yksi.
Zulu (uku)nye. 
HIDDEN
ROOTS
The roots of the word one (un, sim,
prin, cen) are hidden in the following words: inch (from
Lat. uncia), onion, ounce, primal, primate, primitive,
primrose, prince, recent, simple, simulate, sincere (from Lat. sincerus meaning "clean,
pure, sound", derived from the IndoEuropean roots ‘sem’ and ‘ker’,
the underlying meaning of which is: 'of one growth', hence
'pure, clean'), single, unanimous, unicorn, uniform, unify,
union, unique, unit, universe; alone, any, lonely, only, none.
In French: ensemble, oignon, premier, printemps, sanglier,
semblable, sincère. In Spanish: centolla, centollo ('spidercrab',
from Celtic *kintuollos, the largest one < *kĭntu,
first, and *ollos, big). Gaulish person names: Cintullus,
Cintugnatos, 'first born' (< *kĭntu, first
one; cognates: Lat. recens 'new', Gr. kainos 'young,
new'). 

1.41 
is
also called Pythagoras' constant.
is the ratio of diagonal to side length in a square.
≈ 1.4142135623
7309504880 1688724209 6980785696 7187537694 8073176679 7379907324
7846210703 8850387534 3276415727 3501384623 0912297024 9248360558
5073721264 4121497099...
One
of the earliest numerical approximation of 2 was
found on a Babylonian
clay tablet (from the Yale Babylonian Collection), dated
approximately to between 1800 B.C. and 1600 B.C. The annotations
on this tablet give an impressive numerical approximation in
four sexagesimal figures:
1 + 24/60 + 51/60^{2} + 10/60^{3} = 1.41421296...
≈ (P_{n+1 }
P_{n})/P_{n} (P = Pell
numbers)
≈ 17/12
≈ 99/70
≈ 1.0110101000001001111..._{2}
=
2sinus(45°) = 2cosinus(45°)
= 1 + (1 / (2 + (1 / (2 + (1 / (2 + ... ))))))
= (i + ii)
/ i
If
you want to have some fun with 2:
start with the very rough approximation 7/5. Then
(7+5+5)/(7+5) = 17/12
(17+12+12)/(17+12) = 41/29
(41+29+29)/(41+29) = 99/70
(99+70+70)/(99+70) = 239/169
...
continuing closer approximations of 2

posted by Larry Bickford 
Writing
numbers using only square roots of 2:
3 = log_{2}log_{2}((2))
4 = log_{2}log_{2}(((2)))
5 = log_{2}log_{2}((((2))))
6 = log_{2}log_{2}(((((2)))))
... etc.
ISO
paper sizes are all based on a single aspect ratio of the square
root of two, or approximately 1:1.4142. Basing paper upon this
ratio was conceived by Georg Lichtenberg in 1786, and at the
beginning of the 20th century, Dr Walter Porstmann turned Lichtenberg's
idea into a proper system of different paper sizes.

1.62 
is
the Golden Number,
also called Golden Ratio or Phi.
Golden Number property: ( +
1)/ = /1
Fibonacci
number sequence is intimately connected with the Golden
Ratio.
The
fraction 1/998999 contains Fibonacci numbers, i.e.:
1/998999=0.000001001002003005008013021034055089...
Radii
at 0° and approximately 222.49° divide a circle in the
Golden Ratio: B/A = /1
=
(5
+ 1)/2
= 1 + (1 / (1 + (1 / (1 + (1 / (1 + ... ))))))
= (4
+ (4!
 4))/4
= 2sin(666)
≈ F_{n+1} / F_{n }(F = Fibonacci
numbers)
≈ 1.61803
39887 49894 84820 45868 34365 63811...
Remarkably,
you can use Fibonacci
successive terms to convert miles to kilometers:
8 miles ≈ 13 kilometers
13 miles ≈ 21 kilometers
This works because the two units stand in the Golden Ratio (to
within 0.5 percent).
The
3184th Fibonacci number is an apocalypse number (Apocalpyse numbers
are numbers having exactly 666 digits).

1.73 
is
also known as Theodorus' constant (it is named after Theodorus of
Cyrene, who proved that the square roots of the numbers from
3 to 17, excluding 4, 9, and 16, are irrational).
is the diagonal of a cube having 1unit sides.
is the height of an equilateral triangle having 2unit sides.
The
shape 'Vesica
piscis' (fish bladder) has a major axis/minor axis ratio
equal to the square root of 3, this can be shown by constructing
two equilateral triangles within it.
≈ 1.7320508075
6887729352 7446341505 8723669428 0525381038 0628055806 9794519330
1690880003 7081146186 7572485756 7562614141 5406703029 9699450949
9895247881 1655512094...
=
2sinus(60°) = 2sinus(30°)
= 1 + (1 / (1 + (1 / (2
+ (1 / (1 + (1
/ 2 + ...
)))))))
≈ 97/56
≈ 1.1011101101100111101..._{2}

2
⠼⠃

is
the only even prime.
is the first taxicab number
(trivial). 
posted by Charles Rathbone
there
are no integers x, y, and z for which x^{n }+
y^{n} = z^{n} is valid,
when n is greater than 2 (see Fermat's
last conjecture).
n^{2} ± n is
always divisible by 2.
2 +
2 = 2 x 2 = 2^{2}
= 3^{3 } 5^{2}
= 4^{2 } 3^{2}  2^{2 } 1^{2}
= (3^{2} + 4^{2} + 5^{2} + 6^{2} +
7^{2} + 8^{2} + 9^{2})/
(1^{2} + 2^{2} + 3^{2} + 4^{2} +
5^{2} + 6^{2} + 7^{2})
= (2+(2+(2+(2
+ ... )))) ;
Proof: if N = (2+(2+(2+(...
)))) ,
then N^{2} = 2 + (2+(2+(2
+ ... ))) = 2 + N ,
solving N^{2}  N  2 = 0 , we find the
positive solution N = 2
= (3
+ 22)
 (3
 22)
= ^{3}(63
+ 10)  ^{3}(63
 10)
= log_{a} a^{2}
=(1
+ i)(1  i)
=
X/V
2^{7} = 71^{2 }
17^{3}
is
the smallest prime that can grow 7 times by the right:
2 is prime,
29 is prime,
293 is prime,
2939 is prime,
29399 is prime,
293999 is prime,
2939999 is prime.
29399999 is prime.
When
you increase the area of a square of 1 unitsquare, the side n of
this square  for n > 3  increases approximately
of 1/2n. For example: (1^{2} +
12^{2}) ≈ 12 + 1/(2 x 12) ≈ 12.0416... 
G. Sarcone
Curiosity...
An everyday example when 1 + 1 ≠ 2:
1 liter of water + 1 liter of alcohol = 1.926 liters of liquid
"Liangyi" (Number
2), in the Mystical Numbers of Taoism, symbolizes the Twin, the
First Division, the Duality of Opposites (Yin/Yang).
In
Cantonese the number two is fortunate, because it sounds similar
to "easy" in the dialect.
The twosecond
rule is an easy way to make sure you have left enough
following distance between your car and the vehicle in front,
no matter what speed you're travelling at. To check if you
are travelling two seconds behind the vehicle in front:
 watch the vehicle in front of you pass a landmark (such as a
sign, tree, or power pole) at the side of the road,
 as it passes the landmark, start counting 'One thousand and one,
one thousand and two',
 if you pass the landmark before you finish saying those eight
words, you are following too closely. Slow down, pick another landmark
and repeat the words to make sure you have increased your following
distance.  Source Land
Transport NZ.
The
most common twoletter words in order of frequency are: of, to,
in, it, is, be, as, at, so, we, he, by, or, on, do, if, me, my,
up, an, go, no, us, am.
'Skiing'
is the only word in the English language with TWO i.
A
honey bee must tap TWO million flowers to make ONE pound of honey!
"A
man is a person who will pay two dollars for
a onedollar item he wants. A woman will pay one dollar for a
twodollar item she doesn't want..."  William
Binger
TWO in
different languages (© G.
Sarcone)
(Old English twā) 
Reconstructed
protolanguage:  *PAL 
Indoeuropean  *DWI, *DUWO

Sanskrit
 DVÎ 
dvai (m)
/ dvā (f) / dve (n)
ProtoHellenic
 *DWO , Greek, Attic  ΔYO DUO 
Latin  DVO, Æ ,
 BI , Archaic Latin  *DWO ,  *DWI


Italian due;
French deux; Spanish and Catalan dos;
Provençal dous (m), dos (f);
Portuguese dois (m), duas (f);
Romanian doi (m), două (f);
Romansh dus (m), duas (f);
Sardinian dúos (m), dúas (f). 
Old
Celtic  DO  
Breton daou (m), div (f);
Welsh dau (m), dwy (f);
Irish a dó (cardinal), dhá (things), beirt (persons). 
Old
Germanic  TWAIZ  
Dutch twee;
German zwei (often zwo is
used to avoid confusion with drei, 3); Danish and
Norwegian to; Swedish två;
Icelandic tveir. 
Old
Slavic  DUVA, DVE  
Russian два dva;
Czech dva; Slovenian dvá;
Polish dwa. 
Proto
IndoIranian  *DVA:  
Persian دو do;
Hindi दो d̪oː. 
Evolution
from 'seal
script' to modern sinograph 二 :
Old Chinese (pron.)  ñzhi  
Chinese 二 èr (is
used for numbers and in counting) / 两 liǎng (is
used when counting objects or persons). 
ProtoSemitic
 *ThNÂ ;  KIL' 
Semitic
root  ThN , derived verb 'thny', to repeat.
Ancient Egyptian [sn.] sen; Akkadian shénâ;
Punic shnem.

Arabic اِثنان ithna:n;
Hebrew שתיים shtayim;
Maltese: tnejn; Amharic hulät. 
More
languages
Magyar kettő.
Turkish iki.
Mayan ca.
Nahuatl ōme.
Suomi kaksi.
Zulu (isi / ku)bili. 
HIDDEN
ROOTS
The roots of the word two are hidden
in the following words: balance, bezel, bicycle, binary, biscuit,
combine, diploma, diptych, double, doubt, duel, duet, duplex,
duplicate, pinochle; between, twist, twice, twill, twin; Mishnah.
In French: bafouiller, berlue, besace, bévue, bigle,
binocle, bisquer, brouette (< bisrouette). 

2.24 
is
an irrational number involved in the formula for the Golden
ratio.
is also used in statistics when dealing with 5business day weeks.
is the hypothenuse of a right triangle having 1 and 2unit sides.
is the diagonal a rectangular box having 1, 2
and 2unit
sides.
=
e^{i} +
2
≈ 2.2360679774
9978969640 9173668731 2762354406 1835961152 5724270897 2454105209
2563780489 9414414408 3787822749 6950817615 0773783504 2532677244
4707386358 6360121533...
≈ 85/38
≈ 10.0011110001101111..._{2}

e
2.72

Discovered
by the Scottish mathematician John
Napier of Merchistoun.
e stands for exponens (in Latin,
'exponential')
= 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + ...
e has
this mathematical property:
≈ 2.7182818284
5904523536 0287471352 6624977572 4709369995 9574966967 6277240766
3035354759 4571382178 5251664274 2746639193 2003059921 8174135966
2904357290 0334295260...
≈ ^{6}(^{4} + ^{5}) (mathematical
coincidence)
e^{ia} =
cos a + i sin a
Ln x ≡ log_{e }x
log x =
log e · Ln x
Benjamin
Peirce suggested the innovative notation, that looked like a
paper clip, for e and shown
below:
From J. D. Runkin's Mathematical Monthly, vol. I, No. 5, Feb. 1859

3
⠼⠉

is
the only prime 1 less than a perfect square. 
Robin Regan
is the number of spatial dimensions needed to mathematically describe
a solid.
are the primary colors.
are
the geometric constructions you cannot build using just a ruler
and compasses: 1. You cannot trisect  divide into three equal
parts  a given angle; 2. Double a cube; and 3. Square a circle.
A number
is divisible by 3 when the sum of its digits can be divided by
3.
If the
denominator of a rational number is not divisible by 3,
then the repeating part of its decimal
expansion is an integer divisible by 9. Example: 1/7 = 0.142857...
has a repeating part '142857' divisible by 9. Another example
with a larger recurring
decimal: 1/23 = 0.0434782608695652173913... has a repeating
part '0434782608695652173913' divisible by 9.
3 + 2 =
log_{2} 32
= (14
 65)
+ 5 (sum
of two square roots)
=
4! / (4 x 4)
=
XV/V = CL/L = MD/D
=
4 + 4 – 5 =
4^{3} + 4^{3} – 5^{3
}=
17,469 / 5,823 (this division contains all digits
1 through 9 once)
3 x
51249876 = 153749628 (the multiplication uses
all 9 digits once  and so does its product!)
3 x
37 = 111
33 x 3367 = 111,111
333 x 333667 = 111,111,111
3333 x 33336667 = 111,111,111,111
33333 x 3333366667 = 111,111,111,111,111
3 x 1.5 =
3 + 1.5
3^{2 }=
3! + 3
3^{2 }= 5^{2}  4^{2}
3^{3 }= 6^{3}  5^{3}  4^{3}
3^{3 }= 3^{2} + 3^{2} + 3^{2}
3^{4} x
425 = 34425
3
is the minimum colors needed to create camouflage patches, usually
used in military compounds and vehicles. 
posted by George Pantazis
A
3 x 3 alphamagic
square is a magic
square for which the number of letters in the word for each
number generates another magic square, for instance:

five (4) 
twentytwo (9) 
eighteen (8) 
twentyeight (11) 
fifteen (7) 
two (3) 
twelve (6) 
eight (5) 
twentyfive (10) 

A 3 x
6 rectangle has an area equal to its perimeter.
In
one gram of water the number of molecules is about:
3.3 x 10^{2}^{2} = 33000000000000000000000
The balanced
ternary base, is a numeral system which uses 3
values or digits: 1, 0, and 1. It works as follows
(in the example, the symbol 1 denotes the digit 1):
Decimal 
6 
5 
4 
3 
2 
1 
0 
1 
2 
3 
4 
5 
6 
7 
Balanced
ternary 
110 
111 
11 
10 
11 
1 
0 
1 
11 
10 
11 
111 
110 
111 
Ternary
or base3 numbers can
be converted to balanced ternary notation by adding 1111... with
carry, then subtracting 1111... without borrow. For instance:
021_{3} + 111_{3} = 202_{3}, 202_{3} 
111_{3} = 111_{3(bal)} = 7_{10}
This
nonstandard positional numeral system is easily represented as
electronic signals, as potential can either be negative, neutral,
or positive (comparison logic). The balance ternary system is also
useful to solve the classical 2pan
balance puzzle.
The
letters A, F, H, K, N, Y and Z are
made up with 3 lines.
In
SMS language <3 means
'I love you', and <333,
'I love you so much'.
3 hundred
millions of Indians live with less than 1 dollar per day (2004).
Nonpaternity
rates: statistically, one in three men who ask
for paternity
test turn out not to be the biological parent.
An octopus
has 3 hearts.
The
number 3 symbolizes the principle of growth. In Guangdong province,
China, three is associated with living or giving birth.
"Sanqing" (also
known as "Sanxing" or "Sancai"), in the Mystical
Numbers of Taoism, represents the number 3 and symbolizes the
Three Luminaries: Sun, Moon, Stars. It also defines the concept
of "Heaven, Mankind, Earth" as well as "Upper,
Centre, Lower".
Deep
thought: "There are 3 kinds of people: those who can count
and those who can't".
Riddle
1: Spell 'mousetrap' in 3 letters...
Answer: CAT.
Riddle
2: Spell 'water' in 3 letters...
Answer: H2O.
Joke: Chuck
Norris once won a game of Connect
Four in 3 moves!
If
you’re a Simpsons fan, then you problably know about “Blinky”,
the threeeyed
fish found near the nuclear plant where Homer Simpson was
working. As it turns out, the Simpsons were right yet again,
as fishermen in Córdoba, Argentina caught a threeeyed
wolf fish in a reservoir fed by a local nuclear power plan! (Fri,
Oct 28, 2011)
In
the taoist mythology, Erlang Shen (二郎神),
or Erlang is a Chinese God with a third truthseeing
eye in the middle of his forehead.
The
French sentence 'un bonhomme haut comme trois pommes' (a 3appletall
fellow) and the German sentence 'ein Kerlchen drei Käse
hoch' (a 3cheesetall fellow) mean both a
pintsized guy/child.
"Les
fourmis, chacune d'elles ressemble au chiffre 3.
Et il y en a! Il y en a 333333333333... jusqu'à l'infini" (Jules
Renard, 'Histoires naturelles').
Translation: "The ants. Each of them resembles a figure
3. That's, it. There are 333333333333 to infinity".
'Bookkeeper'
and 'bookkeeping' are the only words in the English language
with three consecutive double letters.
The
most common threeletter words in order of frequency are: the,
and, for, are, but, not, you, all, any, can, had, her, was, one,
our, out, day, get, has, him, his, how, man, new, now, old, see,
two, way, who, boy, did, its, let, put, say, she, too, use.
The
name "Mitsubishi" (三菱) consists of two
parts: "mitsu" meaning 'three' and "bishi" meaning
'water caltrop', and hence 'rhombus', which is reflected in the
company's famous logo (it is also translated as 'three diamonds').
Other Japanese family names containing the number 3: Mitsudani,
Mitsugi, Mitsui, Mitsuhashi, Mitsude, Mitsuishi, Mitsumura, Mitsubori,
Mitsumata, Mitsuyama, Mitsuzawa, Mitsuya, Mitsukuchi, Mitsukyou,
Mitsuboshi, Mitsuzima, Mitsue, Mitsuike, Mitsuaichou, Mitsubuchi,
Mitsuse, Mitsuyanagi, Mitsumachi, Mitsukunugi, Mitsuwa, Mitsuzaku,
Mitsumatsu, Mitsuhuzi, Mitsuduka, Mitsuwari.
THREE in
different languages (© G.
Sarcone)
(Old English thrīe) 
Indoeuropean
 *TREYES, *TISORES,
*TRI 
Sanskrit
 TRÎ 
trayaḥ (m)
/ tisraḥ (f) / trīṇi (f)
Greek,
Attic  TPEIΣ, TPIA TRÊS, TRIA 
Latin  TRES, TRIA , Archaic
Latin  *TREIES 

Italian tre;
French trois; Spanish and Catalan tres;
Provençal trei, tres;
Portuguese três; Romanian trei;
Romansh trais; Sardinian très. 
Old
Celtic  TRI  
Breton tri (m), teir (f);
Welsh tri (m), tair (f);
Irish trí (m), teoir (f,
old Irish), triúir (people). 
Old
Germanic  THRIJIZ  
Dutch drie;
German drei; Danish, Norwegian,
and Swedish tre; Icelandic þrír. 
Old
Slavic  TRIJE, TRI  
Russian три tri;
Czech tři; Slovenian trí;
Polish trzy. 
Proto
IndoIranian  *TRAYAS  
Persian سه se;
Hindi तीन t̪iːn. 
Evolution
from 'seal
script' to modern sinograph 三 :
Old Chinese (pron.)  sâm  
Chinese 三 sān. The
sinograph 叁 is used as a replacement for sān on
legal and financial documents to prevent fraud. 
ProtoSemitic
 *SALATh 
Semitic root  ThLTh 
Ancient Egyptian [ḫmt'] khemet;
Akkadian shalash;
Punic shlosht. 
Arabic ثلاثة thalathâ;
Hebrew שלושה shlôshah;
Maltese: tlieta; Amharic sost. 
More
languages
Magyar három.
Turkish üç.
Mayan oxi'.
Nahuatl ēyi.
Suomi kolme.
Zulu (ku)thathu. 
HIDDEN
ROOTS
The roots of the word three are hidden
in the following words: contest, detest, obtest, protest, sesterce,
sitar, teapoy, tercet, tertian, tern, terpolymer, test, testament,
testicle, testify, testimony, trammel, travel, trefoil, trench,
trephine, trey, triad, triangle, triathlon, tribe, trio, triple,
triplex, trine, trinity, trimurti, trivial, triumvir, trocar,
troika; third, thrice. In French: travail, treillis, trémail.
In Spanish: terliz, trabajo. 
Buy
your favorite
Number (3) here.

3.14

=
Perimeter / Diagonal, of any circle.
Pi expanded
to 45 decimal places:
3.14159 26535 89793 23846 26433 83279 50288 41971 69399
Pi expanded
to 52 binary places:
11.0010010000111 1110110101010 0010001000010 1101000111001
You
cannot square a disc using just a ruler and compasses because is
a transcendental
number.
= 4(1/1
 1/3 + 1/5  1/7 + 1/9  1/11 + ... )
= 2(2/1 x 2/3 x 4/3 x 4/5 x 6/5 x 6/7 x 8/7 x 8/9 x ... )
≈ 355/113
(a real good rational approximation of )
≈ (6^{2})/5
In
the late 18th century, James Stirling, a Scottish mathematician,
developed an approximation for factorials using the transcendental
numbers 'Pi' and 'e':
n! ≈ (2n)^{1/2} (n/e)^{n}
The
most famous formula for calculating Pi is Machin's formula:
/4
= 4 arctan(1/5) – arctan(1/239)
This formula, and similar ones, were used to push
the accuracy of approximations to Pi to over 500 decimal places
by the early 18th century (this was all hand calculation!).
Interestingly,
there are no occurrences of the sequence 123456 in the first
million digits of Pi. 
posted by George Pantazis
Bamboozlement
with Pi
Does Pi equal 3? No? Then have a look on the algebraic equation
below:
x = ( +
3)/2
2x = +
3
2x( 
3) = ( +
3)( 
3)
2x
 6x = ^{2} 
9
9  6x = ^{2} 
2x
9  6x + x^{2} = ^{2} 
2x
+ x^{2}
(3  x)^{2} = ( 
x)^{2}
3  x = 
x
3 =
We use
Pi to:
 describe the DNA double helix,
 determining the distribution of primes  the probability that
two randomly selected integers are relatively prime (i.e. have
no common factors) is 6 / p2,
 analyzing the ripples on water,
 checking for accuracy  as there are now millions upon millions
of known decimal places of Pi, by asking a super computer to compute
this many figures its accuracy can be tested,
 in cryptography  the science of coding,
 generate of a random number.
On Pi
Day (March 14 or 314) in 1879, a baby was
born in Ulm, Germany to a German couple whose name meant "one
stone". That baby was Albert Einstein!
occurs
naturally in tables of death, in what is known as a Gaussian
distribution of deaths in a population; that is, when a person
dies, the event 'feels' Pi.
The
symbol for Pi was introduced by the English mathematician William
Jones in 1706.
Mathematician
John Conway pointed out that if you break down the digits of
Pi into blocks of ten, the probability that one of those blocks
will contain ten distinct digits is about one in 40,000. Curiously,
this first happens in the 7th block of ten digits.
There
is the little rhyme to help the memorisation of twentyone digits
of :
Now, I wish I could recollect pi.
"Eureka," cried the great inventor.
Christmas Pudding; Christmas Pie
Is the problem's very center.
Joke:
A round pizza with radius 'z' and thickness 'a' has the volume Pi·z·z·a.

S
3.24 
3.246979603717467...
is the Silver
Number or silver constant given by:
x^{3}  5x^{2} + 6x  1 = 0
S = ^{3}7
+ (7 ·^{ 3}7
+ (7 ·^{ 3}7
+ ... ))

4
⠼⠙

is
the smallest number of colors sufficient to color all planar
maps with no adjoining countries sharing the same color.
are the number of coordinates needed to describe an event in 'spacetime':
t, x, y, z.
Pick
any whole number... If the number is even, divide it by 2; if
it is odd, multiply it by 3, then add 1. By repeating this procedure,
sooner or later you'll arrive at the number 4,
which will give you 2, which in turn gives you 1, and then get
a 4 again! No matter what number you choose, you'll always arrive
at the 421 cycle.
The
word 'four' has 4 letters and is the smallest honest
number. Honest numbers are numbers n that can be
described using exactly n letters in standard mathematical
English.
A 4
x 4 square has an area equal to its perimeter.
The
only solution of x^{y }= xy involving integers is:
x = y = 2, and x^{y} = xy = 4
4^{2 }=
2^{4}, is the solution to the equation a^{b} = b^{a},
for a b.
x^{2 }
y^{2} is divisible by 4 only when (x
 y) is even.
Any
integer of the form n^{4} + 4 is
not a prime, except for n=1, because it can be factorized
as follows:
n^{4} + 4 = (n^{2} 
2n + 2)(n^{2} + 2n + 2) (Aurifeuillean factorization)
Any
prime of the form 4k + 1 is the sum
of two square numbers:
13 = 4 x 3 + 1 = 2^{2} + 3^{2} ;
73 = 4 x 18 + 1 = 3^{2} + 8^{2} ;
137 = 4 x 34 + 1 = 4^{2} + 11^{2}
=
3^{2 } 2^{2}  1^{2
}= (20
 (20
 (20
 (20
 ... ))))
=
1(1/2)^{0} + 2(1/2)^{1} + 3(1/2)^{2} +
4(1/2)^{3} + 5(1/2)^{4} + 6(1/2)^{5} +
...
16/64
= 16/64
= 1/4
If you
multiply the number 21978 by 4, it turns backwards!
= 15,768
/ 3,942 (contains all digits 1 through 9 once)
=
CD/C
The
two equalities 4 x 1738 = 6952 and 4 x 1963 = 7852, use the digits
19 exactly once!
An
intriguing 4 x 4 magic
square (fig. 1) that works just as well upsidedown (fig.
2):
fig.
1 
I8 
99 
86 
6I 
66 
8I 
98 
I9 
9I 
I6 
69 
88 
89 
68 
II 
96 

96 
II 
89 
68 
88 
69 
9I 
I6 
6I 
86 
I8 
99 
I9 
98 
66 
8I 

fig.
2 
4 is
the smallest digit that never occurs in any term of the "lookandsay" sequence.
The Shadok's numbers
are a kind of quaternary or base4 numeration system:
0
is Ga 

4
is Buga 

8
is Zoga 

12
is Meuga 

1
is Bu 

5
is Bubu 

9
is Zobu 

13
is Meubu 

2
is Zo 

6
is Buzo 

10
is Zozo 

14
is Meuzo 

3
is Meu 

7
is Bumeu 

11
is Zomeu 

15
is Meumeu 

The
digit 4 on an LCD calculator is made up of 4 bars.
A
famous riddle: Show how onehalf of five is four!
Answer: Take off the first and last letters and you have the roman
numeral for 4. The Roman numeral for four is IV (whose
letters are onehalf of the four letters in the spelledout word "five").
Why
is the Roman numeral IIII used instead of IV on
clocks and watches?
 using IIII brings more symmetry and balance to the dial. The
IIII offsets the heavy VIII that is found on the other side.
 the strict use of IV instead of IIII wasn't common until after
the middle ages (the practice of placing smaller digits before
large ones to indicate subtraction came into popularity in Europe
after the invention of the printing press), the Romans generally
used IIII. Clocks and watches are patterned after sundials, which
were in use long before the middle ages.
Berger’s
4:9 theory
In
his book "Bauwerk und Plastik des Parthenon, in Antike Kunst" (Basel,
1980). E. Berger presents a study that investigates the way that
the Pythagorean ideas of ratios of small numbers were used in the
construction of the Temple
of Athena Parthenos. In his opinion the ratio 4 : 9 were fundamental
to the construction. A basic rectangle of sides 4 and 9 was constructed
from three rectangles of sides 3 and 4 with diagonal 5 (see drawing).
This form of construction also meant that the 345 Pythagorean
triangle could be used to good effect to ensure that right angles
in the building were accurately determined.
The length of the Temple of Athena Parthenos is 69.5 m, its width
is 30.88 m and the height at the cornice is 13.72 m. To a fairly
high degree of accuracy this means that the ratio width : length
= 4 : 9 while also the ratio height : width = 4 : 9.
 Source: Article by
J.J. O'Connor and E.F. Robertson.
The foursecond
rule is the amount of time that internet user will
wait for a page to load before leaving and going to another
site.
Swear
number:
The phrase "fourletter word" is used
to describe most swear words in the English language.
The Pythagorean
oath, as quoted by the Renaissance magician Cornelius Agrippa,
is as follows:
"I with pure mind by the number four do swear;
That's holy, and the fountain of nature
Eternal, parent of the mind..."
In Japan
and in most Asiatic cultures, the number 4 (sinograph: 四)
is considered unlucky because it is prounounced shi which
sounds like the word 'death'. Due to that, many numbered product
lines skip the number 4. However, in some cases the word yon ('4'
in early classical Japanese) is used instead of shi:
when counting floors in a building, or when you are asked "which
floor?" in an elevator... The aversion or fear of the number
4 is called "Tetraphobia".
Mathtrick:
Four equals three!
Suppose: a + b = c .
This can also be written as:
4a  3a + 4b  3b = 4c  3c .
After reorganizing:
4a + 4b  4c = 3a + 3b  3c .
Take the constants out of the brackets:
4(a + b  c) = 3(a + b  c) .
Remove the same term left and right, then:
4 = 3
Where is the error?
Challenge:
using four 4's and any operations, try to write
equations that have the integers from 1 to 100 as the answer
(see example below):
1 = 44/44
2 = 4/4 + 4/4
3 = (4 + 4 + 4)/4
4 = 4(4  4) + 4, etc...
(click here to
see solutions)
Curiosity:
Think of any number and write it out in WORDS. Count the number
of letters it contains and write that down in WORDS. And so
on:
• TWENTYEIGHT (11 letters) >
• ELEVEN (6 letters) >
• SIX (3 letters) >
• THREE (5 letters) >
• FIVE (4 letters) >
• FOUR (4 letters) > etc.
You will always arrive at FOUR!
A dollar
bill can be double folded (forward and backwards) 4x10^{3} times
before it will tear.
STA4NCE
= For instance!
In Italian,
the expression in quattro e quattr'otto (in
four and four eight) means 'in a trice'.
An amusing
Finnish word that contains 4 y: Yötyöhyöty (advantage
gained from working night shifts with the correspondingly higher
salary).  posted by Juhani Sirkiä
4
rivers are mentioned in the Old Testament, Gen 2, 10:
"And a river went out of Eden ... and parted ... into four heads. The ...
first [is] Pison ... which compasses the whole land of Havilah ... the
second [is] Gihon ... that compasses the whole land of Ethiopia ...
the third [is] Hiddekel ... that goes toward the east of Assyria ...
and the fourth [is] Euphrates that goes eastward to Assyria".
The
number 4 symbolizes the principle of putting ideas into form.
It signifies work and productivity.
1
in 4 people worldwide is Muslim, and 2 out of
3 of the world's Muslims are in Asia (data: 2009).
"Sixiang" (Number
4), in the Mystical Numbers of Taoism, represents the Four Essences:
Earth, Water, Air, Fire.
The
most common fourletter words in order of frequency are: that,
with, have, this, will, your, from, they, know, want, been, good,
much, some, time, very, when, come, here, just, like, long, make,
many, more, only, over, such, take, than, them, well, were.
There
are only 4 words in the English language which
end in 'dous': hazardous, horrendous, stupendous and tremendous.
FOUR in
different languages (© G.
Sarcone)
(from Old English feower) 
Indoeuropean
 *K^{W}ETORES,
*K^{W}ETESRES 
Sanskrit
 CATÙR 
catvāraḥ / catasraḥ / catvāri
ProtoHellenic
 *Q^{W}ET(O)RO 
Greek, Attic  τέτταρες, τέτταρα TETTARES, TETTARA 
Latin  QUATTUOR ,
Archaic Latin  *QUATBORO 

Italian quattro;
French, Provençal and Catalan quatre;
Spanish cuatro; Portuguese quatro;
Romanian patru; Romansh quatter;
Sardinian bàtero. 
Old
Celtic  PETOR  
Breton pevar (m), peder (f);
Welsh pedwar (m), pedair (f);
Irish a ceathair (cardinal), ceathre (things), ceathrar (people). 
Old
Germanic  FITHWOR  
Dutch
and German vier; Danish, Norwegian,
and Swedish fire; Icelandic fjórir. 
Old
Slavic  CETYRIJE, CETYRI  
Russian четыре chetyrye;
Czech čtyři; Slovenian štíri;
Polish cztery. 
Proto
IndoIranian  *K'ATWA:RAS  
Persian چهار chahar;
Hindi चार chaːr. 
Evolution
from 'seal
script' to modern sinograph 四 :
Old Chinese (pron.)  si  
Chinese 四 sí. 
ProtoSemitic
 *RABA' 
Semitic root  RB' 
Ancient Egyptian [ỉfd'] aft'u;
Akkadian or erbe;
Punic 'arbah. 
Arabic أربعة arba'â;
Hebrew ארבעה ârba'ah;
Maltese: erbgħa; Amharic arat. 
More
languages
Magyar négy.
Turkish dört.
Mayan can.
Nahuatl nāhui.
Suomi neljä.
Zulu (ku)ne. 
HIDDEN
ROOTS
The roots of the word four are hidden
in the following words: cadre, cahier, carillon, carnet, carrefour,
casern, czardas, escadrille, petorritum, quadrant, quadriga,
quadroon, quarantine, quarrel, quarry, quarter, quartan, quaternay,
quatrain, quire, squad, square, tetrad, trapezium, trocar;
farthing, filler, firking; rabi, arroba (from Arabic الربع arrub', "quarter").
In French: carreau, écarquiller, écarter, Périgord
(< *petrucorii, 'four troops'), Périgueux,
arrobase. In Italian: quadrivio, squadra, squartare, tessera.
Gaulish person names: Petrullus, Petrogenos, Petrusonia ‘fourth
born’ (< *petuares / petru, 'four'). 
Buy
your favorite
Number (4) here.

5
⠼⠑
V

is
the only prime number that ends in 5.
is the number of Platonic
solids.
is the only prime number that ends in 5.
is a congruent
number because it is the area of a 20/3, 3/2, 41/6 triangle
(a congruent number is an integer that is the area of a right triangle
with three rational number sides).
The
Roman numeral for 5 is V, which comes from a
representation of an outstretched hand.
Any
power of 5 ends in a 5 (except 5^{0}).
= 3^{2} 
2^{2} = 1^{2} +
2^{2}
= 2^{5} 
3^{3}
5^{2} =
25
5^{2} = 3^{2} + 4^{2} = 13^{2} 
12^{2}
= (11
x 11  11)/(11 + 11)
= D/C
19/95
= 19/95
= 1/5
26/65
= 26/65
= 2/5
(5 
1)! + 1 = 0 (mod 5^{2})
Any
number having a abc5abc5 pattern
is divisible by: 5, 73, 137, and 10001
Can
you count in 'dingbong'?
The
inhabitants of 'Fongaponga' use a series of sounds made from this
strange device to represent numbers: 'ding'
with the handbell, 'eek' when squeezing the rubber
bulb of the horn, and 'bong'
when beating the tambourine with the small ball. These very special base5
numerals are then strings made from 3 sounds each
corresponding to an additive numerical value. Looking at the number
list below, we can guess with the help of some logic that 'eek'
is actually a 'function' that indicates subtraction and that every
'ding' equals 5, and every 'bong', 7.
1. dingdingdingeekbongbong
2. bongeekding
3. dingdingeekbong
4. bongbongeekdingding
5. ding
6. dingdingdingdingeekbongbong
7. bong
8. dingdingdingeekbong
9. bongbongeekding
10. dingding
11. bongbongbongeekdingding
12. dingbong
13. dingdingdingdingeekbong
14. bongbong
15. dingdingding
16. bongbongbongeekding
17. dingdingbong
18. bongbongbongbongeekdingding
19. dingbongbong
20. dingdingdingding
21. bongbongbong
22. dingdingdingbong
23. dingdingdingdingdingdingeekbong
24. dingdingbongbong
From number 24 on, all numbers are only
combinations of dings and/or bongs.
25. ddddd
26. dbbb
27. ddddb
28. bbbb
29. dddbb
30. dddddd...

Source:
 Sarcone's "dingbong numbers" are
sequence A102701 in
the 'Encyclopedia of integer sequences'.
 "Fongaponga", Focus
BrainTrainer nr. 7, page 51. 
A fivesided
polygon (pentagon) has 5 diagonals. This is the only shape for
which the number of sides and diagonals is the same (which may
explain why pentagrams, pentacles, and pentangles are
so common and appear so often as iconographic symbols). 
by Patrick Vennebush 
Pentagram, Pentangle and Pentacle are
all names for a 5pointed star. This mystical symbol is supposed
to keep away devils and witches.

Early
Greek coin marked with Quincunx Pattern 
The number
5 is geometrically represented in the Quincunx pattern.
This design is arranged by marking four corners of an imaginary quadrilateral
and a central axis through a series of dots or objects  as noticed
on dice, playing cards, or dominoes. The significance of the quincunx
pattern originates in Pythagorean mathematical mysticism.
A
famous riddle involving 5s: How can you make the following
equation true by drawing only one straight line:
5+5+5=550
Answer:
545+5=550
Another
famous riddle: From a word of 5 letters,
take 2 letters and have 1.
Answer: ALONE  AL = ONE.
• What
English word contains all 5 vowels ONCE?
Answer: auctioned, authorize, dialogue, discourage, education,
housemaid, mensuration, obnubilate, pneumonia, precarious, precaution,
regulation, sequoia, tambourine, ultraviolet, uncopyrightable...
The words ABSTEMIOUS, ANEMIOUS, ARSENIOUS, CAESIOUS and FACETIOUS
contain all 5 vowels appearing in alphabetical order, while in
the words SUBCONTINENTAL and UNORIENTAL they appear in reverse
alphabetical order!  G.
Sarcone.
• What
Italian/French/Spanish word uses all 5 vowels
once?
Answer: aiuole (flowerbeds) / oiseau (bird) / murciélago
(bat).
A list of words that contain all 5 vowels once BOTH in Italian
and in Spanish: ADULTERIO, AURIFERO, CAULIFORME, COMUNICANTE, DEPURATIVO,
DELUSORIA, EDUCATIVO, EQUIVOCA, ESTUARIO, EUFORIA, FERRUGINOSA,
INCESTUOSA, LUTERANISMO, PAUPERISMO, PERUVIANO, SURREALISMO, VITUPEROSA,
VOLUMETRIA...  G.
Sarcone.
In
French, some verbs like 'vouaient' (were dedicating, dedicated),
'jouaient' (were playing, played)... contain 5 different consecutive
vowels!
• "In
this exclamation, there are five i's!" (autoreferential sentence)
1000
+ 5 = 1005 (one thousand five) is the smallest
natural number whose name contains the five vowels a, e, i, o,
u (in any order).
In
French "je te dis un mot de cinq lettres!" (I tell
you a word of five letters) is an exclamation of anger against
the person for whom the insult is intended.
5th
April
At 1:02 AM and 3 seconds on Wednesday, April 5, 2006, it was the
1st hour of the day, the 2nd minute of the hour, the 3rd second
of that minute in the 4th month and the 5th day of '06... or just: 01:02:03 040506 for
short!
For many other places, this coincidental chronological oddity happens
at 1:02 AM May 4.
The
five rivers of Hades are:
• Acheron (the river of woe. Etymologically,
the name probably means 'marshlike': cf. Greek word akherousai, 'marshlike
water'),
• Cocytus (the river of lamentation; from Greek kokutos,
'lamentation'),
• Phlegethon (the river of fire; from Greek present
participle of phlegethein, 'to blaze'),
• Lethe (the river of forgetfulness; from Greek lethe,
'forgetfulness')
• Styx (the river of hate; cognate with Greek
words stygos 'hatred' and stygnos 'gloomy').
Five
is a very popular number in Chinese culture since it occupies
the central position (one through nine) and also reflects the
'Five Elements Philosophy' (Wuxing)  Wood, Fire, Earth, Metal
(or Gold), and Water (in Chinese: 木, mù; 火,
huǒ; 土, tǔ; 金, jīn; 水, shǔi).
In Switerland,
the banking sector employs about 5% of the workforce
(data: 2005).
According
to a research by Commtouch, quoted by NYT, only 5 countries:
China, South Korea, Russia, USA and Brazil generate 99% of spams.
The
name Pontius (Pilatus), in early Italic language means
'the 5^{th}'. We
can find the Indoeuropean root penkwro, the '5th',
in the word finger (finger, from Germanic fingwraz, "one
of five").
 by Gianni A. Sarcone 
Macuilxochitl was
the god and patron of art, games, beauty, dance, flowers, and
song in Aztec mythology. His name contains the Nahuatl words xochitl ("flower")
and macuil (five), and hence means "Fiveflower" (but
he could also be referred to as Chicomexochitl, "Sevenflower").
FIVE in
different languages (© G.
Sarcone)
(from Old English fîf) 
Indoeuropean
 *PENK^{W}E 
Sanskrit  PAÑCA 
Greek, Attic  ΠENTE PENTE 
Latin  QVINQVE , Archaic Latin  *PENQVE 
Vulgar Latin  *CINQUE  
Italian cinque;
French cinq; Spanish and Portuguese cinco;
Provençal and Catalan cinc;
Romanian cinci; Romansh tschintg;
Sardinian chímbi. 
Old
Celtic  PEMPE  
Breton pemp;
Welsh pump; Irish cúig (things), ciúigear (persons). 
Old
Germanic  FIMFI  
Dutch vijf;
German fünf; Danish, Norwegian,
and Swedish fem; Icelandic fimm. 
Old
Slavic  PE^{n}TI  
Russian пять p'jat';
Czech pět; Slovenian pét;
Polish pięć. 
Proto
IndoIranian  *PANK'A  
Persian پنج panj;
Hindi पाँच panch. 
Evolution
from 'seal
script' to modern sinograph 五 :
Old Chinese (pron.)  nguo  
Chinese 五 wǔ. 
ProtoSemitic
 *KhAMSh 
Semitic root  KhMSh 
Ancient Egyptian [dỉ'] t'uau;
Akkadian khamish;
Punic khamsh. 
Arabic خمسة khamsâ;
Hebrew חמשה khamishah;
Maltese: hamsa; Amharic amist (pron.
amst). 
More
languages
Magyar öt.
Turkish bes¸.
Mayan ho.
Nahuatl mācuīlli.
Suomi viisi.
Zulu (isi)hlanu. 
HIDDEN
ROOTS
The roots of the word five are hidden
in the following words: cinquain, Cinquecento, cinquefoil,
keno, pachisi, Pentecost, pentesmon, pentagon, pentameter,
pentathlon, pinkster flower, Pontius, punch, Quentin, quincunx,
quinial, quintain, quintessence, quintet, quintuple; femto,
fin, finger, fist, foist; khamsin. In French: esquinter, quinquagénaire.
In Spanish: quintar. In German: Pfinztag, Quentchen. 

6
⠼⠋

is
a congruent
number because it is the area of a 3, 4, 5 triangle (a congruent
number is an integer that is the area of a right triangle with
three rational number sides).
is the smallest perfect
number, that is a number whose divisors add up to itself, e.g.:
1 x 2 x 3 = 1 + 2 + 3 = 6
The
probability that a number picked at random from the set of integers
will have no repeated prime divisors is 6/^{2}.
 Source:
Chartres
n^{3}  n is
divisible by 6. That is, any product of 3 consecutive integers
is divisible by 6.
The equation x^{n}  y^{m} = ±6 with n, m > 1
has NO solution. In other words, 6 cannot be a
difference of two powers!
=
3(1/1 + 1/2 + 1/3 + 1/6) = 6^{2}(1/1  1/2  1/3)
= 1/2^{0} + 3/2^{1} + 5/2^{2} + 7/2^{3} +
9/2^{4} + 11/2^{5 }+ 13/2^{6} + ... (sum
of consecutive odd numbers with reciprocal powers
of 2)
= 4! / 2^{2} = 4! / 2!^{2}
= ((10
 10/10))!
= ((1
+ 3)
+ (1
 3))^{2}
= (Log(10 x 10 x 10))!
= 10_{2} + 10_{2} + 10_{2} = 110_{2}
= DC/C
6^{2} =
36
6^{2} = 1^{3} + 2^{3} + 3^{3}
6^{3 }= 3^{3} + 4^{3} + 5^{3}
6 is
the smallest number of colors needed to color the regions on
a map on a Möbius
strip. A Möbius strip is a continuous closed surface
with only one side; formed from a rectangular strip by rotating
one end 180 degrees and joining it with the other end.
6
circles of the same size (try this with 6 coins of the same denomination)
will always perfectly surround, all touching, without gaps, 1
circle of that same size.  Posted by Aaron
Pyle
The
probability to get one 6 with 6 dice is 0.665...
The probability to get two 6's with 2 x 6 = 12
dice is 0.619...
The probability to get three 6's with 3 x 6 =
18 dice is 0.597...
Arithmetical
nut with 6: "From six take nine; from nine
take ten; from forty take fifty, and have six left" (see
below)
A base6
(senary or heximal) numeral system is used by the Ndom people
of the Frederik Hendrik Island, near New Guinea. For example,
in Ndom language the number 7 is mer abo sas (6 + 1),
and the number 17, mer an thef abo meregh (6 x 2 + 5).
"This
exclamation has unexpectedly six 's',
six 'i' and six 'x'!" (autoreferential
sentence)  G. Sarcone.
Any
one of us is only about 6 acquaintances away from anyone else
in the world.
Brazilians
have two different names for six: seis or meia (short
for meia duzia, 'half dozen').
In
old French, the word 'hasart' meant 6 at the
game of dice. The earliest meaning of HAZARD (<hasart) was,
however, 'stroke of luck (or bad luck)'. In the past, the dice
featured on one face a flower pattern. Thus the Arabs called
gaming dice "flowers", in Arabic 'azzahr'.
the "sixth
sick sheik's sixth sheep's sick" is said to be
the hardest tongue twister in English.
Riddle:
can you transform the Roman number IX into 6 by
drawing only one line?
Answer: SIX (yes, the line is curvilinear...).
Joke.
Solving the equation by one dumbo:
Chinese
people like the number 6. One possible reason is because it is
the largest number on a dice, and when gambling, one wins if
the number six is thrown. When playing mahjong, the host is
the most likely to be the one who throws the number six, and
who therefore has a better chance of winning. Reflecting this,
the Chinese have a saying, "double six makes you the happiest".
For Chinese businessmen the number six means "a smooth business".
"Liùhe" (六和),
in the Mystical Numbers of Taoism, represents the number 6 and
symbolizes the Six Harmonies:
體 合 於 心 (Body in harmony with Heart),
心 合 於 意 (Heart in harmony with Intent),
意 合 於 氣 (Intent in harmony with Qi),
氣 合 於 神 (Qi in
harmony with Spirit),
神 合 於 動 (Spirit in harmony with Motion),
動 合 於 空 (Motion in harmony with Nothing).
When
a Yoruban
man in Nigeria get really attracted to a woman, he sends six shells
to her. In fact, the Yoruban word efa means both 'six'
and 'attracted'. If this chat up line works, the girl replies
with eight shells  ejo meaning both 'eight' and 'I
agree'!
6
persons
Whenever two people meet, there are really six people
present. There is each man as he sees himself, each man as the
other person sees him, and each man as he really is.
 William James.
Italian
natural ambigram
9 , 111 111 = 6 uno dopo un 9 (this
Italian sentence means "6 ones after
a 9" and can be read the same upside down).
6
Photo Snuff is an Indian tobacco brand.
• English
words which contain strings of 6 consonants: "bergschrund", "borschts", "eschscholtzia", "latchstring", "weltschmerz".
6 x
10^{5} is the number of engineers China produced in 2005.
In comparison, India produces nearly 5 x 10^{5} technical
graduates annually! (data 2008)
SIX in
different languages (© G.
Sarcone)
(Old English siex) 
Indoeuropean
 *SEKS   *SWEKS 
Sanskrit  S'AS' 
Greek, Attic  'EΞ HEX 
Latin  SEX  
Italian sei;
French six; Spanish and Portuguese seis;
Provençal sièis;
Catalan and Romansh sis; Romanian şase;
Sardinian ses. 
Old
Celtic  SUEK(O)S ,  SVEK(O)S  
Breton c’hwec’h;
Welsh chwech; Irish sé (things), seisear (persons). 
Old
Germanic  SEKS  
Dutch zes;
German sechs; Danish and Norwegian seks;
Swedish and Icelandic sex. 
Old
Slavic  S^{h}ESTI  
Russian шесть shest';
Czech šest; Slovenian šést;
Polish sześć. 
Proto
IndoIranian  *(K)SWACSH  
Persian شش shesh;
Hindi छः ch'eh. 
Evolution
from 'seal
script' to modern sinograph 六 :
Old Chinese (pron.)  lyuk  
Chinese 六 liù. 
ProtoSemitic
 *ShIDTh 
Semitic root  ShSh ,  ShT 
Ancient Egyptian [sỉs' or ỉs' (?)] sas;
Akkadian shishshu;
Punic shish. 
Arabic ستة sittâ;
Hebrew ששה shishah;
Maltese: sitta; Amharic sidist (pron.
sədəsətə). 
More
languages
Magyar hat.
Turkish altı.
Mayan uac.
Nahuatl chicuacē.
Suomi kuusi.
Zulu isithupha. 
HIDDEN
ROOTS
The roots of the word six are hidden
in the following words: bisextil, hexad, hexagon, Seicento,
semester, senary, sestet, sestina, sext, sextant, sextile,
sextuple, siesta, Sistine. In French: setier, sexagénaire,
sextuor, sizain. In Italian: sciamito [< Greek (he)xámiton],
seienne, staio. 

06  712  1323  2469  70200  201684  5H0P
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Encyclopedia
of Numbers © G. Sarcone, Archimedes Lab,
Genoa, Italy

