Posted on 10/25/2004 1:46:25 AM PDT by accipter
It's a common misconception to think of zero as "nothing." It's actually very far from "nothing:" it is merely a different notation for 1 (one). You hardly think of 1 as nothing --- do you not?
Zero with respect to addition is EXACTLY the same thing as one with respect to multiplication. When we use both --- as in the case of numbers --- and in order not to confuse the two, we designate them with different symbols. That's all.
Who cares what that totals? It's irrelevant. Of the 30, the clerk got 25+2 and 3 was returned.
I can visualize a "look-see" proof, but it's hard to describe in words, but a Flash animation would make it really obvious.
Consider that (1+2+3+...)^2 can be represented as a square array of cubes (think children's blocks). Start with one cube representing (1)^2. To move up to (1+2)^2, add a stripe two blocks wide along the right and top sides of the first block, kind of like this:
2 2 2 2 2 2 1 2 2Adding the third term, we get:
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 3 3 3 2 2 2 3 3 3 1 2 2 3 3 3In each case, after adding "N", the upper right corner of the array is an NxN square of new blocks labeled with number "N". And the remaining new stripe on the upper left and lower right consist of just enough blocks (of number "N") to stack on the NxN upper right "corner" to make a cube NxNxN (left as an exercise for the reader).
So all the "1" cubes make a 1x1x1 cube, and all the "2" cubes make a 2x2x2 cube, etc. And this is exactly the number represented by your 1^3+2^3+3^3...N^3.
So yes, (1 + 2 + 3)^2 = (1^3 + 2^3 + 3^3) (and also for the general case), because of the geometric properties of squares and cubes.
Again, this would be a lot more obvious with animated children's blocks being layered and stacked in some sort of visualization -- like it looks in my head :-)
That wasn't Gödel....I believe that equation is from Euler. Magnificent, nonetheless...SSZ
thanks for the math help. I was actually attempting to understand it using the "blocks" method in my head, but to no avail. It looks like I was close to the right track but not quite there. The induction method also looks interesting.
I'll take a look at these later when I have the chance.
My solution to the problem you pose: I decided to give it to you. Feel free to keep it!
Ah, I see you beat me to it. It's Euler's identity, BTW.
Here http://www.cut-the-knot.org/induction.shtml is a link to mathematical induction with links to examples.
I agree that the Euler equtation is the most beautiful as it as well can be described to most people. If you remove the latter constraint, then there are other candidates.
The liberal's favorite: a != a.
time for a magic trick...
Get a number in your head...
Got it?
Now double it..
Add 4 to it.
Now cut this number in half..
Subtract the number you started with from this number..
got it?
sure?
2! :)
Balance.
22/7 =
Base case:
S(1) = 1 = C(1)
Assume that:
S(N)^2 = C(N)
And show that implies
S(N+1)^2 = C(N+1)
Proof:
Start with
(i) N^2 + 2N + 1 = (N+1)^2
(ii) N^2 + N + N + 1 = (N+1)^2
Now note that (see following note if you don't recognize this common identity in (iii))
(iii) S(N) = (N^2 + N)/2
(iv) 2S(N) = N^2 + N
Thus by (ii) and (iv)
(v) 2S(N) + (N + 1) = (N+1)^2
Multiplying (v) by (N+1)
(vi) 2S(N)(N+1) + (N+1)(N+1) = (N+1)^3
Now adding (vi) to the assumption that S(N)^2 = C(N)
(vii) S(N)^2 + 2S(N)(N+1) + (N+1)^2 = C(N) + (N+1)^3
Noting that S(N+1)^2 = (S(N) + (N+1))^2 = S(N)^2 + 2S(N)(N+1) + (N+1)^2, and that C(N+1) = C(N) + (N+1)^3, we have
(viii) S(N+1)^2 = C(N+1)
Note:
To prove that S(N) = (N^2 + N)/2, we can also this do inductively:
S(1) = (1^2 + 1)/2 = 1
Assuming that S(N) = (N^2 + N)/2
Show that S(N+1) = ((N+1)^2 + (N+1)) / 2
Taking
S(N+1) = S(N) + N + 1
= (N^2 + N)/2 + N + 1
= (N^2 + N)/2 + (2N + 2)/2
= (N^2 + 3N + 2) / 2
= (N^2 + 2N + 1 + N + 1) /2
= ((N+1)^2 + (N+1)) / 2
Festival of Elegant Equations
You passed the exam.
Strictly speaking, this is true for systems that contain ordinary arithmetic. Pressburger arithmetic (addition, but not multiplication) and Euclidean Geometry, are complete. Ordinary arithmetic is not.
COOL! F=ma is my favorite. While F=ma is not as well known to the general public as E=mc2 (and when is the last time YOU used that one?), F=ma is more practical and when saying it, it kind of rolls off the tongue; Force = mass × accelartaion. [maybe it's a guy thing with the word FORCE :-) ]
My 2nd fave is a2 + b2 = c2, but the ones I used to use the most often in work are;
a sin C / sin ABut now with AutoCAD®, angles, angle sides and properties of a circle are all accurately measured and calculated for you, so the math is minimal - not like 'the olden days'.
b sin C / sin B
a sin B / sin A
pr2
Which reminds me - I better start DOING some work :-)
Our human minds, and what we conceptualize as hard sciences, are limited by three dimensions of space and linear time.
The Author of the universe is not limited to these constraints.
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