1. =3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 ...
2. A sphere has two sides. However, there are one-sided surfaces.
3. There are shapes of constant width other than the circle. One can even drill square holes.
4. There are just five regular polyhedra
5. In a group of 23 people, at least two have the same birthday with the probability greater than 1/2
6. Everything you can do with a ruler and a compass you can do with the compass alone
7. Among all shapes with the same perimeter a circle has the largest area.
8. There are curves that fill a plane without holes
9. Much as with people, there are irrational, perfect, complex numbers
10. As in philosophy, there are transcendental numbers
11. As in the art, there are imaginary and surreal numbers
12. A straight line has dimension 1, a plane - 2. Fractals have mostly fractional dimension
13. You are wrong if you think Mathematics is not fun
14. Mathematics studies neighborhoods, groups and free groups, rings, ideals, holes, poles and removable poles, trees, growth ...
15. Mathematics also studies models, shapes, curves, cardinals, similarity, consistency, completeness, space ...
16. Among objects of mathematical study are heredity, continuity, jumps, infinity, infinitesimals, paradoxes...
17. Last but not the least, Mathematics studies stability, projections and values, values are often absolute but may also be extreme, local or global.
18. Trigonometry aside, Mathematics comprises fields like Game Theory, Braids Theory, Knot Theory and more 19. One is morally obligated not to do anything impossible
20. Some numbers are square, yet others are triangular
21. The next sentence is true but you must not believe it
22. The previous sentence was false
23. 12+3-4+5+67+8+9=100 and there exists at least one other representation of 100 with 9 digits in the right order and math operations in between
24. One can cut a pie into 8 pieces with three movements
25. Program=Algorithms+Data Structures
26. There is something the dead eat but if the living eat it, they die.
27. A clock never showing right time might be preferable to the one showing right time twice a day
28. Among all shapes with the same area circle has the shortest perimeter
Did you know that..., part 2.
29. Curves of infinite length may enclose finite areas.
30. Falsity implies anything.
31. There is order in chaos.
32. To get cafe au lait one should carry coffee to milk and not milk to coffee.
33. Sets may be thick, thin and normal.
34. In some circumstances index equals the content.
35. In other circumstances, an index may have a content of its own.
36. There are things distant yet near. There are others that are near yet distant.
37. There are three plane regions that share exactly the same boundary.
38. A continuous linear function must have the form f(x)=ax. Discontinuous linear functions look dreadful.
39. A continuous function may grow considerably virtually without changing.
40. You can't add apples and oranges but you can add their shapes.
41. There are many things that can be added: numbers, vectors, matrices, spaces, shapes, sets, functions, equations, strings, chains...
42. Among any two integers or real numbers one is larger, another smaller. But you can't compare two complex numbers.
43. The only triangle with rational sides and angles is equilateral.
44. 0!=1
45. One is morally obligated to do everything impossible.
46. The word 'fraction' derives from the Latin fractio - to break. However, there are continuous fractions.
47. For every object there is a distance at which it looks its best.
48. At any given time in New York there live at least two people with the same number of hairs.
49. Sometimes in order to add one has to take the difference.
50. Demographic tests show that the person least likely to buy Wired magazine is an American schoolteacher.
Did you know that..., part 3.
51. Complex numbers are in a sense perfect while there is little doubt that perfect numbers are complex.
52. You can position 10 defenders of a square castle so that on every side there will be 5 men.
53. There are many things that can be multiplied: numbers, vectors, matrices, functions, equations, sets, pegs...
54. A torus may be brushed smooth but a sphere can not.
55. A circle may be quite useful in drawing straight lines.
56. In the sequence of all integers, there are arbitrary long runs with no primes.
57. With just one caveat, anything you can do with a compass and a ruler you can do with the ruler alone.
58. There are really impossible things.
59. You can add apples and oranges.
60. Complex number to a complex power may be real.
61. Irrational number to an irrational power may be rational.
62. There are trisectable angles that are not constructible.
63. There exist triangular numbers that are also square.
64. No two integers are equidistant from the square root of 2
65. Almost every integer has a digit 3 in it
66. C0 - C0 = [-1, 1]