Thursday, 8 November 2007

Reading Mathematics

Taking a break from my "study mode"...

i = (k+1)(2k+1)
= 1

"When you add consecutive numbers starting with 1, and the number of numbers you add is odd, the result is equal to the product of the middle number among them times the last number."

All right, I didn't come up with something as ingenious as this (that would be attributed to Levi Ben Gershon, back in the 13th century). The notation (the way the equation is written) is modern though.

Crash course
What's ? The big bent E is pronounced "sigma" (Greek equivalent of "S"), and means - mathematically - "summation" or "addition".

Summation of what? Generally, it's the summation of a list of terms.

{0, 1, 2, 3, ...} is a list.

{3, 9, 12, 15, ...} is also a list.

For the above example, the list is {1, 2, 3, ...}. See that little i below the ? It tells us where to start. So we start at i=1.

If we have a starting point, then we should also have an ending point, right? That's what 2k+1 tells us: where to stop. For notations like these, k generally means a positive integer, e.g. 2, 21, 865, etc. (Well, 2k+1 is used when we want to say "an odd number". Try it out with k=4 and k=9.)

Crash course ends
Oops, so I didn't explain why the list is {1, 2, 3, ...}, did I? Hmmm... to put it simply, well, it's because the formula is i. If it had been 3i , we would have {3, 6, 9, ...}. I think that paints a rough picture.

And since it's a summation, it means 1+2+3+...+(2k+1).

Crash course really ends

I was finding out more about one particular lecturer when I bumped into his online article on How to Read Mathematics. Here's a spoon-sized bite of the article:

Reader: That's interesting. I wouldn't have guessed that. You mean that in my class with 30 students, there's a pretty good chance that at least 2 students have the same birthday?

Professional: You might want to take bets before you ask everyone their birthday. Most people don't think that a duplicate will occur. That's why some authors call this the birthday paradox.

This formula:

i = (k+1)(2k+1)
= 1

is also taken from the article. And I couldn't help agreeing with the author: "You should take as much time in unraveling the 2-inch version as the 2-sentence version."

So the next time you're going nuts over your Math textbook/notes, try slowing down.

Interested for more?
It's here: How to Read Mathematics (by Shai Simonson and Fernando Gouvea)

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