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-   -   0=-1 (Calculus Puzzle) (http://www.chiefdelphi.com/forums/showthread.php?t=104907)

RoboDesigners 21-03-2012 16:05

0=-1 (Calculus Puzzle)
 
Here's a neat puzzle I've come up with/discovered... See if you can find the error/oversight.

∫tan(x) dx = ∫sin(x)/cos(x) dx = ∫sin(x)sec(x) dx

integration by parts:
u = sec(x)
du = sec(x)tan(x) dx

dv = sin(x) dx
v = -cos(x) dx

∫tan(x) dx = -cos(x)sec(x) - ∫-cos(x)sec(x)tan(x)dx

∫tan(x) dx = -1 + ∫tan(x)dx

Now subtract ∫tan(x) dx from both sides and...

0 = -1

Ether 21-03-2012 16:19

Re: 0=-1 (Calculus Puzzle)
 
Quote:

Originally Posted by RoboDesigners (Post 1147417)
∫tan(x) dx = -1 + ∫tan(x)dx

Now subtract ∫tan(x) dx from both sides and...

0 = -1

∫tan(x)dx - ∫tan(x)dx is not equal to 0.

It's an arbitrary constant.



RoboDesigners 21-03-2012 18:32

Re: 0=-1 (Calculus Puzzle)
 
Quote:

Originally Posted by Ether (Post 1147427)
∫tan(x)dx - ∫tan(x)dx is not equal to 0.

It's an arbitrary constant.

Yep... that's basically it...

I had a feeling you'd be the first one to respond to this thread, and you'd have the answer... :)

//Andrew

Dusk Star 21-03-2012 18:51

Re: 0=-1 (Calculus Puzzle)
 
Well, and you left off the "+ C" that you always need...
(though that may very well be what Ether meant)

Peck 21-03-2012 19:07

Re: 0=-1 (Calculus Puzzle)
 
now make 1=2 :)

Ether 21-03-2012 19:12

Re: 0=-1 (Calculus Puzzle)
 
Quote:

Originally Posted by Peck (Post 1147507)
now make 1=2 :)

0 = -1

add 2 to both sides:

2 = 1



Peck 21-03-2012 19:39

Re: 0=-1 (Calculus Puzzle)
 
Let a and b be equal non-zero quantities

a = b

2. Multiply through by a

a^2 = ab

3. Subtract b^2

a^2 - b^2 = ab - b^2

4. Factor both sides

(a - b)(a + b) = b(a - b)

5. Divide out (a - b)

a + b = b

6. Observing that a = b

b + b = b

7. Combine like terms on the left

2b = b

8. Divide by the non-zero b

2 = 1

jblay 21-03-2012 20:09

Re: 0=-1 (Calculus Puzzle)
 
1/3=.33333...
2/3=.66666...
1/3+2/3=.99999...
1=.99999...

Ether 21-03-2012 20:23

Re: 0=-1 (Calculus Puzzle)
 
Quote:

Originally Posted by Peck (Post 1147520)
Let a and b be equal non-zero quantities
5. Divide out (a - b)

Divide by zero.



Ether 21-03-2012 20:31

Re: 0=-1 (Calculus Puzzle)
 
Quote:

Originally Posted by jblay (Post 1147526)
1=.99999...

The ellipsis means infinitely repeating decimal so

.99999... means limit(sum(9/10^i,i,1,n),n,inf) = 1

there is no contradiction.



Peck 21-03-2012 20:58

Re: 0=-1 (Calculus Puzzle)
 
Has this turned into a thread of trying to find a math trick that Ether doesn't know how to work it? If so, i want a chance to disprove it before he dose :).

Brian Ha 21-03-2012 21:44

Re: 0=-1 (Calculus Puzzle)
 
Peck, sorry to tell you but i doubt ull win haha

Although ether, can you prove or disprove Riemann Sums? Now that would be cool.

PAR_WIG1350 21-03-2012 21:44

Re: 0=-1 (Calculus Puzzle)
 
167 = -53

No illegal operations were performed, slightly non-standard

DonRotolo 21-03-2012 22:13

Re: 0=-1 (Calculus Puzzle)
 
How about this one:

13 * 7 = 28

Here is the explanation (BTW, the guy in the glasses is Ether as a young man :p )

Ether 21-03-2012 22:21

Re: 0=-1 (Calculus Puzzle)
 
Quote:

Originally Posted by Brian Ha (Post 1147589)
Although ether, can you prove or disprove Riemann Sums?

fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet




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