Did you know that...

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

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.

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]

67. The length of the diagonal of the unit square equals the square root of 2

68. Every composite number is the product of some factors and also the some of the same numbers

69. Simple quadrilaterals tessellate the plane

70. There is a simple solution to the affirmative action problem

71. Two simple polygons of equal area can be dissected into a finite number of congruent polygons

72. cos(36°) = (1 + √5)/4

73. 1/3 + 1/4 = 7/12

74. Σ2-n = Σn•2-n

How we view history

We view the history of mathematics from our own position of understanding and sophistication. There can be no other way but nevertheless we have to try to appreciate the difference between our viewpoint and that of mathematicians centuries ago. Often the way mathematics is taught today makes it harder to understand the difficulties of the past.

There is no reason why anyone should introduce negative numbers just to be solutions of equations such as x + 3 = 0. In fact there is no real reason why negative numbers should be introduced at all. Nobody owned -2 books. We can think of 2 as being some abstract property which every set of 2 objects possesses. This in itself is a deep idea. Adding 2 apples to 3 apples is one matter. Realising that there are abstract properties 2 and 3 which apply to every sets with 2 and 3 elements and that 2 + 3 = 5 is a general theorem which applies whether they are sets of apples, books or trees moves from counting into the realm of mathematics.

Negative numbers do not have this type of concrete representation on which to build the abstraction. It is not surprising that their introduction came only after a long struggle. An understanding of these difficulties would benefit any teacher trying to teach primary school children. Even the integers, which we take as the most basic concept, have a sophistication which can only be properly understood by examining the historical setting.

Brilliant discoveries?

It is quite hard to understand the brilliance of major mathematical discoveries. On the one hand they often appear as isolated flashes of brilliance although in fact they are the culmination of work by many, often less able, mathematicians over a long period.

For example the controversy over whether Newton or Leibniz discovered the calculus first can easily be answered. Neither did since Newton certainly learnt the calculus from his teacher Barrow. Of course I am not suggesting that Barrow should receive the credit for discovering the calculus, I'm merely pointing out that the calculus comes out of a long period of progress starting with Greek mathematics.

Now we are in danger of reducing major mathematical discoveries as no more than the luck of who was working on a topic at "the right time". This too would be completely unfair (although it does go some why to explain why two or more people often discovered something independently around the same time). There is still the flash of genius in the discoveries, often coming from a deeper understanding or seeing the importance of certain ideas more clearly.

Notation and communication

There are many major mathematical discoveries but only those which can be understood by others lead to progress. However, the easy use and understanding of mathematical concepts depends on their notation.

For example, work with numbers is clearly hindered by poor notation. Try multiplying two numbers together in Roman numerals. What is MLXXXIV times MMLLLXIX? Addition of course is a different matter and in this case Roman numerals come into their own, merchants who did most of their arithmetic adding figures were reluctant to give up using Roman numerals.

What are other examples of notational problems. The best known is probably the notation for the calculus used by Leibniz and Newton. Leibniz's notation lead more easily to extending the ideas of the calculus, while Newton's notation although good to describe velocity and acceleration had much less potential when functions of two variables were considered. British mathematicians who patriotically used Newton's notation put themselves at a disadvantage compared with the continental mathematicians who followed Leibniz.

Let us think for a moment how dependent we all are on mathematical notation and convention. Ask any mathematician to solve ax = b and you will be given the answer x = b/a. I would be very surprised if you were given the answer a = b/x, but why not. We are, often without realising it, using a convention that letters near the end of the alphabet represent unknowns while those near the beginning represent known quantities.

It was not always like this: Harriot used a as his unknown as did others at this time. The convention we use (letters near the end of the alphabet representing unknowns) was introduced by Descartes in 1637. Other conventions have fallen out of favour, such as that due to Viète who used vowels for unknowns and consonants for knowns.

Of course ax = b contains other conventions of notation which we use without noticing them. For example the sign "=" was introduced by Recorde in 1557. Also ax is used to denote the product of a and x, the most efficient notation of all since nothing has to be written!

A challenge

If you think that mathematical discovery is easy then here is a challenge to make you think. Napier, Briggs and others introduced the world to logarithms nearly 400 years ago. These were used for 350 years as the main tool in arithmetical calculations. An amazing amount of effort was saved using logarithms, how could the heavy calculations necessary in the sciences ever have taken place without logs.

Then the world changed. The pocket calculator appeared. The logarithm remains an important mathematical function but its use in calculating has gone for ever.

Here is the challenge. What will replace the calculator? You might say that this is an unfair question. However let me remind you that Napier invented the basic concepts of a mechanical computer at the same time as logs. The basic ideas that will lead to the replacement of the pocket calculator are almost certainly around us.

We can think of faster calculators, smaller calculators, better calculators but I'm asking for something as different from the calculator as the calculator itself is from log tables. I have an answer to my own question but it would spoil the point of my challenge to say what it is. Think about it and realise how difficult it was to invent non-euclidean geometries, groups, general relativity, set theory, .... .