I remembered many years ago, in my Secondary Four days, a student came forward and presented a chalkboard full to equations and concluded that 1 + 1 = something else. I cannot remember, but the whole class surely could not find any error in the logic based on maths theories we are familiar with.
Is there anyone out there who could help to refresh these equations?
I am still puzzled, but could not review them, as chalk marks on a chalk board are cleared, as soon as we rub them off with a duster.
Hope to rekindle the old fun in maths for myself.
Well, here is one version
Start with a=b
(multiply both sides by a)
a^2 = ab
(subtract b squared from both sides)
a^2-b^2 = ab-b^2
(factorise)
(a+b)(a-b) = b(a-b)
(divide both sides by a-b)
a+b = b
Since a=b, for any given value of a, then a=1/2a (2=1, 4=2, 8=4 etc)
See if you can spot the flaw in the algebra... 
That's called an invalid proof. There are many ways to prove that 1=0, or something to that effect. I can prove that 1+1=1 in very numerous ways. Bikerman gave a basic way to do it using algebra. Once you figure out the mistake in his proof, you will be able to find the mistake in many different invalid proofs (because many of them use that same mistake). Here's some more.
This one uses calculus:
Let x=1
(d/dx)x=(d/dx)1
1=0
1+1=1
Q.E.D
The problem with that proof is that it treats x as a variable, when x is defined as a constant.
Proof that -1=1
-1=-1
1/-1=-1/1
sqrt(1/-1)=sqrt(-1/1)
sqrt(1)/sqrt(-1)=sqrt(-1)/sqrt(1)
multiply both sides by sqrt(1)sqrt(-1)
sqrt(1)sqrt(1)=sqrt(-1)sqrt(-1)
1=-1
This proof is invalid because sqrt(a/b)=sqrt(a)/sqrt(b) only applies when a and b are positive real numbers (where as the sqrt(-1) is imaginary).
If you find these interesting, then just say so and I can post some more. I know quite a bit of these. Most of these are quit famous and can be found in numerous math sources. As a matter of fact, I did a quick google search for "invalid proof" and I found both of these on wikipedia.
http://en.wikipedia.org/wiki/Invalid_proof | Bikerman wrote: |
Well, here is one version
Start with a=b
(multiply both sides by a)
a^2 = ab
(subtract b squared from both sides)
a^2-b^2 = ab-b^2
(factorise)
(a+b)(a-b) = b(a-b)
(divide both sides by a-b)
a+b = b
Since a=b, for any given value of a, then a=1/2a (2=1, 4=2, 8=4 etc)
See if you can spot the flaw in the algebra... |
Is it that since a=b, you can't divide by (a-b) since that is 0? | saratdear wrote: |
| Is it that since a=b, you can't divide by (a-b) since that is 0? |
Quite correct - well spotted.
It is actually quite obvious in this example, if you look closely, but, believe me, the same mistake has been made by even great physicists such as Einstein. | shenyl wrote: |
I remembered many years ago, in my Secondary Four days, a student came forward and presented a chalkboard full to equations and concluded that 1 + 1 = something else. I cannot remember, but the whole class surely could not find any error in the logic based on maths theories we are familiar with.
Is there anyone out there who could help to refresh these equations?
I am still puzzled, but could not review them, as chalk marks on a chalk board are cleared, as soon as we rub them off with a duster.
Hope to rekindle the old fun in maths for myself. |
1 + 1 = 10 in binary...
1+1=x
(1+1)^2=x^2
(1+1) * (1+1) = x^2
1 + 1 + 1 + 1 = x^2
4 = x^2
2=x
Epic failure, but I did find a very unique way to add 1+1... | Bikerman wrote: |
| saratdear wrote: | | Is it that since a=b, you can't divide by (a-b) since that is 0? |
Quite correct - well spotted.
It is actually quite obvious in this example, if you look closely, but, believe me, the same mistake has been made by even great physicists such as Einstein. |
Thank you.
And I did find a way to prove that 1+1=1 in some old magazine. It's really quite silly and not even a mathematic proof. It was like...how if 1 river and 1 river would combine, the resultant would not be two rivers, but 1 river. 
| saratdear wrote: |
And I did find a way to prove that 1+1=1 in some old magazine. It's really quite silly and not even a mathematic proof. It was like...how if 1 river and 1 river would combine, the resultant would not be two rivers, but 1 river. :roll: |
That's just moronic. If you have one liter of water and mix it with one liter of water you get two liters of water (even though you have one pitcher). I would have burned that nonsense and then regretted buying the magazine.So many answers to my long lost remembrance of the chunk of equations my school mate has given to our class.
Really thankful, this does open my understanding, that many of the maths theories does have assumptions and restrictions, which are often overlooked.
Thanks to all for clearing my forgetfulness.
With regards. I will PDF this thread for my future remembrance of 1 + 1 = 1 problem.
OK, here's an honest to goodness proof that 1+1=1 - no tricks, no mistakes.
If you take a solid sphere it is theoretically possible to divide it into 5 parts, then join them together to form two solid spheres, each equal in size to the original sphere.
This mind blowing result is called the Banach-Tarski paradox.
Unfortunately, it involves some fairly advanced maths, and assumes the axiom of choice. For those who have heard of the axiom of choice, this is the most startling consequence I have come across.
For non-mathematicians, the beginning of the article on the Banach-Tarski paradox is still worth looking at.
I should point out that the decomposition of the sphere in the paradox is a purely theoretical mathematical exercise:
1- Because the sphere must be really solid (not made out of atoms with lots of spaces between them)
2- The use of the axiom of choice means that the decomposition of the sphere is theoretically possible, but you can't actually give a specific way of doing it.
Have fun! Truly this is way beyond a casual paradox.
It is good to discover this, and moreover NO MISTAKE!
Thanks, but I will sure have to take a deep dive into many more maths issues, before I can appreciate this paradox.
So it is indeed way beyond - in terms of my time, and possibly intellectual knowledge.
But I can safely said - 1 + 1 = 1. and point others to the URL. With best regards.
| infinisa wrote: |
| OK, here's an honest to goodness proof that 1+1=1 - no tricks, no mistakes. |
i don't know how well the claim of "no tricks, no mistakes" holds up. ^_^; You are, after all, dealing with infinities again, and it has already been pointed out that everything breaks down when you deal with infinities.
You can use similar logic to point out that a 1 unit long line is the same size as a 2 unit long line (the set of points in the first line is the same size as the set of points in the second line), thus giving you that 1 = 2. It's the same type of logic as using the Banach–Tarski paradox to claim that 1 + 1 = 1, but easier to grasp for non-mathematicians. Would anyone care to point out the flaw? ^_^Yap, that will be great if some great cracker can illuminate some lights, that the others have a better understanding of these great theorems.
Yap, I will appreciate that as well, anyone can help?
| Indi wrote: |
| infinisa wrote: | | OK, here's an honest to goodness proof that 1+1=1 - no tricks, no mistakes. |
i don't know how well the claim of "no tricks, no mistakes" holds up. ^_^; You are, after all, dealing with infinities again, and it has already been pointed out that everything breaks down when you deal with infinities.
You can use similar logic to point out that a 1 unit long line is the same size as a 2 unit long line (the set of points in the first line is the same size as the set of points in the second line), thus giving you that 1 = 2. It's the same type of logic as using the Banach–Tarski paradox to claim that 1 + 1 = 1, but easier to grasp for non-mathematicians. Would anyone care to point out the flaw? ^_^ |
How would the set of points in the 1 unit long line be equal to the set of points in the 2 unit point line?Hello All
The other day I mentioned the Banach-Tarski paradox:
| Quote: |
If you take a solid sphere it is theoretically possible to divide it into 5 parts, then join them together to form two solid spheres, each equal in size to the original sphere.
|
What I'm going to do now is show how this result (which is correct Mathematics) leads to the result 1+1=1.
Let's call the original solid sphere S1, and the 5 parts P1, P2, P3, P4 & P5.
Obviously, the volume of the sphere S1 is equal to the sum of the volumes of the parts.
We can write this as an equation as follows:
vol(S1) = vol(P1) + vol(P2) + vol(P3) + vol(P4) + vol(P5)
Now the rearrangement of the parts to form S2 & S3 obviously involves actions like moving and rotating the parts. These actions do not change the size of the parts in any way. Mathematically, we say these are isometrys (distance preserving). Naturally, such actions do not change the volume of the parts in any way.
So after the rearrangement, P1 & P2 fit together to form S2, so we get:
vol(S2) = vol(P1) + vol(P2)
And P3, P4 & P5 fit together to form S3, so we get:
vol(S3) = vol(P3) + vol(P4) + vol(P5)
Adding the last two equations, we get:
vol(S2) + vol(S3) = vol(P1) + vol(P2) + vol(P3) + vol(P4) + vol(P5)
Comparing this with the first equation, we see that:
vol(S2) + vol(S3) = vol(S1)
Now we may assume that vol(S1) is 1 (if it's not, just scale the sphere up or down as necessary),
and so vol(S2) & vol(S3) are also both 1 (That's what the Banach-Tarski paradox states)
Substituting these values in the last equation, we get:
1 + 1 = 1
QED! Hi Indi
| Indi wrote: |
| infinisa wrote: | | OK, here's an honest to goodness proof that 1+1=1 - no tricks, no mistakes. |
i don't know how well the claim of "no tricks, no mistakes" holds up. ^_^; You are, after all, dealing with infinities again, and it has already been pointed out that everything breaks down when you deal with infinities.
You can use similar logic to point out that a 1 unit long line is the same size as a 2 unit long line (the set of points in the first line is the same size as the set of points in the second line), thus giving you that 1 = 2. It's the same type of logic as using the Banach–Tarski paradox to claim that 1 + 1 = 1, but easier to grasp for non-mathematicians. Would anyone care to point out the flaw? ^_^ |
You are absolutely right when you say that "a 1 unit long line is the same size as a 2 unit long line (the set of points in the first line is the same size as the set of points in the second line)". But here you are using the word "size" in a completely different way from me: I'm talking about measure (where distances matter), but you're just talking about cardinality.
The transformations involved in the the Banach–Tarski paradox are all isometries (distance preserving transformations).
So the challenge to you is: can you transform a 1 unit long line into a 2 unit long line with an isometry?
Good try & good luck! infinisa - I don't have a clue what you're talking about, but I'll have to trust you... is there any variation on that proof that would seem sensible to a lay man?
Well, as one professor used to tell me.:"1+1 is not equal to 2 in all VECTORIAL SPACES"
| Quote: |
The other day I mentioned the Banach-Tarski paradox:
Quote:
If you take a solid sphere it is theoretically possible to divide it into 5 parts, then join them together to form two solid spheres, each equal in size to the original sphere.
|
Here I see that theoretically and practically do not equate.
It is the Banach-Tarski paradox's claim that is to be proven wrong.
I am not going to pursue this further, as it is getting very cloudy.
Thanks anyway "infinisa" for the trememdous effort to bring this paradox to our attention.Many are saying 1 + 1 = 11. And theres a way we could find out why. Maybe someone could share how it happened. 
| shenyl wrote: |
It is the Banach-Tarski paradox's claim that is to be proven wrong.
|
The claim is correct.
The point that has not been made yet is the nature of the pieces.
What the result says is start with a solid ball. Then divide this into 5 pieces. Then reassemble the 5 pieces, without squishing any of them, into two balls of the same size as the original.
It should be obvious that the pieces are not like the pieces we would get if we cut up a real ball. Nor can we use arguments about volume, because the pieces needed in the problem consist of infinite sets of points, not finite solid chunks.
The point here is that what you mean by the volume of a piece of a sphere makes sense for finite pieces, but doesn't make sense in the same way when you consider an infinite set of discrete points. So, suppose we have a sphere, and now we take the same sphere, but minus the point at the centre. What volume does this ball have? Using usual meaning of volume, we would say the same! Okay, that is just removing one point, what if we removed an infinite number of points (all points on a line through the ball, for example). Using normal meaning of volume, we would say the same, since a line has no volume. And so on.Hi Speaker_To_Animals
| Speaker_To_Animals wrote: |
| shenyl wrote: |
It is the Banach-Tarski paradox's claim that is to be proven wrong.
|
The claim is correct.
The point that has not been made yet is the nature of the pieces.
What the result says is start with a solid ball. Then divide this into 5 pieces. Then reassemble the 5 pieces, without squishing any of them, into two balls of the same size as the original.
It should be obvious that the pieces are not like the pieces we would get if we cut up a real ball. Nor can we use arguments about volume, because the pieces needed in the problem consist of infinite sets of points, not finite solid chunks.
The point here is that what you mean by the volume of a piece of a sphere makes sense for finite pieces, but doesn't make sense in the same way when you consider an infinite set of discrete points. So, suppose we have a sphere, and now we take the same sphere, but minus the point at the centre. What volume does this ball have? Using usual meaning of volume, we would say the same! Okay, that is just removing one point, what if we removed an infinite number of points (all points on a line through the ball, for example). Using normal meaning of volume, we would say the same, since a line has no volume. And so on. |
You hit the nail on the head - and very well explained, if I may say so!
I must confess I was leading you guys up the garden path, but felt justified in doing so because this whole topic (seeking a proof that 1+1=1) is clearly a hopeless quest (at least, anyone with faith in Maths would hope so!).
So it's time for me to come clean:
1. The Banach-Tarski paradox is correct, as I've mentioned previously. However, it does depend on the axiom of choice, as I also mentioned earlier. Don't worry about the axiom of choice for the now; I'll come back to it in a minute
2. The "proof" I gave that the "Banach-Tarski paradox implies 1+1=1" (Sun Jan 04, 2009 9:46 pm) looks innocent enough, but is in fact wrong.
3. The error arises from the fact that in the Banach-Tarski paradox, the 5 parts that the original sphere are divided into are so inconceivably weird that you cannot meaningfully measure their volume. That being the case, none of the equations in my proof makes any sense!
Speaker_To_Animals gave some examples of objects where the idea of volume seems to lead to trouble. Not so however; these examples are relatively innocent:
1. A sphere minus the point at the centre: Since a single point has volume zero, removing this doesn't change the volume.
2. A sphere minus all points on a line through the ball: Since a line is a one dimensional object, it also has volume zero, so removing it doesn't change the volume.
Well, the kind of object with no well defined volume is far weirder than these examples. In fact, it is impossible can to describe clearly and unambiguously how to make one! (This is consequence of having to rely on the axiom of choice, as we shall see below)
In mathematical language, a 3-dimensional object whose volume cannot be measured is called a non-measurable set. For a long time people found this idea so weird that they thought no such thing could exist.
Perhaps the simplest example of a non-measurable set is the Vitali set, which is set of points chosen from the interval from 0 to 1 [0,1]. The important thing here is that the construction depends on our old friend, the axiom of choice. In fact, it turns out that without assuming the axiom of choice, it is possible that non-measurable sets don't exist.
So the paradoxes we've been looking at basically illustrate the paradoxical consequences of the axiom of choice. This doesn't make it wrong; much of important mathematics relies on it being true.
Given its importance to all this, I should like to be able to explain what the axiom of choice is all about, but I think I'll leave this for another occasion! Let me just say this: It allows you to say that certain objects exist without actually showing how to make one. Amazing. This turned into a thread with fun problems.
I have one to contribute but it's not much of a toughie.
Ten envelopes (with different names on each) , ten letters (with the same names). The letters are put in the ten envelopes randomly. If nine are placed correctly, what is the chance that the other is placed incorrectly? Solve mathematically.
Hi guissmo
If 9 of the letters are correctly placed, then the 10th must be too.
So the probability is 1.
(I don't see the connection between this problem and 1+1=1, but maybe you can explain )
Cheers No one's caught on to the plain - non-mathematical - objection to the claim that the Banach-Tarski paradox proves 1 + 1 = 1? ^_^;
Here, i'll repeat the clue: "You can use similar logic to point out that a 1 unit long line is the same size as a 2 unit long line (the set of points in the first line is the same size as the set of points in the second line), thus giving you that 1 = 2."
Here's an even more blatant clue: Exactly what are you counting as "1" in the Banach-Tarski paradox example?
Hi Indi
| Indi wrote: |
No one's caught on to the plain - non-mathematical - objection to the claim that the Banach-Tarski paradox proves 1 + 1 = 1? ^_^;
Here, i'll repeat the clue: "You can use similar logic to point out that a 1 unit long line is the same size as a 2 unit long line (the set of points in the first line is the same size as the set of points in the second line), thus giving you that 1 = 2."
Here's an even more blatant clue: Exactly what are you counting as "1" in the Banach-Tarski paradox example? |
I believe I already showed that there's a crucial difference between the Banach-Tarski paradox and your unit line analogy in my post on Sun Jan 04, 2009 10:08 pm:
| Quote: |
You are absolutely right when you say that "a 1 unit long line is the same size as a 2 unit long line (the set of points in the first line is the same size as the set of points in the second line)". But here you are using the word "size" in a completely different way from me: I'm talking about measure (where distances matter), but you're just talking about cardinality.
The transformations involved in the the Banach–Tarski paradox are all isometries (distance preserving transformations).
So the challenge to you is: can you transform a 1 unit long line into a 2 unit long line with an isometry? |
Then I explained the real error in my argument in my post of Fri Jan 09, 2009 8:41 pm:
The problem is not with the Banach-Tarksi paradox, but with my subsequent "proof" (Posted: Sun Jan 04, 2009 9:46 pm) that this leads to 1+1=1
As for the question "what are you counting as "1" in the Banach-Tarski paradox example", I am talking about the volumes of 3D objects: technically this is called measure in Mathematics, and generalizes the usual notion of length, area and volume. As I said, in the decomposition of the Banach-Tarski paradox, the pieces are so weird that we cannot measure their volume, which is where my argument breaks down.
Getting back to the line of unit length, you can divide it into an infinite number of pieces of equal size (measure) called Vitali sets. This is crazy, because if the measure of each piece is 0, this makes the unit line also have measure 0; whereas if the measure of each piece is greater than zero, this makes the unit line have infinite measure. Conclusion: These pieces have no meaningful size: technically they're not measurable.
Hope this clears things up. | infinisa wrote: |
Hi Indi
| Indi wrote: | No one's caught on to the plain - non-mathematical - objection to the claim that the Banach-Tarski paradox proves 1 + 1 = 1? ^_^;
Here, i'll repeat the clue: "You can use similar logic to point out that a 1 unit long line is the same size as a 2 unit long line (the set of points in the first line is the same size as the set of points in the second line), thus giving you that 1 = 2."
Here's an even more blatant clue: Exactly what are you counting as "1" in the Banach-Tarski paradox example? |
I believe I already showed that there's a crucial difference between the Banach-Tarski paradox and your unit line analogy in my post on Sun Jan 04, 2009 10:08 pm:
| Quote: |
You are absolutely right when you say that "a 1 unit long line is the same size as a 2 unit long line (the set of points in the first line is the same size as the set of points in the second line)". But here you are using the word "size" in a completely different way from me: I'm talking about measure (where distances matter), but you're just talking about cardinality.
The transformations involved in the the Banach–Tarski paradox are all isometries (distance preserving transformations).
So the challenge to you is: can you transform a 1 unit long line into a 2 unit long line with an isometry? |
Then I explained the real error in my argument in my post of Fri Jan 09, 2009 8:41 pm:
The problem is not with the Banach-Tarksi paradox, but with my subsequent "proof" (Posted: Sun Jan 04, 2009 9:46 pm) that this leads to 1+1=1 |
Er, ok, first of all, how is that explanation "the plain - non-mathematical - objection"?
Second, weren't you the one who proposed the paradox? Why are you answering your own question? Why not give someone else a chance to test their understanding?Hi Indi
| Quote: |
| Er, ok, first of all, how is that explanation "the plain - non-mathematical - objection"? |
I don't think there is a "plain - non-mathematical - objection" to the Banach-Tarski paradox example. I tried to show that yours does not apply to my argument - it's too simplistic.
| Quote: |
| Second, weren't you the one who proposed the paradox? Why are you answering your own question? Why not give someone else a chance to test their understanding? |
As I mislead people into thinking that the problem was with the Banach-Tarski paradox, which is a very technical piece of mathematics, I thought I ought to set the record straight.
Maybe I should have waited longer after Speaker_To_Animals answer before giving the game away. | infinisa wrote: |
| I don't think there is a "plain - non-mathematical - objection" to the Banach-Tarski paradox example. I tried to show that yours does not apply to my argument - it's too simplistic. |
That is because it was not an objection to the paradox. It was pointing out that the paradox does not apply in this case. Which, coincidentally, was exactly what you said in much less "simplistic" terms: | infinisa wrote: |
| The problem is not with the Banach-Tarksi paradox, but with my subsequent "proof" |
So when you did this:
| Quote: |
| I believe I already showed that there's a crucial difference between the Banach-Tarski paradox and your unit line analogy... |
You were just repeating the obvious answer in less "simplistic", highly technical terminology. Yes there is a crucial difference between the paradox and the unit line analogy. That was the point.Ahh… I will never forget my first class, where we got a long and confusing proof that one plus one equals two… 
| biljap wrote: |
| Ahh… I will never forget my first class, where we got a long and confusing proof that one plus one equals two… |
Really? I'm impressed! I never saw such a proof at school, but once read a book (in German!) in which this result was proudly presented as theorem no. 50!Very nice thread o.o
It has definitely increased my interest in mathematics 
Hello All
| infinisa wrote: |
| biljap wrote: | | Ahh… I will never forget my first class, where we got a long and confusing proof that one plus one equals two… |
Really? I'm impressed! I never saw such a proof at school, but once read a book (in German!) in which this result was proudly presented as theorem no. 50! |
This has given me an idea: Let's start a counter-thread: Can anyone prove that 1 plus 1 equals 2?
Please note that this is not a joke
Good luck to you all!
