FRACTALS See
http://www.youtube.com/results?search_query=Mandelbrot+fractal+tour+guide
Also see
http://www.scientificamerican.com/article/mathematicsramanujan/
and
http://en.wikipedia.org/wiki/Selfsimilarity and also
http://en.wikipedia.org/wiki/Fractal and
http://en.wikipedia.org/wiki/Chaos_theory
In October 1992, when I was President of the Society of
Actuaries, I had a video presentation of
pictures of fractals and some background at the Annual Meeting of the
Society of Actuaries.
Robert Devaney developed the script. See
Chaos video, but start about 40% of the way through.
(OOPs is copywrite protected so will only play on my computer) If
interested, read Don's
Presidential
Address.
Julia Sets See
http://www.youtube.com/watch?v=2AZYZL8m9Q
Mandelbrot Sets See
http://www.youtube.com/watch?v=8ma6cV6fw24 Also see discussion by
Mandelbrot at
http://www.youtube.com/watch?v=ay8OMOsf6AQ
http://en.wikipedia.org/wiki/Mandelbrot_set
Newton Fractal See
http://vimeo.com/9770779
FIBONACCI SERIES See
http://www.youtube.com/watch?v=SjSHVDfXHQ4
The
first two terms are 1 and 1. Then use the formula
The first 21 Fibonacci numbers F_{n} for n = 0, 1, 2,
..., 20 are:

F_{0} 
F_{1} 
F_{2} 
F_{3} 
F_{4} 
F_{5} 
F_{6} 
F_{7} 
F_{8} 
F_{9} 
F_{10} 
F_{11} 
F_{12} 
F_{13} 
F_{14} 
F_{15} 
F_{16} 
F_{17} 
F_{18} 
F_{19} 
F_{20} 
0 
1 
1 
2 
3 
5 
8 
13 
21 
34 
55 
89 
144 
233 
377 
610 
987 
1597 
2584 
4181 
6765 
These numbers also give the solution to certain enumerative problems.
The most common such problem is that of counting the number of
compositions of 1s and 2s that sum
to a given total n: there are F_{n+1} ways to
do this. For example F_{6} = 8 counts the eight compositions:
1+1+1+1+1 = 1+1+1+2 = 1+1+2+1 = 1+2+1+1 = 2+1+1+1 = 2+2+1 = 2+1+2 =
1+2+2,
all of which sum to 6−1 = 5.
Golden Ratio is the only number that is one more than its
resiprocal. See http://en.wikipedia.org/wiki/Golden_ratio
The Fibonacci numbers have a
closedform solution. It is known as
Binet's formula:
where
Coin Game You
have a pile of coins, and you can take up to and including twice what your
opponent took last time…uses a winning strategy of leaving your opponent on
a Fibonacci number or on the sum of two or more Fibonacci numbers. The
person who picks up the last coin wins. On your first turn you can’t
take all.
Start with 11 coins.
Take 3, leaving 8 and you win, as 8 is a Fibonacci number.
Start with 12 coins.
Take 4 and the other guy can take 8 and you lose. So reduce the pile to the
sum of two Fibonacci numbers 8 +3 = 11 and take 1 reducing the 12 to 11.
Your opponent can now take 1 or 2. If he takes 1, you take 2 and he has 8
to choose from. If he takes 2, you take 1 and he has 8 to choose from.
Both are bad.
Whatever he chooses,
say 2, you take 1 and leave him with 5. He can take 1 or 2. If he takes 1,
you take 1… leaving him with 3. He can take 1 or 2…and you take what’s
left.
BINARY NUMBERS
The modern binary number system was invented by
Gottfried Leibniz in 1679.
Decimal pattern 
Binary numbers 
0 
0 
1 
1 
2 
10 
3 
11 
4 
100 
5 
101 
6 
110 
7 
111 
8 
1000 
9 
1001 
10 
1010 
11 
1011 

Nim The traditional game of
Nim used three rows of 3, 4, and 5 coins. On your turn, you can take any number from only
one row. The person to pick the last coin wins. The winning
strategy uses the Binary System to matched up pairs of powers of 2's.
3 = 2 +
1, 4 = 4, 5
= 4 +1
The person who goes first will win by taking 2 from the 3 pile to
create matches.
If the game was four
rows of 8, 13,
17, and 20:
20 =
16 + 4, 17 =
16 + 1, 13 =
8 + 4 +
1, 8
= 8
The person going first loses as we have
matches
If the game was five
rows of 5, 6,
7, 8, 10:
10 =
8 + 2, 8 = 8,
7 = 4 + 2 + 1, 6 =
4 + 2, 5 = 4 + 1
The person who goes first will win Take 2 from 5 pile, leaving 2 and 1.
The name is probably derived from
German nimm meaning "take
[imperative]", or the obsolete English verb nim of the same meaning.
Wythoff's game is a twoplayer
mathematical
game of strategy, played with two piles of
counters. Players take turns removing counters from one or both piles; in
the latter case, the numbers of counters removed from each pile must be
equal. The game ends when one person removes the last counter or counters,
thus winning.
Martin Gardner claims that the game was
played in China under the name 捡石子 jiǎn shízǐ ("picking stones"). The
Dutch mathematician
W. A. Wythoff published a mathematical
analysis of the game in 1907.
The Last Biscuit game is played by removing
cookies from two jars, either from a single jar, or the same number from
both jars. This game is also called the Puppies and Kittens game.
The strategy of winning Wythoff's Nim is to reduce the piles to a
number pair.. If the starting pile numbers are safe, the first player loses.
He is certain to leave an unsafe pair of piles, which his opponent can
always reduce to a safe pair on his next move. If the game begins with
unsafe numbers, the first player can always win by reducing the piles to a
safe pair and continuing to play to safe pairs.
Let us take the safe pairs in sequence, starting with the pair
nearest 0/0, and arrange them in a row with each smaller number above its
partner, as in the table below. Above the pairs write their "position
numbers." The top numbers of the safe pairs form a sequence we shall call A.
The bottom numbers form a sequence we shall call B.
Position (n) 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
A 
1 
3 
4 
6 
8 
9 
11 
12 
14 
16 
17 
19 
21 
22 
24 
B 
2 
5 
7 
10 
13 
15 
18 
20 
23 
26 
28 
31 
34 
36 
39 
You cannot win with 21, 34. a) If
you take 6 from the 34 pile leaving 21, 28 your opponent will take 4
from the 21 pile leaving you with 17, 28. b) If you take 7 from
both piles leaving 14, 27 your opponent will take 4 from the 27 pile leaving
you with 14, 23.
You cannot win with 21, 34. a) If you take 6
from both piles leaving 15, 28 your opponent will take 19 from the 28 pile
leaving you with 9, 15. b) If you take 7 from the 34 pile leaving your
opponent with 21, 27 your opponent will take 12 from each pile leaving you
with 9, 15.
The two sequences,
each one strictly increasing, have so many remarkable properties that dozens
of technical papers have been written about them. Each B number is the
sum of its A number and its position number. If we add an A number
to its B number, the sum is an A number that appears in the A sequence at a
position number equal to B. (An example is 8 + 13 = 21. The 13th number of
the A sequence is 21.)
Can we generate the sequences by a recursive
algorithm that is purely numerical? Yes.
Start with 1 as the top number of the first safe
pair. Add this to its position number to obtain 2 as the bottom number. The
top number of the next pair is the smallest positive integer not previously
used. It is 3. Below it goes 5, the sum of 3 and its position number. For
the top of the third pair write again the smallest positive integer not yet
used. It is 4. Below it goes 7, the sum of 4 and 3. Continuing in this way
will generate series A and B.
There is a bonus. We have discovered one of the
most unusual properties of the safe pairs. It is obvious from our procedure
that every positive integer must appear once and only once somewhere in the
two sequences.
Is there a way to generate the two sequences
nonrecursively? Yes. Wythoff was the first to discover that the numbers in
sequence A are simply multiples of the golden ratio rounded down to
integers! (He wrote that he pulled this discovery "out of a hat.")
Also see:
http://www.fq.math.ca/Scanned/173/hoggatt.pdf
COLORS Matt Parker, a Mathematician/Comedian, has
written a program to convert pictures into an Excel Spread Sheet as that
what a digital picture is. See
http://makeanddo4d.com/spreadsheet/
NUMBERPHILE & PI See
https://www.patreon.com/numberphile
SYMMETRY and ITERATION See
www. rmmsmsp.ucdenver.edu/instructormaterial/geometrydaisies.pps
Scientists strive to find
mathematical patterns. However, Sir Francis Bacan said: "There is no beauty
that hath not some strangeness in the proportion." Since beauty lies
in the eyes of the beholder, are the most beautiful creations short lyrics
or long symphonies?
David Wells said we might
conclude that the beauty of the Mandelbrot set is "romantic"
Hardy said: "A
mathematician, like a painter or poet, is a maker of patterns. If his
patterns are more permanent than theirs, it is because they are made with
ideas." Hardy also hoped that "nothing he had ever discovered
would have any practical use.
Paul Erdős said, "Why are numbers
beautiful? It's like asking why is
Beethoven's Ninth Symphony beautiful. If
you don't see why, someone can't tell you. I know numbers are
beautiful. If they aren't beautiful, nothing is".
Some see beauty in mathematical results that establish connections
between two areas of mathematics that at first sight appear to be unrelated.
These results are often described as deep. One example are
often cited is
Euler's identity:

This is a special case of
Euler's formula which states that, for any
real number x,



 Physicist
Richard Feynman called Euler's formula
"our jewel" and "the most remarkable formula in mathematics".
Modern examples include the
modularity theorem, which establishes an
important connection between
elliptic curves and
modular forms (work on which led to the
awarding of the
Wolf Prize to
Andrew Wiles and
Robert Langlands), and "monstrous
moonshine", which connects the
Monster group to
modular functions via
string theory for which
Richard Borcherds was awarded the
Fields Medal.
An example of mathematical
elegance is Leibnitz's series:


BIG NUMBERS See
http://www.guardian.co.uk/world/2009/mar/25/trilliondollarrescueplan
MATH TERMS
See
http://www.cuttheknot.org/glossary/stop.shtml
Repunits are numbers where all digits are the same, and
were named repunits by Albert Beiler.
In particular, the
decimal (base10) repunits that
are often referred to as simply repunits are defined as

 The problem of discovering which repunits are prime, and factoring
others, is similar to the analyzing Mersenne numbers (named after the
French monk
Marin Mersenne who studied them in the
early 17th century) of the form
. The first four Mersenne primes are 3, 7, 31 and 127. If n
is a
composite number then so is 2^{n} − 1.
The definition is therefore unchanged when written
where p is assumed prime. More generally, numbers of the form
without the primality requirement, are called Mersenne numbers.
Repunits
therefore have exactly
decimal digits.
Amazingly, the squares of the repunits
give the
Demlo numbers,
,
,
Ramchandra Dattaraya Kaprekar (1905–1986) studied the
Demlo numbers, named after a
train station 30 miles from Bombay on the then
G. I. P. Railway
where he had the idea of studying them.^{
}These are the numbers 1, 121, 12321, …,
which are the squares of the
repunits 1,
11, 111, ….^{
} He was an
Indian
recreational mathematician
who described several
classes of natural numbers
including the
Kaprekar,
Harshad
and
Self numbers
and discovered the
Kaprekar constant,
named after him. Despite having no formal postgraduate training and working
as a schoolteacher, he published extensively and became well known in
recreational mathematics circles.^{
}The following table contains some repunits.
3 x 37 = 111
6 x 37 = 222
9 x 37 = 333
12 x 37 = 444
15 x 37 = 555
18 x 37 = 666
21 x 37 = 777
24 x 37 = 888
27 x 37 = 999 
1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
111111 x 111111 = 12345654321
1111111 x 1111111 = 1234567654321
11111111 x 11111111 = 123456787654321
111111111 x 111111111=12345678987654321 
1 x 9 + 2 = 11
12 x 9 + 3 = 111
123 x 9 + 4 = 1111
1234 x 9 + 5 = 11111
12345 x 9 + 6 = 111111
123456 x 9 + 7 = 1111111
1234567 x 9 + 8 = 11111111
12345678 x 9 + 9 = 111111111
123456789 x 9 +10= 1111111111 
2519 Mod 2 = 1
2519 Mod 3 = 2
2519 Mod 4 = 3
2519 Mod 5 = 4
2519 Mod 6 = 5
2519 Mod 7 = 6
2519 Mod 8 = 7
2519 Mod 9 = 8
2519 Mod 10 = 9 
0 x 9 + 8 = 8
9 x 9 + 7 = 88
98 x 9 + 6 = 888
987 x 9 + 5 = 8888
9876 x 9 + 4 = 88888
98765 x 9 + 3 = 888888
987654 x 9 + 2 = 8888888
9876543 x 9 + 1 = 88888888
98765432 x 9 + 0 = 888888888
987654321 x 9  1 = 8888888888
9876543210 x 9  2 = 88888888888 
1 x 8 + 1 = 9
12 x 8 + 2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
12345 x 8 + 5 = 98765
123456 x 8 + 6 = 987654
1234567 x 8 + 7 = 9876543
12345678 x 8 + 8 = 98765432
123456789 x 8 + 9 = 987654321 


142857 x 2 = 285714
142857 x 3 = 428571
142857 x 4 = 571428
142857 x 5 = 714285
142857 x 6 = 857142 
91

times 
1 
= 
0 
9 
1 
7x7=49 
91 
times 
2 
= 
1 
8 
2 
67x67=4489 
91 
times 
3 
= 
2 
7 
3 
667x667=44889 
91 
times 
4 
= 
3 
6 
4 
6667x6667=44448889 
91 
times 
5 
= 
4 
5 
5 
66667x66667=4444488889 
91 
times 
6 
= 
5 
4 
6 
666667x666667=444444888889 
91 
times 
7 
= 
6 
3 
7 
6666667x6666667=44444448888889 
91 
times 
8 
= 
7 
2 
8 
etc. 
91 
times 
9 
= 
8 
1 
9 

1x9+2 = 11 
9 x 9 + 7 = 88 
9 x 9 = 81 
6 x 7 = 42 
12x9+3 = 111 
98 x 9 + 6 = 888 
99 x 99 = 9801 
66 x 67 = 4422 
123x9+4 = 1111 
987 x 9 + 5 = 8888 
999 x 999 =
998001 
666 x 667 = 444222 
1234x9+5 =
11111 
9876 x 9 + 4 = 88888 
9999 x 9999 =
99980001 
6666 x 6667 = 44442222 
12345x9+6 =
111111 
98765 x 9 + 3 = 888888 
etc 
etc 
123456x9+7 =
1111111 
987654 x 9 +2 =
8888888 


1234567x9+ =
11111111 
9876543x9+1= 88888888 


12345678x9+9=111111111 
98765432x9 = 888888888 


1x1=1 
4x4=16 
11x11=121 
34x34=1156 
111x111=12321 
334x334=111556 
1111x1111=1234321 
3334x3334=11115556 
11111x11111=123454321 
33334x33334=1111155556 
111111x111111=12345654321 
etc. 
1111111x1111111=1234567654321 

11111111x11111111=123456787654321 

111111111x111111111=12345678987654321 

Santa's Options Assuming Rudolph was in front, there are 40,320
ways to arrange the other eight reindeer.
Geometry
Are the eight balls
moving in a circle or a straight line?
http://showyou.com/v/ypNe6fsaCVtI/crazycircleillusion?u=multimotion
Roman Numerals The original Roman year had 10 named months
Martius "March", Aprilis "April", Maius "May", Junius
"June", Quintilis "July", Sextilis "August", September
"September", October "October", November "November",
December "December", and probably two unnamed months in the dead of
winter when not much happened in agriculture. The year began with Martius
"March". Numa Pompilius, the second king of Rome circa 700 BC, added the two
months Januarius "January" and Februarius "February". He also
moved the beginning of the year from Marius to Januarius and
changed the number of days in several months to be odd, a lucky number.
After Februarius there was occasionally an additional month of
Intercalaris "intercalendar". This is the origin of the leapyear day
being in February. In 46 BC, Julius Caesar reformed the Roman calendar
(hence the Julian calendar) changing the number of days in many months and
removing Intercalaris.
The numerals are: 1=I (unus); 5=V (quinque); 10=X (decem); C=100 (centum),
and M=1000 (mille). They also used 50=L (quinquaginta); and
500=D (quingenti).
When a small number comes before a larger
number, the smaller number is subtracted. 4 = IV or 51. When a smaller
number follows a larger one, the two are added together: 7 = VII, or 5 + 2
and 19 = XIX, or 10 + 9. Although the Romans used a decimal
system for whole numbers, they used a duodecimal system for fractions
because the divisibility by 12 made it easier to handle the common fractions
of 1/3 and 1/4.
Chinese Multiplication
See
http://www.youtube.com/watch?v=8iIU9EDC2GQ and
http://www.youtube.com/watch?v=maRN2fUOF0o
Leonhard Euler (17071783): One of his
many contributions was called "Euler's Formula". The formula
states that, for any real number x,
where e is the base of the natural logarithm, i is the
imaginary unit, and cos and sin are the trigonometric functions, with the
argument x given in radians. The formula is still valid if x
is a complex number. Richard Feynman called Euler's formula "our jewel"
and "one of the most remarkable, almost astounding, formulas in all of
mathematics".
Euler Line
In the 18th century, the Swiss mathematician Leonhard Euler noticed that
three of the
centers of a triangle are always
collinear (they always lie on a straight line). The three
centers that have this surprising property are the triangle's
centroid (where the three
medians of the triangle's sides meet),
circumcenter (where the
perpendicular bisectors of the triangle's sides meet) and the
orthocenter (where the three
altitudes to the vertices of the triangle meet). The distance from the orthocenter
to the centroid is two times the distance from the centroid to the
circumcenter. (Another center, the incenter,
where the bisectors of the three angles meet, is not on this line.)

Acute Triangle 

Obtuse Triangle 

Equilateral Triangle 
Triangle centers
Euler and the Nine Point Circle
The ninepoint circle is a circle that can be constructed for any
given triangle. It is so named because it passes through nine significant
concyclic points defined from the triangle. These nine points are:
The ninepoint circle is also known as Feuerbach's circle,
Euler's circle, and Terquem's circle.
To construct the nine point circle of a triangle, see
http://jwilson.coe.uga.edu/EMT668/EMT668.Folders.F97/Anderson/geometry/geometry1project/construction/construction.html
.1. Draw a triangle ABC and construct the midpoints of the three sides.
Label them as L, M, N.
2. Construct the feet of the altitudes of the triangle
ABC. Label them as D, E, F. Label the point of intersection of the three
altitudes as H. This is also called the orthocenter.
3. Construct the midpoints of the segments AH, BH, CH. Label
them as X, Y, Z.
4. Notice the nine points, L,M,N,D,E,F,X,Y, Z, lie in a circle called the
NinePoint Circle..
5. Construct the circumscribed circle for triangle
LMN. Label the center of that circle U.
The center U of the circumscribed circle for triangle LMN will also be
the center of the NinePoint Circle.
More on NinePoint Circle at
http://en.wikipedia.org/wiki/Triangle_center where we learn
the following:
Let A, B, C denote
the vertex angles of the reference triangle, and let x : y : z be
a variable point in trilinear
coordinates;
then an equation for the Euler line is

Center 
Trilinear Coordiates 
On Euler Line? 
Orttocenter 
Secant A: Secant B : Secant C 
Yes 
Centroid 
Cosecant A: Cosecant B: Cosecant C 
Yes 
Circumcenter 
Cosine A: Cosine B: Cosine C 
Yes 
Nine Point Circle 
Cosine (B  C): Cosine (C  A): Cosine (A
 B) 
Yes 
In Center 
1:1:1 
Only if Isosceles 
Euler and polyhedrons
A platonic solid is a
regular,
convex
polyhedron with
congruent
faces of
regular
polygons and the same number of faces
meeting at each
vertex. There are five regular
polyhedrons that meet those criteria, and each is named after its
number of faces.:
Euler's formula for polyhedrons is: V  E + F = 2 That
is the number of vertices, minus the number of edges, plus the number of
faces, is equal to two.
More on Euler
In
analytical mathematics,
Euler's identity (also known as Euler's equation) is the
equality:
Euler's
number e
is an important
mathematical constant,
approximately equal to 2.71828, that is the base of the
natural logarithm.^{
} It is the
limit of
(1 + 1/n)^{n} as
n becomes large, an expression that
arises in the study of
compound interest,
and can also be calculated as the sum of the infinite
series^{:}
e is the
unique
real number such that
the value of the
derivative (slope of
the
tangent line) of the
function f(x) = e^{x}
at the point x = 0 is equal to 1. The
function e^{x} so defined
is called the
exponential function,
and its
inverse is the
natural logarithm, or
logarithm to
base
e.
The number e is
of eminent importance in mathematics, alongside
0,
1,
π
and
i.
All five of these numbers play important and recurring roles across
mathematics, and are the five constants appearing in one formulation of
Euler's identity.
Like the constant π, e
is
irrational: it is not
a ratio of
integers; and it is
transcendental: it is
not a root of any nonzero
polynomial with
rational coefficients. The numerical value of e
truncated to 50
decimal places is
2.71828182845904523536028747135266249775724709369995...
An unexpected
Wallislike formula for
is given by the
Pippenger product
Another Pippinger product for
given by
This is analogous to the products due to Guillera (Sondow 2006).
and
e and compound interest
Let
be the
principal (initial investment), be
the annual compounded rate, the
"nominal rate," be
the number of times
interest is compounded per year (i.e.,
the year is divided into conversion
periods), and be
the number of years (the "term"). The
interest rate per
conversion period is then
If interest is compounded times
at an annual rate of (where,
for example, 10% corresponds to ),
then the effective rate over the
time (what an investor would earn if he did not redeposit his interest after
each compounding) is
The total amount of holdings after
a time when
interest is reinvested is then
Note that even if interest is compounded continuously, the return is still
finite since
where
e is the base of the
natural logarithm.
The time required for a given
principal to double (assuming
conversion period) is given by solving
or 
where
ln is the
natural logarithm. This function can
be approximated by the socalled
rule of 72:
Pi or
The number π is a mathematical constant, the ratio of a
circle's circumference to its diameter, commonly approximated as 3.14159. It
has been represented by the Greek letter "π" since the mid18th century,
though it is also sometimes spelled out as "pi" .
Being an
irrational number, π
cannot be expressed exactly as a
fraction Also,
π is a
transcendental number – a number that is
not the
root of any nonzero
polynomial having
rational
coefficients.
The earliest known use of the Greek letter
π to represent the ratio of a circle's circumference to
its diameter was by Welsh mathematician
William Jones in his 1706 work
Synopsis Palmariorum Matheseos; or, a New Introduction to the
Mathematics. ^{
}The Greek letter first appears there in the
phrase "1/2 Periphery ( π)" in the
discussion of a circle with radius one. Jones may have chosen
π because it was the first
letter in the Greek spelling of the word periphery.
However, he writes that his equations for
π are from the "ready pen of the truly ingenious Mr. John
Machin", leading to speculation that
Machin may have employed the
Greek letter before Jones. ^{[105]}
It had indeed been used earlier for geometric concepts.
William Oughtred used
π and δ, the Greek letter
equivalents of p and d, to express ratios of periphery and
diameter in the 1647 and later editions of Clavis
Mathematicae. After Jones introduced the Greek letter in
1706, it was not adopted by other mathematicians until
Euler started using it,
beginning with his 1736 work
Mechanica. Before then,
mathematicians sometimes used letters such as c or p
instead.^{[105]}
Because Euler corresponded heavily with other mathematicians in
Europe, the use of the Greek letter spread rapidly. In 1748,
Euler used π in his widely read
work
Introductio in analysin infinitorum
(he wrote: "for the sake of brevity we will write this number as
π; thus π
is equal to half the circumference of a circle of radius 1") and
the practice was universally adopted thereafter in the
Western world.
Look at using continued fractions to represent pi at
https://www.bing.com/videos/searchq=pi+represented+by+series+and+continuous+fractions&qpvt=pi+represented+by+series+and
+continuous+fractions&FORM=VDRE
NUMBER THEORY
Numbers are classified according to
type. The first type of number is the first type you ever learned about: the
counting, or "natural" numbers:
1, 2, 3, 4, 5, 6, ...
The next type is the "whole"
numbers, which are the natural numbers together with zero:
0, 1, 2, 3, 4, 5, 6, ...
Then come the "integers", which
are zero, the natural numbers, and the negatives of the naturals:
... 6, 5, 4, 3, 2, 1, 9, 1,
2, 3, 4, 5, 6, ...
The next type is the "rational", or
fractional, numbers, which are technically regarded as ratios
(divisions) of integers. In other words, a fraction is formed by dividing
one integer by another integer. a way to generate all the rational
numbers with no repeats is at :
https://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree And a 1999
paper by Calkin and Wilf at
https://www.math.upenn.edu/~wilf/website/recounting.pdf
Sums of Natural Numbers
The sum of the first n natural numbers is: 1 + 2 + 3 + 4 ....... + n
= n(n+1)/2 Gauss developed this formula when in primary school:
the average of the first number and the last number times the number of
numbers!
If n = 6: 1 + 2 + 3 + 4 + 5 + 6 = 21 = (6x7)/2
Sums of Even Numbers The sum of the first k even natural
numbers is: 2 + 4 + 6 ..... + 2k = k(k+1).
If k = 3: 2 + 4 + 6 = 12 = (3x4)
Sums of Odd Numbers The sum of the first k odd natural
numbers is: 1 + 3 + 5 ..... + (2k  1) = k^{2}
If k = 3: 1 + 3 + 5 = 9
= (3x3) ^{
}

Sums of Squares The sum of the squares of
the first n natural numbers is:
30 = (1^{2} + 2^{2} + 3^{2} + 4^{2}) =
1 + 4 + 9 + 16
Also see:
http://www.takayaiwamoto.com/Sums_and_Series/sumsqr_1.html and
http://www.math.utah.edu/~palais/sums.html
365 = ( 10^{2} + 11^{2} + 12^{2}) = (13^{2}
+ 14^{2})
Sums of Cubes The sum of the cubes of the first n natural numbers is:

As you can see, the sum of the
cubes of the first n natural numbers
is equal to the square of the sum of the first n natural numbers.
Divisor Function: D(x) = number of
divisors in a number including 1 and x. If m and n are relatively
prime, then D(mn) = D(m) x D(n)
m 
n 
mn 
D(m) 
D(n) 
D(mn) = D(m) x D(n) 
Divisors 
2 
3 
6 
2 
2 
4 
1, 2, 3, 6 
4 
9 
36 
3 
3 
9 
1, 2, 3, 4, 6, 9, 12,18, 36 
15 
1 
15 
4 
1 
4 
1, 3, 5, 15 
28 
1 
28 
6 
1 
6 
1, 2, 4, 7, 14, 28 
15 
28 
420 
6 
4 
24 
1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20,
21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420 
Take any integer. Examples: 15 and 28. Write its factors.
Underneath each factor write the D(x) for each factor:
Factors of 15 


15 
5 
3 
1 
D(Factors of 15) 


4 
2 
2 
1 
Factors of 28 
28 
14 
7 
4 
2 
1 
D(Factors of 28) 
6 
4 
2 
3 
2 
1 
Liouville's Theorem says: The sum
of the cubes of the numbers in the second line equals the square of the sum
of those same numbers: 64 + 8 + 8 + 1 = 81. 4 + 2 + 2 + 1 = 9,
and 9 squared = 81. The sum of the cube of the numbers in the fourth line equals the square of
the sum of those same numbers: 216 + 64 + 8 + 27 + 8 + 1 =
324.
6 + 4 + 2 + 3 + 2 + 1 = 18, and18 squared = 324.
A Conjecture is: Pick
small numbers and the sum of their cubes is less than the square of their
sums. But if you pick large numbers the converse is true. See examples
using four numbers are below:





Their Sum 
Square of their Sum 
Sum of their Cubes 
Square of their
Sum minus the Sum of their Cubes 
A 
1 
1 
1 
3 
6 
36 
30 
6 
B 
1 
1 
2 
2 
6 
36 
18 
18 
C 
2 
2 
3 
4 
11 
121 
107 
14 
D 
1 
2 
2 
4 
9 
81 
81 
0 
E 
1 
1 
2 
5 
9 
81 
135 
54 
F 
1 
2 
6 
8 
17 
289 
737 
448 
G 
3 
6 
7 
8 
24 
576 
1098 
522 
Sum of 1  2 + 3  4 +
5  6 ... =
http://en.wikipedia.org/wiki/1_%E2%88%92_2_%2B_3_%E2%88%92_4_%2B_%C2%B7_%C2%B7_%C2%B7
In a 1749 report,
Leonhard Euler admits that the series diverges but prepares to
sum it anyway:
...when it is said that the sum
of this series 1−2+3−4+5−6 etc. is ^{1}⁄_{4},
that must appear paradoxical. For by adding 100 terms of this series, we
get −50, however, the sum of 101 terms gives +51, which is quite different
from ^{1}⁄_{4} and
becomes still greater when one increases the number of terms. But I have
already noticed at a previous time, that it is necessary to give to the
word sum a more extended meaning....
Euler proposed a
generalization of the word "sum" several times; see
Euler on infinite series. In the case of 1 − 2
+ 3 − 4 + ..., his ideas are similar to what is now known as
Abel summation:
...it is no more
doubtful that the sum of this series 1−2+3−4+5 + etc. is
^{1}⁄_{4}; since it
arises from the expansion of the formula ^{1}⁄_{(1+1)2},
whose value is incontestably ^{1}⁄_{4}.
The idea becomes clearer by considering the general series 1 − 2x +
3x^{2} − 4x^{3} + 5x^{4} − 6x^{5}
+ &c. that arises while expanding the expression
^{1}⁄_{(1+x)2},
which this series is indeed equal to after we set x = 1.
There are many ways
to see that, at least for
absolute values x < 1, Euler is right in that

One can take the
Taylor expansion of the righthand side, or apply the formal
long division process for polynomials. Starting from the lefthand side,
one can follow the general heuristics above and try multiplying by (1+x)
twice or
squaring the
geometric series 1 − x + x^{2}
− .... Euler also seems to suggest
differentiating the latter series term by term.
In the modern view, the series 1 − 2x + 3x^{2} − 4x^{3}
+ ... does not define a
function at x = 1, so that value
cannot simply be substituted into the resulting expression. Since the
function is defined for all x < 1, one
can still take the limit as x approaches 1, and this is the
definition of the Abel sum:

Leonhard Euler is most famous for the
"Euler Identity":
The special case, with x =
π
gives the beautiful identity:
,
which involves 0, 1, i, e and
π.
We can easily see this result by looking at the diagram of a
circle.
The x axis contains the real numbers
(Re) and the y axis contains
the imaginary numbers (Im).
The radius of the circle is
i.
The identity is when the angle equals
π 

Triangular
Numbers
The numbers which can be arranged in a compact triangular pattern are
termed as triangular numbers. The triangular numbers are formed by partial
sum of the series 1+2+3+4+5+6+7......+n. So
T_{1} = 1
T_{2} = 1 + 2 = 3
T_{3} = 1 + 2 + 3 = 6
T_{4} = 1 + 2 + 3 + 4 = 10
So the n^{th} triangular number can be obtained as T_{n}
= n(n+1)/2, where n is any natural number. In other words triangular numbers
form the series 1,3,6,10,15,21,28.....
n^{2} = the sum of two consecutive triangular numbers,
because T_{n} + T_{n1 }= n(n+1)/2 +
(n1)(n)/2 = n^{2}
^{Cubics, Quartics, and Quintics}
Niccolo Tartaglia, who solved the cubic, failed miserably for the
rest of his life (mainly because he spent it trying to discredit Cardano).
See
http://www.storyofmathematics.com/16th_tartaglia.html 
Giralamo Cardano, who
stole Tartaglia's solution, is also credited with solving the cubic. 
Lodovico Ferrara, solved the general quartic,
was poisoned, probably by his sister, over an inheritance dispute. 
Evariste Galois, who showed the general quintic
was unsolvable, died in a duel at the age of 29. 
Niels Henrik Abel, who duplicated and extended Galois' proof
independently, finally managed to receive his first faculty position.
The notification letter arrived a few days after Abel had died of pneumonia.
He was 29. 
Brocard's Problem: N factorial + 1 = X squared. This is
true for X = 5, 11, and 71, but that may be all. Pierre Rene Jean Baptiste Henri Brocard: (1845  1922).
Bruce Berndt and William Galway used a computer in 2000 to show there are no
other solutions up to N = one billion.
Primes, other than 2 or 3 are either of the form 6n + 1 or 6n  1.
Lychrel Numbers. Most numbers become a palindrome by reversing their
digits and adding repeatedly. (349 + 943 = 1292, 1292 + 2921 = 4213,
4213 + 3124 = 7337 a palindrome. Those that do not convert, are Lychrel Numbers. The name "Lychrel" was coined by Wade Van Landingham: a
rough anagram of his girlfriend's name Cheryl.
Catalan's conjecture (occasionally now referred to as
Mihăilescu's theorem) was conjectured by the mathematician Eugene
Charles Catalan in 1844 and proven in 2002 by Preda
Mihăilescu.
To understand the conjecture, notice that 2^{3} and 3^{2}
(i.e. 8 and 9) are two
powers of
natural numbers,
whose values 8 and 9 respectively are consecutive. The conjecture states
that this is the only case of two consecutive powers.
That is to say, that the only
solution in the natural numbers
of
 x^{a} − y^{b} = 1
for x, a, y, b > 1 is x = 3, a
= 2, y = 2, b = 3.
SQUARE ROOT The square root of the number 81 is 9. 81 is the only
number whose square root is the sum of its digits.
What did Pythagoras say when he was first confronted with the square root of
2? "There has to be a rational explanation for this."
PARTITIONS See
http://en.wikipedia.org/wiki/Partition_(number_theory)
In
number theory and
combinatorics, a partition of
a positive
integer n, also called an integer
partition, is a way of writing n as a
sum of positive integers. Two sums that
differ only in the order of their summands are considered the same
partition. If order matters, the sum becomes a
composition. For example, 4 can be
partitioned in five distinct ways:
 4
 3 + 1
 2 + 2
 2 + 1 + 1
 1 + 1 + 1 + 1
The orderdependent composition 1 + 3 is the same partition as 3 + 1,
while 1 + 2 + 1 and 1 + 1 + 2 are the same partition as 2 + 1 + 1.
A summand in a partition is also called a part. The number of
partitions of n is given by the partition function p(n).
So p(4) = 5. The notation λ ⊢
n means that λ is a partition of n.
Partitions can be graphically visualized with
Young diagrams (boxes) or
Ferrers diagrams
(dots). They occur in a number of branches of
mathematics and
physics, including the study of
symmetric polynomials, the
symmetric group and in
group representation theory in general.
Young diagrams associated to the
partitions of the positive integers 1 through 8. They are arranged so that
images under the reflection about the main diagonal of the square are
conjugate partitions. (below)
GAMMA FUNCTION
The (complete) gamma function
is defined to be an extension of the
factorial to
complex and
real number arguments. It is related to the
factorial by
a slightly unfortunate notation due to Legendre which is now universally
used instead of Gauss's simpler
There are no points
at which
.
The gamma function can be defined as a
definite integral
for (Euler's
integral form) as
or or
The complete gamma function can
be generalized to the upper
incomplete gamma function and
lower
incomplete gamma function .

Plots of the real and imaginary parts of in
the complex plane are illustrated above.
Below we see
the
gamma function along part of the real axis:
Integrating = by
parts for a
real argument, it can be seen that =
If is
an
integer = so
the gamma function reduces to the
factorial for a
positive integer argument.
A beautiful relationship between and
the
Riemann zeta function is
given by:
for
The gamma function can also be defined by an
infinite product form:
where is
the
EulerMascheroni constant
The Euler limit form is so, =
The reciprocal of the gamma function is
an
entire function expressed as
where is
the
EulerMascheroni constant
and is
the
Riemann zeta function
