Perpetuating the myth that all women chess players are hot is the new Women's Chess Champion, 24-year-old Alexandra Kosteniuk from Russia. Who knew they played chess over there? She was a grandmaster at the age of 14 and last year she beat China's Hou Yifan in Nalchik, somewhere in Russia, to become the first Russian to hold the women's championship title since the fall of the Soviet Union. Because of some of her "fashion" modeling portfolio, Kosteniuk has been compared to Anna Kournikova, but she points out that she has in fact won individual events, a feat that has eluded Kournikova. Her rating is 2516, making her the 690th best player in the world, well behind former women's champ Judit Polgar at #36. Kosteniuk hosts a charming chess instruction video webcast at chesskillertips.com. [Insert "positions worth analyzing" joke here.]Saturday, January 24, 2009
The Pale White Queen of Chess
Perpetuating the myth that all women chess players are hot is the new Women's Chess Champion, 24-year-old Alexandra Kosteniuk from Russia. Who knew they played chess over there? She was a grandmaster at the age of 14 and last year she beat China's Hou Yifan in Nalchik, somewhere in Russia, to become the first Russian to hold the women's championship title since the fall of the Soviet Union. Because of some of her "fashion" modeling portfolio, Kosteniuk has been compared to Anna Kournikova, but she points out that she has in fact won individual events, a feat that has eluded Kournikova. Her rating is 2516, making her the 690th best player in the world, well behind former women's champ Judit Polgar at #36. Kosteniuk hosts a charming chess instruction video webcast at chesskillertips.com. [Insert "positions worth analyzing" joke here.]
Friday, January 2, 2009
The Number Devil in All of Us
When I was a math teacher, one day I was helping my students check a column of numbers of the form n2 - 1 to see if any were prime. I impressed them by quickly factoring the whole list: I rattled off some big ones like, "Nope, 182 - 1 is 17 x 19, 192 - 1 is 18 x 20," and so on, and most of the students caught on to the pattern: n2 - 1 = (n - 1)(n+ 1). It's a specific case of the formula which should be known to anybody taking the SATs: (a + b)(a - b) = a2 - b2.
I knew the formula could be used for multiplying two numbers conveniently placed on either side of a nice round number, like 88 x 92 = (90 - 2)(90 + 2) = 902 - 22 = 8,100 - 4 = 8,096. What I didn't know until recently was it could be used for squaring big numbers:
Start with the formula
a2 - b2 = (a + b)(a - b)
Add b2 to both sides and you get
a2 = (a + b)(a - b) + b2
Now instead of squaring a number like 27 by multiplying it by itself, you can use a nice round number, in this case 3o, which is 27 + 3. Use this to solve 272.
272 = (27 + 3)(27 - 3) + 32 = 30 x 24 + 9 = 720 + 9 = 729.
The way I would have done this previously is to use the formula
(a - b)2 = a2 - 2ab + b2
272 = (30 - 3)2 = 302 - 2(30)(3) + 32 = 900 - 180 + 9 = 720 + 9 = 729.
What Arthur Benjamin uses in the video below to solve 57,6832 is the related formula
(a + b)2 = a2 + 2ab + b2
(57,000 + 683)2 = 57,0002 + 2(57,0000)(683) + 6832
Which one could argue is simplifying the job a bit, but each term of the above expression still makes me reach for my calculator.
Some folks get really good at using these binomial expansions to square big numbers. I was impressed by the job Art does in this video introducing his calculating tricks.
He made a couple of mistakes (the square of 722 is actually 521,284) but his performance is a great bit of publicity for his terrific book Secrets of Mental Math. Learning a few "tricks" can certainly serve to lessen math anxiety, so try it out!
I knew the formula could be used for multiplying two numbers conveniently placed on either side of a nice round number, like 88 x 92 = (90 - 2)(90 + 2) = 902 - 22 = 8,100 - 4 = 8,096. What I didn't know until recently was it could be used for squaring big numbers:
Start with the formula
a2 - b2 = (a + b)(a - b)
Add b2 to both sides and you get
a2 = (a + b)(a - b) + b2
Now instead of squaring a number like 27 by multiplying it by itself, you can use a nice round number, in this case 3o, which is 27 + 3. Use this to solve 272.
272 = (27 + 3)(27 - 3) + 32 = 30 x 24 + 9 = 720 + 9 = 729.
The way I would have done this previously is to use the formula
(a - b)2 = a2 - 2ab + b2
272 = (30 - 3)2 = 302 - 2(30)(3) + 32 = 900 - 180 + 9 = 720 + 9 = 729.
What Arthur Benjamin uses in the video below to solve 57,6832 is the related formula
(a + b)2 = a2 + 2ab + b2
(57,000 + 683)2 = 57,0002 + 2(57,0000)(683) + 6832
Which one could argue is simplifying the job a bit, but each term of the above expression still makes me reach for my calculator.
Some folks get really good at using these binomial expansions to square big numbers. I was impressed by the job Art does in this video introducing his calculating tricks.
He made a couple of mistakes (the square of 722 is actually 521,284) but his performance is a great bit of publicity for his terrific book Secrets of Mental Math. Learning a few "tricks" can certainly serve to lessen math anxiety, so try it out!
Subscribe to:
Posts (Atom)
