these pp is on mathmeticians

5. Mathematics have come a great length. What people do not realize is that
mathematics are used everywhere; GPS, cellular phones, computers, the
weather, our television cable guide, sound waves (use a sinusoidal wave or
sine wave), clocks, speed limits, shopping, balancing our checkbooks, etc.
Math is used everywhere. Another thought; if trigonometry was discovered
first, why is it we take Algebra before Trigonometry? Or how did they do
Trigonometry without knowing Algebra first? Hmm.
A couple of fun facts:
111111111 12345679 x 9 = 111111111
x111111111 and
12345678987654321 12345679 x 8 = 98765432

6. Unsolved Math Problems
1. Are there infinitely many primes of the form n^2+1?
2. Is every integer larger than 454 the sum of seven or fewer positive cubes?
3. Start with any positive integer. Halve it if it is even; triple it and add 1 if it is
odd. If you keep repeating this procedure, must you eventually reach the
number 1?
For example, starting with the number 6, we get: 6, 3, 10, 5, 16, 8, 4, 2, 1.
4. Let f(n) be the maximum possible number of edges in a graph on n vertices in
which no two cycles have the same length. Determine f(n).
5. Prove: If G is a simple graph on n vertices and the number of edges of G is
greater than n(k-1)/2, then G contains every tree with k edges
http://math.whatcom.ctc.edu/content/Links.phtml?cat=60&c=0

10. Euclid's Geometry

11. What is Euclid’s Geometry 1.1
• Euclidean geometry, the study of plane and solid
figures on the basis of axioms and theorems
employed by the Greek
mathematician Euclid (c. 300 BCE). In its rough
outline, Euclidean geometry is the plane and solid
geometry commonly taught in secondary schools.
Indeed, until the second half of the 19th
century, when non-Euclidean geometries attracted
the attention of mathematicians, geometry meant
Euclidean geometry.

12. What is Euclid’s Geometry 1.2
• It is the most typical expression of general
mathematical thinking. Rather than the
memorization of simple algorithms to solve equations
by rote, it demands true insight into the subject,
clever ideas for applying theorems in special
situations, an ability to generalize from known facts,
and an insistence on the importance of proof. In
Euclid’s great work, the Elements, the only tools
employed for geometrical constructions were the
ruler and the compass—a restriction retained in
elementary Euclidean geometry to this day.

13. Difference between Axioms and
Postulates.
 Postulates - The assumptions
that were specific to geometry are
called postulates.
Axioms - The assumptions that
are used throughout mathematics
and not specifically linked to
geometry are called Axioms.

14. All Axioms and Postulates.
Axioms:-
• Things that equal the same thing also equal one another.
• If equals are added to equals, then the wholes are equal.
• If equals are subtracted from equals, then the remainders
are equal.
• Things that coincide with one another equal one another.
• The whole is greater than the part.
• Things which are double of the same things are equal to
one and other.
• Things which are halves of the same things are equal to one
another.

15. Postulates
1. A straight line segment can be drawn joining any two
points.
2. Any straight line segment can be extended indefinitely in a
straight line.
3. Given any straight line segment, a circle can be drawn
having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. if a straight line falling on 2 straight lines makes the
interior angles on the same side of it taken together less
than 2 right angles, then 2 straight lines, if produced
indefinitely, meet on that side on which the sum of angles is
less than 2 right angles.

16. Euclid’s 1 to 4 Postulates

17. Euclid of Alexandria
• Euclid of Alexandria is the most
prominent mathematician of antiquity
best known for his treatise on
mathematics The Elements. The long
lasting nature of The Elements must
make Euclid the leading mathematics
teacher of all time. However little is
known of Euclid's life except that he
taught at Alexandria in Egypt.

18. Srinivasa Ramanujan
Srinivasa Ramanujan was one of India's
greatest mathematical geniuses. He made
substantial contributions to the analytical
theory of numbers and worked on elliptic
functions, continued fractions, and infinite
series.
Ramanujan was shown how to solve cubic
equations in 1902 and he went on to find his own
method to solve the quartic. The following
year, not knowing that the quintic could not be
solved by radicals, he tried (and of course failed)
to solve the quintic.

19. Rene Descartes
Rene Descartes was a great philosopher and
thinker, many overlook his contribution to math
because of his overwhelming additions to the field
of philosophy, however we would like to point out
this mans work on mathematics so that he gets
even more credit to his name. By the way he
passed away from a cold, away from him native
France, and could have probably made an even
bigger impact on modern science if he had not
passed away in a relatively early age.

20. Thales
Thales, an engineer by trade, was the first of the Seven Sages, or
wise men of Ancient Greece. Thales is known as the first
Greek philosopher, mathematician and scientist. He founded the
geometry of lines, so is given credit for introducing abstract
geometry.
Thales is credited with the following five theorems of geometry:
A circle is bisected by its diameter.
Angles at the base of any isosceles triangle are equal.
If two straight lines intersect, the opposite angles formed are
equal.
If one triangle has two angles and one side equal to another
triangle, the two triangles are equal in all respects. (See
Congruence)
Any angle inscribed in a semicircle is a right angle. This is
known as Thales' Theorem.

21. Q: Why do mathematicians, after a dinner at a Chinese restaurant,
always insist on taking the leftovers home?
A: Because they know the Chinese remainder theorem!
Teacher: "Who can tell me what 7 times 6 is?"
Student: "It's 42!"
Teacher: "Very good! - And who can tell me what 6 times
7 is?"
Same student: "It's 24!"

22. Q: What does the zero say to the the eight?
A: Nice belt!
Q: What is the difference between a Ph.D. in mathematics and a
large pizza?
A: A large pizza can feed a family of four...
Q: How does a mathematician induce good behavior in her children?
A: `I've told you n times, I've told you n+1 times...'
Q: What do you get if you divide the cirucmference of a jack-
o-lantern by its diameter?
A: Pumpkin Pi!
Q: What is a topologist?
A: A person who cannot tell a doughnut from a coffee mug.

23. Parallelogram
I have:
2 sets
of parallel sides
2 sets of equal sides
opposite angles equal
adjacent angles supplementary
diagonals bisect each other
diagonals form 2 congruent triangles

24. Is a square a rectangle?
Some people define categories exclusively, so that a rectangle is a
quadrilateral with four right angles that is not a square. This is
appropriate for everyday use of the words, as people typically use
the less specific word only when the more specific word will not
do. Generally a rectangle which isn't a square is an oblong.
But in mathematics, it is important to define categories
inclusively, so that a square is a rectangle. Inclusive categories
make statements of theorems shorter, by eliminating the need
for tedious listing of cases. For example, the visual proof that
vector addition is commutative is known as the "parallelogram
diagram". If categories were exclusive it would have to be
known as the "parallelogram
(or rectangle or rhombus or square) diagram"!

25. Is a square a rectangle?
Some people define categories exclusively, so that a rectangle is a quadrilateral
with four right angles that is not a square. This is appropriate for everyday use
of the words, as people typically use the less specific word only when the more
specific word will not do. Generally a rectangle which isn't a square is an
oblong.
But in mathematics, it is important to define categories inclusively, so that a
square is a rectangle. Inclusive categories make statements of theorems
shorter, by eliminating the need for tedious listing of cases. For example, the
visual proof that vector addition is commutative is known as the "parallelogram
diagram". If categories were exclusive it would have to be known as the
"parallelogram (or rectangle or rhombus or square) diagram"!

26. Parallelogram
I have:
2 sets
of parallel sides
2 sets of equal sides
opposite angles equal
adjacent angles supplementary
diagonals bisect each other
diagonals form 2 congruent triangles