I have heard comments about boys generally being better than girls at math. I've even heard "scientific" reasons; something about girls having more synapses (connections) between brain hemispheres so that boys do better in logical, more linear things, while girls develop the more broad based "intuitive" style of thinking.
I don't know if any of that reasoning is true or not, but as a math major in college, I found that my best qualities in math were seeing the "overall picture" and grasping similarities between two widely different areas -- the more feminine style described above. Granted, I also needed the "linear" quality to grind out the answer or the proof (friends often mistakenly say, during a game, "let Stan keep score - he's a mathematician"; I tell them I'm not an arithmatician, I can't add for the life of me!), but the things that made me good at math are not at all the things I hear about why boys are supposed to be better.Return to the Loose Cannons -- Mathematics subject list
I once overheard a conversation in which one person was trying to console another by saying "Count your blessings." I'm not sure the other person felt very consoled, but it got me thinking about a mathematical question.
Interpolating from the overheard conversation, let me define a blessing as something that keeps you from thinking about bad things. When someone says "Count your blessings," they are attempting to keep you from thinking about bad things. So merely having blessings to count is, in itself, a blessing.
As an example, suppose a person has a nice house, a good job and a loving mate. This person would have these three "concrete" blessings, but she would also have the more abstract blessing of having blessings to count, making a total of four blessings.
Now suppose there is a person who is severely lacking in concrete blessings. In fact, he cannot even think of one. My question is: Can he still say that he has the abstract one? For if he could count it, he would have it; but does the act of counting it initiate its existence? Or must it exist before it can be counted? (I suppose just this contemplation itself would keep one from thinking about bad things -- unless, of course, this contemplation itself, is a bad thing:-)
I've tried to analyze this to see if it is a problem with definitions, but I can't seem to get behind it all. Any ideas?Return to the Loose Cannons -- Mathematics subject list
The four colour map theorem was, for a long time, one of the famous unsolved problems in mathematics. The conjecture was that a "map" of arbitrary "countries" requires at most four colours to fill in the countries so that no two countries sharing a common border are coloured the same. It has apparently been proved (in a most aesthetically unsatisfactory way) using huge computer searches. In the past, the theorem had countless "proofs" that later turned out to be wrong or incomplete, so I hate to add to the mess; but here is an avenue that I find interesting (although it doesn't lead very far with my limited background).
The idea is basically another observation of a progression (see "Curve Space Three Ways"). A zero dimensional point requires at most one colour. A one dimensional line broken into segments requires at most two colours. It is conjectured that a two dimensional plane requires at most four colours. This pattern (0 uses 1, 1 uses 2, 2 uses 4) seems to be an exponential series of powers of two. This suggests that if it looked like three dimensional space required at most "two-cubed" or eight colours, then I would really have something here. Well, after thinking about it for a bit, I realized that one could easily construct a three dimensional "map" of solid touching shapes that would require any number of colours. My guess that eight colours might suffice quickly shattered. However (and this is where I hit the ceiling of my abilities), I can imagine that there might be some kind of subset of three dimensional maps wherein eight colours do suffice, and that the subset, extrapolated from three to two dimensions, would still end up being the general case in two dimensions. If such a case could be found and could be generalized to "n" dimensions, this would possibly provide a more aesthetically appealing proof. But I'm not holding my breath.Return to the Loose Cannons -- Mathematics subject list
The first (right-most) place in a base-10 (decimal) number may be one of ten possible digits -- 0 through 9. The next place has ten possibilities for each of the original ten, making 102 or 100 numbers. So on for each additional digit --103, 104, etc. In base 2 (binary), the progression is 21, 22, 23, 24, etc. In general, the progression is a power sequence of the base number.
What about the following progression:
The first place (one's) has a binary character, using the digits 0 and 1. The next place (ten's) takes on an additional digit to form a trinary character using 0, 1, and 2. The next place (hundred's) uses 0, 1, 2, and 3, and so on. The progression of numbers would look like this:
0, 1, 10, 11, 20, 21, 100, 101, 110, 111, 120, 121, 200, 201, 210, 211, 220, 221, 300, 301, 310, 311, 320, 321, 1000 . . .
Now instead of the progression being powers of a base number, it becomes a factorial series, i.e. 1x2=2, 1x2x3=6, 1x2x3x4=24, 1x2x3x4x5=120, etc. I don't know if this idea has any usefulness or not, but it strikes me that these numbers might be represented somehow by the positional nature of the "digits." The first digit relates to its own position or not (on/off); the second digit can relate (on/off) to its own position and the position of the first digit. The third digit can relate (on/off) to itself, the second, and the first digits, and so on. This factorial representation allows larger and larger numbers to become more and more compact compared with a normal power-based system, while still retaining the on/off convenience of a binary system. Perhaps something of this type could be useful in computers and computing methods. (Don't ask me how, though -- I haven't developed the idea that far.)Return to the Loose Cannons -- Mathematics subject list
I have noticed something about digits that is surely mere coincidence, but in addition to being an interesting tidbit, is also the cause of a common error. If you look at the pairs of digits that are five apart (i.e. 1/6, 2/7, 3/8, 4/9, 0/5) you will see that with the exception of the pair 0/5, the pairs of numbers can eaily be mistaken for each other when jotted down quickly.
Thus, a quickly scribbled 4 can often be mistaken for a 9 (and visa versa), a 2 for a 7, a 6 for a 1 (if the 'circle' part of the six is small or incomplete), and, admittedly less often a 3 for an 8 (although the similarity is visually obvious).
The curious thing about these similarities is that if the errors are made for a series of numbers to be totaled, the sum is always 5 off. Whenever I notice this difference of 5, I immediately look for this type of error. It is frustrating, though that I can't look for a single pair of numbers. Eight of the ten digit are in conspiracy together to cause this confusion.Return to the Loose Cannons -- Mathematics subject list