Maths
- 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
- 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