Question

\[\Huge 1 = 2 \text{ via Continued Fractions}\]

Answer 1

This is true of course that \(1=\dfrac{2}{3-1}\) Now, let's substitute this very expression for 1 in the denominator:
\[1=\dfrac{2}{3-\dfrac{2}{3-1}}\]
We can do that one more time:
\[1 = \dfrac{2}{3-\dfrac{2}{3-\dfrac{2}{3-1}}}\]
And one more time to make sure there is no misunderstanding of the construction,
\[1 = \dfrac{2}{3-\dfrac{2}{3-\dfrac{2}{3-\dfrac{2}{3-1}}}}\]
At this point I am assuming that further steps could be performed by any reader who got this far. To indicate that possibility I'll use the ellipsis:

\[1 = \dfrac{2}{3-\dfrac{2}{3-\dfrac{2}{3-\dfrac{2}{\ddots}}}}\]
Well, we also know that \(2 =\dfrac{2}{3-2}\). With this as a starting point, we follow in the footsteps of the previous example. Replacing 2 in the denominator with that expression gives
\(2=\dfrac{2}{3-\dfrac{2}{3-2}}\)

To continue as before:
\(2=\dfrac{2}{3-\dfrac{2}{3-\dfrac{2}{3-2}}}\)

And finally,
\(2=\dfrac{2}{3-\dfrac{2}{3-\dfrac{2}{3- \dfrac{2}{\ddots } }}}\)

But this is exactly the same continued fraction. By necessity we conclude that 1=2.

Answer 2

We can think of this as:
\[1=\frac{ 2 }{ 3-\frac{2 }{ x } }\] And, we still get 1=2.

Answer 3

True.

Answer 4

Oh my god, we cheated math.

Answer 5

I await someone to disprove this.

Answer 6

@Hero @dumbcow @robtobey @blues @ajprincess @Yahoo! @AccessDenied

Answer 7

Goddamn, 1 is NOT equal 2. PERIOD.

Answer 8

Well, can you disprove it?

Answer 9

It's not the same continued fraction... The scheme is different...
Are you happy now?

Answer 10

Any thoughts, @LogicalApple ?

Answer 11

I know x = 2/(3 - x) yields 1 and 2 as solutions.

I have never worked with continuous fractions before but the little bit of research tells me that a rational number has a finite continued fraction and an irrational one has an infinite continued fraction. Although what you have displayed does not disprove either fact.

Answer 12

Maybe continuous fractions are not unique?

Answer 13

These are generalized continued fractions, by the way. In continuous fractions, every numerator has to be 1.

Answer 14

Also, an infinite continued fraction represents by definition an irrational number, which 1 and 2 are not.

Answer 15

also, the continued fraction represents the decimal part of the number, which in the case of 1 and 2 should be 0

Answer 16

you're redefining continued fraction.

Answer 17

You basically said the same thing as @micahwood50 did.

Thank for post, though.

Answer 18

no matter how far you continue the recursive substitution, at the very bottom they will be unique
3-1 not equal 3-2
and to infer infinite regression implies an irrational number as was posted earlier, thus your conclusion is false since you can't equate an integer to an irrational number


plus, any 4 yr old will tell you 1 of something is not the same as 2 of something :)