1 = 2.

Seaneh · 07-04-2003 8:31pm #1

I will now demonstrate that 1 is equal to 2.

Ahem.
If a = 1, and b = 1, then...

a = b = 1

Multiplying both sides by "a" gives...

a2 =ab

Now, let's subtract b2 from both sides...

a2 - b2 = ab - b2

Factoring out both sides leads us to...

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

Next, we divide both sides by (a-b) and get...

a + b = b

Substituting 1 for A and B brings us...

1 + 1 = 1

Therefore...

2=1.

There you have it folks!
Wasn't that just the fun-est thing you've done in weeks?
I already know that it's a flawed equation. But do any of you know why?

Seaneh · 07-04-2003 8:40pm

considering you can't divide a-b, since a-b=0 and a zero in the denominator gives no real solution, it's rather impossible for 1=2, or for you to get a real solution from that equation

Snaga · 07-04-2003 8:40pm

http://www.mathmojo.com/interestinglessons/1equals2/1equals2.html

DeVore · 07-04-2003 8:45pm

Originally posted by Gandalf/Sean
Next, we divide both sides by (a-b) and get...

Thats your error... a-b = 0

Anything divided by 0 gives an error.

Alternatively (if you tried to use a cancelation argument to avoid being hoist by the division-by-zero petard) I can argue:

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

Since (a-b)=0 this can be rewritten (a+b)(0) = b(0)
which reduces to the trivial 0=0.

you can annoy mathematicians in a better way by suggesting:

since infinity is the largest number known to man infinity + 1 must equal infinity and then subtract "infinity" from both sides leaving 1=0.

Be prepared to get studiously ignored or hit on the head (I prefer the hitting approach).

DeV.

DeVore · 07-04-2003 8:47pm

Bah snaga... some of us use the aul grey matter!

*mutter* computers... never catch on ... what is this interweb malarkey anyway *mutter*

DeV.

DeVore · 07-04-2003 8:48pm

Must learn to write shorter replies.

DeV.

ykt0di9url7bc3 · 07-04-2003 8:49pm

hehe infinty is not number/variable?

kamobe · 07-04-2003 8:50pm

a = b = 1

Multiplying both sides by "a" gives...

a2 =ab

Multiplying both sides by a gives you (a)(a)=(a)(b)
which equates to (1)(1)=(1)(1)

and 1=1

a*a is not equal to 2a

DeVore · 07-04-2003 8:51pm

I think he was using a2 to mean a^2 ("a" squared)...

DeV.

Sir Digby Chicken Caesar · 07-04-2003 8:52pm

i doodled something like this in math class years ago, it freaked me out when I saw it at first. i had to go to the teacher to ask wtf I had done hehe

kamobe · 07-04-2003 8:54pm

I think he was using a2 to mean a^2 ("a" squared)...

DeV.

I was thinking that having looked at the link

leeroybrown · 07-04-2003 9:14pm

But b cannot be 1:

a + b = b

Therefore a = 0

Equally you have proved that 0 = 0.

Making b = 0

But that ain't exactly proper maths either

Snaga · 07-04-2003 9:31pm

Originally posted by DeVore
Bah snaga... some of us use the aul grey matter!
*mutter* computers... never catch on ... what is this interweb malarkey anyway *mutter*

DeV.

Ah yes, I should have titled it 'The internet makes _me_ stupid', or at least very very lazy

(/me huggles google)

honeymonster · 07-04-2003 10:27pm

Me maths teacher gave me this one

4 - 10 = 9 - 15
4 - 10 + 25/4 = 9 - 15 + 25/4
(2 - 5/2)² = (3 - 5/2)²
2 - 5/2 = 3 - 5/2
2=3

Congoose · 08-04-2003 12:26pm

(2 - 5/2)² = (3 - 5/2)²
2 - 5/2 = 3 - 5/2

That is class!

I think this one is about positive and negative square roots.

R.H.S.
(3-5/2) is a positive number (1/2). Square it and you get 1/4. Here, the square root is taken to be (3 - 5/2 = +1/2), so you are taking the positive square root of 1/4.

L.H.S.
(2-5/2) is really a negative number (-1/2). When you square this you get 1/4. Here the square root is deemed to be (2 - 5/2 = -1/2), which is the negative square root of 1/4.

When taking the square root of both sides of an equation, you must take both positive square roots or both negative square roots. This would give you: 1/2 = 1/2 or -1/2 = -1/2. So it is true!

This example takes negative root on the L.H.S. and positive root on the R.H.S., so it's not allowed. Does this make sense or am I going around in circles?

seamus · 08-04-2003 1:47pm

Originally posted by SearrarD
hehe infinty is not number/variable?

Infinity isn't a number in the normal sense of the word. Normal operations have no effect on it.

Infinity +10 = Infinity
Infinity -10 = Infinity
Infinity/10 = Infinity
Infinity * 10 = Infinity

It's generally accepted that

10/Infinity = 0
Infinity/Infinity = Infinity (I think)
Infinity * Infinity = Infinity
Infinity + Infinity = Infinity
Infinity - Infinity = Infinity (I think)

Gordon · 08-04-2003 1:59pm

but multiply infinity by 8 and you get a nice 4 leaf clover.

*ahem*

daveirl · 08-04-2003 8:54pm

This post has been deleted.

Epitaph · 10-04-2003 5:52pm

Infinity/Infinity = Infinity (I think)

I don't think so. I'd imagine you should get 1, it all depends on the "scale" of infinity being discussed.

Fecking maths => TEH DODGY!!

Sev · 10-04-2003 6:36pm

Hehe, me and Davey (PHB) have this running joke now about the value of 0/0

Some say its 0 as its 0 over a number. Some say its infinity because its a number over 0, and some say its 1. Its essentially infinity/infinity, which is the same as 6/6 which is the same as 5/5 which equals 1, so as far as im concerned the answer is 1.

But the question is so utterly stupid and trivial, I like to think of it as a question of faith. What do you believe?

Gordon · 10-04-2003 6:55pm

ummm http://mathforum.org/library/drmath/view/55722.html
So it is "undefined"?

daveirl · 10-04-2003 7:53pm

This post has been deleted.

bonkey · 10-04-2003 11:03pm

Originally posted by Congoose
This example takes negative root on the L.H.S. and positive root on the R.H.S., so it's not allowed. Does this make sense or am I going around in circles?

Dont be confused.

If a² = b², then a = +/- b.
or, if you prefer :
If a² = b², then abs(a) = abs(b).

Not

If a² = b², then a = b.

Its more or less what you mentioned, though....just slightly more formal.

jc

Sev · 13-04-2003 11:20am

Originally posted by daveirl
Infinity isn't a numeric value it's a concept.
Anything divided by 0 is undefined BTW

Infinity/Infinity, technically, is not 'undefined' but 'indeterminate'. Mathematics breaks down with these numbers, so dont get caught up on definitions and technicalities and what some smartey maths dude says is correct. It's all nonsense and 'undefined' anyway, and the values you might assign to these expressions will ultimately depend on their practical application. But nevermind, you seem to have just missed the jest of my post.

daveirl · 13-04-2003 2:20pm

This post has been deleted.

Sev · 13-04-2003 5:30pm

Sure they are, 0 is a number, a number smaller the any other number. Infinity could be considered its reciprocal: a number greater than any other number.

But yes, Infinity can never really physically exist. Anyway I think it's pretty trivial debating this any further.

ecksor · 13-04-2003 7:16pm

Talking about physical existence is irrelevant.

If infinity is a number, as you claim, then which set does it belong to? And how are the arithmetic operations (or just division) defined on that set?

From your comments that this all depends on practical explanations and not to believe what smartey maths people tell you, I have a feeling that you're talking about some sort of notion of infinity that exists in your head rather than anything you can express mathematically, which is fine, but you should make it clear that this is the case.

Bard · 13-04-2003 7:23pm

Originally posted by Sev
Sure they are, 0 is a number, a number smaller the any other number.

Really? Oh...

thanks for clearing that up, 'cos... y'see... I thought negative numbers were 'smaller' than zero.

Infinity could be considered its reciprocal: a number greater than any other number.

Wouldn't 'minus infinity' be infinity's reciprocal?

As for zero, the opposite of 'nothing' isn't necessarily 'everything'. It could merely be 'something'.

But yes, Infinity can never really physically exist. Anyway I think it's pretty trivial debating this any further.

As ecksor said physical existence has nothing to do with it. Infinity exists in mathematics - that is what counts.

Sev · 13-04-2003 7:31pm

Oh sweet jesus. Exactly, thats all Im saying.

Surely there will be procedure and rules governing how operations involving infinity will be carried out in different circumstances. And there will be strict mathematical standards and definitions. But these are subjective and really bare no relevance if you have a concrete grasp of the idea of infinity and know how it might apply to your particular calculation.

Sev · 13-04-2003 7:41pm

Originally posted by Bard
thanks for clearing that up, 'cos... y'see... I thought negative numbers were 'smaller' than zero.

No. Negativity simply donates direction. -9999999999999 is a very large (in magnitude) number

Wouldn't 'minus infinity' be infinity's reciprocal?

No.

As for zero, the opposite of 'nothing' isn't necessarily 'everything'. It could merely be 'something'.

I didnt say zero was the opposite of 'everything'.

As ecksor said physical existence has nothing to do with it.

To do with what?

ecksor · 13-04-2003 7:47pm

Physical existence has nothing to do with how an operation is defined (or not) on a particular set.

But then, you seem to view the actual mathematics of this as irrelevant, which is odd since you also assert that infinity cannot physically exist.

The question of whether 0 is the smallest number that can exist is actually an good example of how such things can be confusing. Bard claims that negative numbers are smaller, which is correct if you take a set such as the integers or the reals and the operation < (less than), but you then correct him and reveal that you're actually using a different set of goalposts (the set of natural numbers, or non-negative reals or some such, who knows).

Anyway, I don't think this a useful discussion to continue.

Sev · 13-04-2003 7:57pm

All I said, is that an infinite tactile quantity cannot exist. Which is obviously true.

The question of whether 0 is the smallest number that can exist is actually an good example of how such things can be confusing. Bard claims that negative numbers are smaller, which is correct if you take a set such as the integers or the reals and the operation < (less than), but you then correct him and reveal that you're actually using a different set of goalposts (the set of natural numbers, or non-negative reals or some such, who knows).

Thankyou for just reiterrating the point Ive been trying to get across this entire thread.

Originally posted by ecksor
Anyway, I don't think this a useful discussion to continue.

As I said.

ecksor · 13-04-2003 8:09pm

Originally posted by Sev
Thankyou for just reiterrating the point Ive been trying to get across this entire thread.

:rolleyes:

I find that ironic, since the point I was trying to make is that you're not defining the context in which your speaking in a way that makes the contentious statements you've made make any sense.

Yes, I appreciate that "it depends on context" was one of your points, but you made some very definite assertions about what infinity is that you didn't make any attempt to qualify by providing ANY context in which what you claim (that infinity is a number) makes any sense. I'd settle for just one, since if you could provide one it would be something interesting that I haven't come across before.

I know I said I didn't want to continue this, but I find it immensely frustrating when people continue to post "no, that's not what I'm talking about. no, that's not what I'm talking about either", since I never know if there is an interesting point that I'm missing or not. If this doesn't have anything to do with maths, then just come out and say so (although how you can attempt to resolve something like infinity or, more importantly, why you would want to would be an interesting discussion in itself).

Sev · 13-04-2003 8:52pm

Well I apologise for any confusion, I would have thought the majority would understand my point of view instead of doing everything they can to find faults (that are completely subjective) with everything I say. Yes, most of everything I have said is based on perfect common sense and logic. I am just trying to separate standardised abstract mathematical axioms from practical understanding.

You are trying to strictly quantify and qualify everything I say. My point is that the maths is entirely subjective. Yes, infinity can be a number if I want it to be. I find no difficulty in contemplating a number which is infinitely big, whether "technically" that does not constitute the expression of infinity in the way that official international mathematical standards defines makes no odds to me. Although you seem quite hung up on that.

Mathematics at its heart is simply a tool, a way of simulating or predicting physical events by the application of logic. I just take exception to the strict definitions or governing rules that only have any relevance in an entirely non physical situation.

If I asked you, what is a number that is bigger than any other number? you would tell me infinity, or that no such number exists.

ecksor · 13-04-2003 9:09pm

Originally posted by Sev
Well I apologise for any confusion, I would have thought the majority would understand my point of view instead of doing everything they can to find faults (that are completely subjective) with everything I say.

I'm not the majority, I tend to take a purely math based view of such things. If it looks like I'm nitpicking, it's because I am nitpicking, but the difference is that I think such things are important, and you don't, which is fair enough, I now understand your point of view.

You are trying to strictly quantify and qualify everything I say. My point is that the maths is entirely subjective.

mathematics is not subjective. It is a means to express and reason about concepts in a formal and objective manner.

I find no difficulty in contemplating a number which is infinitely big, whether "technically" that does not constitute the expression of infinity in the way that official international mathematical standards defines makes no odds to me. Although you seem quite hung up on that.

This is a hang up? Good grief, I had no idea. The word 'standard' makes it sound like infinity is overseen by a committee

Mathematics at its heart is simply a tool, a way of simulating or predicting physical events by the application of logic. I just take exception to the strict definitions or governing rules that only have any relevance in an entirely non physical situation.

It can be a tool that is applied, or a purely abstract thing. However, I'm not sure it does much good in either case if it does not demand rigour, which was what I was looking for.

If I asked you, what is a number that is bigger than any other number? you would tell me infinity, or that no such number exists.

I'd probably say that no such number exists. Then again, it might depend on how you frame the question or what set we're talking about in other words. Mathematics has the notion of the well-ordering principle that states that if a set has at least one member, then it has a least member. There is no equivilent for a greatest member, for obvious reasons.

DeVore · 13-04-2003 9:10pm

Originally posted by Sev
I am just trying to separate standardised abstract mathematical axioms from practical understanding.

I cant imagine why...

You are trying to strictly quantify and qualify everything I say. My point is that the maths is entirely subjective. Yes, infinity can be a number if I want it to be. I find no difficulty in contemplating a number which is infinitely big, whether "technically" that does not constitute the expression of infinity in the way that official international mathematical standards defines makes no odds to me. Although you seem quite hung up on that.

Details are important in mathematics. You are of course quite free to imagine infinity as a "giant pink elephant" in your "subjectivity" but then you are breaking from the correct usage of the mathematical term. If thats what you are doing then I consider you a fwthong[1].

Mathematics at its heart is simply a tool, a way of simulating or predicting physical events by the application of logic. I just take exception to the strict definitions or governing rules that only have any relevance in an entirely non physical situation.

Because 1 apple and 1 apple doesnt make 2 apples?

requoted from above:
My point is that the maths is entirely subjective.

Um no its not, in fact one of the most important things that a mathematician strives for is to remove subjectivity from his/her logic.

2+2=4 regardless of how you define terms. We commonly agree these "gutteral tones" to convey mental images but beyond that the mathematics of that equation are ALWAYS TRUE.

I cant wander around believing that for me 2+2=5 ... well I can, I'm just wrong.

Now, what I CAN DO is redefine my notation so that *I'm* using the symbol "5" to mean what the rest of you refer to as "4". If I do that then 2+2 DOES equal 5 under my notation.

That, however, is not subjectivity. The notation may change but the logic doesnt.

DeV.

[1] I will define "fwthong" however I see fit

Sev · 13-04-2003 9:37pm

I didnt say that maths is entirely subjective, I said that 'the' maths was entirely subjective: the aforementioned very abstract and particular 'laws' regarding infinity. Notice the subjective use of the word the

As for the point im trying to make, which I still havnt seemed to convey, is that the concept of mathematics I perceive, is not based on rules, rigour or standardised doctrine. But the fundamental application of logic.

Details are important in mathematics. You are of course quite free to imagine infinity as a "giant pink elephant" in your "subjectivity" but then you are breaking from the correct usage of the mathematical term.

But that is exactly what I don't need, "mathematical terms" and "details" if I can visualise, comprehend and achieve a solution with my own understanding of what is going on, without any need to categorise elements of my calculations into 'sets', 'subsets' or whatever jargon you require to reconcile a problem with your notion of mathematics.

Im going to let this rest now, cos Im finding this extremely difficult to explain and would appear to have a completely distinct and incompatible mindset approaching this whole debate. But I shall continue working with my giant pink elephant.

Bard · 13-04-2003 9:49pm

Ok I take back what I said about negative numbers being "smaller" than 0. They're not, they're just 'lesser'.

But that is exactly what I don't need, "mathematical terms" and "details"

Maths is a precise science. It's very important to be precise and stick to specific definitions. That's not to say that you can't use allegories to better explain these definitions... and that's not to say that conventional wisdom cannot be challenged - we'd have no innovation otherwise.

Infinity is an impractically non-existant number of undefinably massive magnitude. It's what you could imagine getting if you tried dividing anything by zero. I say you "could imagine getting" because there is no possible answer.

Allegorically here, the question could be posed: "if people were to queue up to get some cake from you and the portion you gave to each person was zero, how many people could you serve?"... there, the answer would be 'infinity'... but technically the answer could be 'it is impossible to answer this question'.

DeVore · 13-04-2003 9:59pm

As for the point im trying to make, which I still havnt seemed to convey, is that the concept of mathematics I perceive, is not based on rules, rigour or standardised doctrine. But the fundamental application of logic.

Ok, I'm sorry, but that just crashed my brain.

Russell's Principia Mathematica is a good but weighty read on the topic of mathematics as it corresponds to logic (or rather, vice versa).

I think I can safely say that most scientists and mathematicians to consider "logic" without "rigour" to be "opinion".

There is such a thing as fuzzy logic, but theres no such thing as sloppy logic.

DeV.

Talliesin · 14-04-2003 6:12pm

Freedom is the freedom to say 2+2=4. If that is granted, all else follows.
- George Orwell, 1984

1 = 2.

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