# Thread: 1=2. How does this work?

1. ## 1=2. How does this work?

someone posted this on another site, and I have no idea how it works. I've been stretching my knowledge of algebra but I can't seem to find the flaw in it.

a = b

Multiply both sides by a

a^2 = ab

Subtract b^2 from each side

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

Factor

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

Divide by a-b

a+b = b

But a = b, so

b + b = b

So

b = 2b

Divide by b

1 = 2

2. Originally Posted by parallaxicality
someone posted this on another site, and I have no idea how it works. I've been stretching my knowledge of algebra but I can't seem to find the flaw in it.
Divide by a-b

STOP!

Since a = b, this is division by zero.

3. The flaw there is diving by zero. You start out with the statement a = b, which means (a - b) = 0. All the rest is just "smoke and mirrors" to make it look like you're doing something complicated, like many puzzles depend on.

When an equation provides a nonsense result, like 1 = 2, that means that some step was invalid. A more complex expression was really the same nonsense, you just reduced it to something more obvious. The nonsense here is dividing by zero.

-Richard

4. a^2 = ab ...........................OK

Subtract b^2 from each side
a^2 - b^2 = ab - b^2 ...........Red flag warning: [0=0]

Factor
(a+b)(a-b) = b(a-b) .............Danger! heading for a cliff: (a+b)x(0)= (b)x(0)

Divide by a-b
a+b = b .............................Over the edge!

5. You've also got to be careful how you handle i (Imaginary numbers)

1) -1 = -1
2) 1/-1 = -1/1
3) sqrt(1/-1) = sqrt(-1/1)
4) sqrt(1)/sqrt(-1) = sqrt(-1)/sqrt(1)
5) 1/i = i/1
6) 1/i = i
7) 1 = i * i
8) 1 = -1

6. ## Re: 1=2. How does this work?

1=2. How does this work?
Simple. Just add 1. Stir and serve.

7. Originally Posted by RobA
You've also got to be careful how you handle i (Imaginary numbers)
Not so much i as sqrt, as sqrt((2)2)=sqrt((-2)2) has a related problem when "cancelling" the square. The complex numbers do introduce more possibilities.

8. And this my friends is what is our downfall. As long as there are people whom are willing to argue that one is two. We are doomed. Doomed...
I am certain that at no time has or does one ever equal two. What ever value you attribute to one. The two is twice of it. Do not throw the now famous line at me that one zero is equal to two zero's, I may become hysterical
As has been clearly stated here.. No amount of fiddling with the very impressive numerics of the highest levels of algebra will or can altar this truth. 1 does not equal 2.
Eight posts and we are off on a tangent to the OP. -- So the answer is no. Its nonsense.

9. Originally Posted by astromark
Eight posts and we are off on a tangent to the OP. -- So the answer is no. Its nonsense.
What was the question again?

10. Originally Posted by astromark
As has been clearly stated here.. No amount of fiddling with the very impressive numerics of the highest levels of algebra will or can altar this truth. 1 does not equal 2.
Sometimes, Winston. Sometimes it equals five. Sometimes it equals three. Sometimes it equals all of them at once. You must try harder. It is not easy to become sane.

11. I recall back in the old days of non-precision cosmology we sued to say that 1=10. The offered proof was this:

11! = 11x10!

11!/! = 11x10!/!

11/11 = 11x10/11

1 = 10

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## Thoughtless Experiments

The math (?) examples above demonstrate nicely my objections to 'thought experiments'. Some well meaning person (for UT, probably an ATM'er) proposes a complicated thought experiment, usually about SR or GR, and the mainstreamers get to spend boodles of time finding the divide by zero, or other flaw.

Publius ought to receive the UT Medal of Honor for patience.

13. Originally Posted by antoniseb
11!/!
Unrecognized. What does 11!/! mean? Or even 1/!?

Or is the step just: "divide both sides by an exclamation mark"?

14. yes, but does 0.999... = 2?

<ducks and runs>

15. Originally Posted by 01101001
Unrecognized. What does 11!/! mean? Or even 1/!?

Or is the step just: "divide both sides by an exclamation mark"?
Yes, I think that's what it means-- it's a joke about taking algebra rules too dogmatically. There's no substitute for the old maxim, "know thy function"-- but modern graphical routines sure make that a lot easier to do!

16. Originally Posted by 01101001
Unrecognized. What does 11!/! mean? Or even 1/!?
Or is the step just: "divide both sides by an exclamation mark"?
This was never taken seriously. The exclamation mark is the Factorial symbol, which I presume you knew. The proof was very tongue-in-cheek, and presented to astronomy students to make the point that 1=10, but for some reason 1<>100.

17. Falls of chair... bangs head on floor, see's stars,. Oh yes astronomy.

18. Originally Posted by antoniseb
This was never taken seriously. The exclamation mark is the Factorial symbol, which I presume you knew. The proof was very tongue-in-cheek, and presented to astronomy students to make the point that 1=10, but for some reason 1<>100.
Yes, I realize it is not serious and ! is factorial

Please explain the notation. What does 11!/! mean? What does 1/! mean? Is this division by punctuation? I don't get it.

What was a student supposed to think was happening here? This use of symbols is unfamiliar to me. I am a stubborn student.

Did Ken G guess right?

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Whe I teach mathematical induction, I offer up a "proof" that n = n + 1, by intentionally leaving out a crucial step in showing that we can find a "starting point" (usually by letting n = 1)

I do this to see if a student will question the teacher as to the utter nonsense of the statement and to try to instill the importance of a first step in the method of induction.

Reminds me also of a funny joke which was circulating a while back:

Q: Expand (x+y)^10
A: ( x + y )^10
( x + y )^10
( x + y ) ^ 10

Heh, math teachers are funniest people on the planet (even funnier than proctologists)

Pete

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Originally Posted by parallaxicality
someone posted this on another site, and I have no idea how it works. I've been stretching my knowledge of algebra but I can't seem to find the flaw in it.
Other people have pointed out the flaw, but if you get stuck on something like this, you can just try plugging in actual numbers. Pick a=b=3, and see where the equalities stop being true. You find it pretty quickly that way.

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Originally Posted by peter eldergill
Whe I teach mathematical induction, I offer up a "proof" that n = n + 1, by intentionally leaving out a crucial step in showing that we can find a "starting point" (usually by letting n = 1)
Can you clarify? Do you mean you use the normal rules of algebra to show that if n = n+1, then n+1 = n+2?

22. Originally Posted by 01101001
Please explain the notation. What does 11!/! mean? What does 1/! mean? Is this division by punctuation? I don't get it.

What was a student supposed to think was happening here? This use of symbols is unfamiliar to me. I am a stubborn student.
The technical term for this line of reasoning is joke.

Fred

23. Originally Posted by Nowhere Man
The technical term for this line of reasoning is joke.
Kinda like cancelling sixes and getting the right answer:

16/64

24. Originally Posted by Nowhere Man
The technical term for this line of reasoning is joke.

Fred
Well, duh. Yes, it's a "funny" non-proof that 1 = 10.

If you understand the joke, you tell me what 11!/! means in English. I don't speak nonsense. Is it too much to ask for an explanation? Is it division by punctuation? What kind of students would tolerate a moment of that? I reallly want to know.

If you like that joke, you'll howl at:

2-#%

19=+

4/\$?22

Ha! Get it?

25. Originally Posted by 01101001
If you understand the joke, you tell me what 11!/! means in English.
Sorry, I forgot that some readers don't know what factorial means. Factorial is the product of all integers greater than one and less than or equal to the number given. The symbol '!' indicates the factorial operation. Thus:

2! = 2
3! = 6
4! = 24
5! = 120
6! = 720
7! = 7 * 6! = 5040
etc.

The numbers go up very quickly.
Factorials are very important in expressing series solutions to complicated math problems.

26. OK, you know that the ! operator means factorial.

By the basic rules of algebra, 11x/x = 11, because the xes cancel out.

If you're symbol-minded, you might get carried away and claim that 11!/! = 11 by the same rule. The absurdity of this statement is what makes it funny.

If that doesn't make you laugh, than maybe I should fall into an open sewer and die*.

Fred

*Mel Brooks reference

27. Another simpler way of expressing that joke would be:

1*2/*=12

28. A joke my high school math teacher always told made fun of a student, who when asked what 5/5 was, said "the fives cancel, so it's nothing".

29. Originally Posted by pghnative
yes, but does 0.999... = 2?

<ducks and runs>
*chases pghnative around with a sickle*

30. Originally Posted by Nowhere Man
OK, you know that the ! operator means factorial.
Just as I said.

Originally Posted by Nowhere Man
By the basic rules of algebra, 11x/x = 11, because the xes cancel out.
Because x/x is 1, when x nonzero. But, colloquially called cancelling out. OK. That's semi-justifiable.

Originally Posted by Nowhere Man
If you're symbol-minded, you might get carried away and claim that 11!/! = 11 by the same rule. The absurdity of this statement is what makes it funny.
Yeah, funny to some. I recognized and appreciated the "cancelling" humor.

However, I'm still wondering about how the punctuation got into the denominator. The step before sticks in my craw. I take it, that it's some handwaving like: "Divide both sides by exclamation mark," that I and Ken G guess at, and I really haven't seen confirmed. I'll assume it.

Fine. OK. I don't know about others' cultures, but in mine, students were expected to interrupt a teacher's exposition if they didn't understand something. This is where I would have asked what "divide by exclamation mark" means. What do the comedic teachers do then?

I really want to know. I am serious. What? Does the teacher just try to wink at this point, and escape on charm? Is there anything to say? Hope no one asks? If one does, just give up and not bother with the conclusion? Maybe, with smart students, everyone always sees what's up and goes along for the ride.

Maybe I'm curious because Antoniseb only provided the text, not the banter that goes with it, at a minimum, the justifications for each step, at least the justification for the erroneous step.

And, again, please, please, I do get that this is a bogus proof, using unreasonable steps to reach an unsupportable answer. I've enjoyed many such in my life.

I have advanced degrees in symbol manipulation. In college, I've taught it, and mathematical algorithms, and generic problem solving. I don't recall, but I've probably done bogus proofs in class. And, I've been paid to produce written humor. I know how to laugh. I know what's up.

I'm sorry, but this isn't one of the good ones, because, unless I'm missing something, the bogus step lacks any subtlety, has no essential surprise factor, is simply symbolic gibberish, contains no ironic twist. Frankly, I'd sooner laugh at someone falling into an open sewer and dying.

One might as well employ, more directly, with more subtlety and less bumbling:

given: 1 = 1
since x + 0 = x; add 0 to the righthand side: 1 = 1 0
therefore: 1 = 10

Now that's funny. And less boring.

Look, I'm not trying to over-over-overanalyze this. It's not thesis-defense time, here. I wanted to find the humor. Perhaps I might want to use this some day. There are always children with vulnerable young minds about to challenge.

I respect Antoniseb, and the wisdom and intelligence inherent therein. And I know Antoniseb can't hit every one out of the park, too. I was just curious about the missing plot details, and I felt like my questions were being intentionally evaded or frustratingly misunderstood.

I appreciate someone finally beginning to address my question as asked. I knew I shouldn't relax my standards. Thanks. Really.

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