Does the square root of negative one lead to a contradiction?

[Johnny5]

The title is self explanatory.

 

I would like to hear different people's answer to this.

[matt grime]

Does the question even have any meaning? I'd say not. How about: does Table Mountain lead to any contradiction?

 

The square root of minus one is simply, when it exists, and element that squares to give minus one. Implicit in that is that we are working in some field or such. It is just an object it leads to nothing; reasoning about it may do.

 

For instance the statement "F is a finite Field with p elements" and the statement "there is necessarily an element that squares to give -1" are contradictory, but "the square root of minus one" on its own?

[Johnny5]

Does the question even have any meaning? I'd say not. How about: does Table Mountain lead to any contradiction?

 

That is one way to avoid the issue altogether.

 

Do numbers exist?

 

Well?

[matt grime]

What issue, Johnny? You haven't raised an issue. What on are you getting at? sqrt(-1) is just some object that may or may not exist within some field.

 

If you were to say, does sqrt(-1)=1 lead to any contradiction (in any field except one of characteristic two) then yes, because you're saying 1=-1. But just (sqrt(-1)? it is not a well formed question. an object can not lead to contradictions until you try to relate it to other things.

[Johnny5]

What issue' date=' Johnny? You haven't raised an issue. What on are you getting at? sqrt(-1) is just some object that may or may not exist within some field.

 

If you were to say, does sqrt(-1)=1 lead to any contradiction (in any field except one of characteristic two) then yes, because you're saying 1=-1. But just (sqrt(-1)? it is not a well formed question. an object can not lead to contradictions until you try to relate it to other things.[/quote']

 

I thought you would never get around to answering my question as to whether or not numbers exist... and you seem to have avoided it. Don't worry I wasn't trying to confuse you, I already know the answer. I wanted to see what you would say.

 

So now, let me read through your paragraph here, to see what you have said.

 

 

This part here is quite good...

 

an object can not lead to contradictions until you try to relate it to other things.

 

Let me see what the hell am I really asking hmm...

 

Well lets say we have a set of consistent axioms, the field axioms.

 

We don't want to do anything to disrupt the set of consistent axioms. They actually have to do with mental operations, that we ourselves perform using symbols.

 

So the last thing in the world we want, is to have a conflicting set of rules. That would be bad.

[matt grime]

I avoided the existential point about numbers since it is completely immaterial in the context you stated it. When I said that the square root of minus one may or may not exist it was in the mathematical sense of: the square root of minus 1 does not exist (ie is not an element of) in the set of real numbers, similarly, in Z_4, the integers mod 4 there is no element that squares to give 3, but there is one in Z_2.

[Johnny5]

I avoided the existential point about numbers since it is completely immaterial in the context you stated it. When I said that the square root of minus one may or may not exist it was in the mathematical sense of: the square root of minus 1 does not exist (ie is not an element of) in the set of real numbers, similarly, in Z_4, the integers mod 4 there is no element that squares to give 3, but there is one in Z_2.

 

Yes i know what sense you used the word Matt, don't worry.

 

 

There is a conceptual difference between

 

existence and 'at least oneness'

 

In some contexts to exist means to be in the current moment in time.

 

But for anyone using set theory, the existential quantifier translates best as "there is at least one"

 

How many men are there?

 

Right now, the current answer is something like four billion.

 

How many men are there throughout the whole of time?

 

The answer to this question was always... at least one.

[Johnny5]

Don't go into modulo yet, really i want to settle this issue about i, for myself.

 

I remember a proof that i leads to some conflict with ordering.

 

It was shown to me, probably a decade ago, but i never bothered with it.

 

But you know, if someone else formulated the argument, it means someone else was thinking about this issue, the one you say i didnt raise.

 

When I said "square root of minus one on its own" i was being deliberately vague.

 

I am concerned about ordering of real numbers.

 

You cannot have two parts of your knowledge that conflict.

 

Argand plane

 

You have one axis real, the other imaginary.

 

So can you order imaginary numbers, as the imaginary axis seems to suggest?

 

i,2i,3i...

 

I wish I could remember how that proof went.

 

Because now i know just a bit more than i did back then.

[matt grime]

So, you're asking "is there an ordering on the complex numbers that respects the multiplicative nature of C, a la the ordering of R, eg given two numbers either x<y, y<x or x=y, and if z>0, then x<y implies xz<yz?"

 

The answer is "of course not" since R is the unique complete ordered field and R and C are not isomorphic as fields, never mind as totally ordered ones, and they are both complete.

 

If this is what you thought the issue was then it is a strange thing to say "the square root of -1 leads to the contradiction". Certainly if one assumes i>0, the i^2=-1>0, contradiction, and if i<0, then i*i>0*i, -1>0 again, a contradiction.

 

However, you see this only is a contradiction to the assertion "there is a an ordering of C compatible with some conditions as above", which is a false assertion.

 

You surely can't have expected people to come to that conclusion that that is what you were after based upon the title of the thread which you have said is

 

a) self explanatory

 

and

 

b) deliberately vague.

 

For if that were so then every true proposition would lead to a contradiction - it would contradict the assertion that some false proposition were true.

 

Putting it another way, you're saying "are there any statements about C that are false owing to the fact that i^2=-1", and obviously there are: for instance the statement, "there is no element that squares to give -1" is contradicted by the existence in C of i. This is hardly earth shattering.

[Johnny5]

If this is what you thought the issue was then it is a strange thing to say "the square root of -1 leads to the contradiction". Certainly if one assumes i>0' date=' the i^2=-1>0, contradiction, and if i<0, then i*i>0*i, -1>0 again, a contradiction.

[/quote']

 

 

That's the one.

 

Thats how the proof would have to go.

 

if i>0 then problem.

if i= 0 then problem.

if i<0 then problem.

 

Suppose i>0.

By a theorem, i*i>0.

but, i*i=-1, hence -1 can be substituted for i*i, so

-1>0, which is false.

 

Suppose i<0.

 

Product of two negative numbers is positive and a positive number is greater than zero. If (-1)^1/2 is less than zero then i*i>0, and again -1>0, which is false.

 

And of course if i=0 then i*i=0 whence -1=0, which is again false.

 

So you start out with a set U.

 

Then you have a bunch of axioms which hold on the set.

 

You have an undefined binary relation <

 

And the following statement is true:

 

For any x,y elements of U

 

x=y or x<y or x>y

 

Then you encounter i= (-1)^1/2

 

Since the above statement is true, root -1 isn't an element of U.

 

Then, what is done, is to postulate a superset C.

 

So that U is a subset of C.

 

From this it follows that any element of U is also an element of C, but the converse is false.

 

Clearly the axioms/theorems of the real numbers do not all hold on C.

 

Some axioms hold but not all.

 

What really I mean, is that the binary relation < does not have general meaning across the complex numbers.

 

Let me google this and see what I find. I should have thought of that sooner.

 

Here is the first article I came across: Dr Math

 

 

To much GD nonsense on the web.

 

Actually here's something that seems familiar...

 

The nature of complex numbers

 

I'm going to have a look at that.

 

 

Complex numbers are ordered pairs of real numbers for which multiplication is defined in a special way. Let (a,b) and (c,d) to ordered pairs of real numbers. The product of (a,b) with (c,d) is defined as:

 

(a,b)*(c,d) = (ac-bd,ad+bc).

 

The work at that site all looks familiar. I think it was in my complex variables book.

[kriminal99]

What issue' date=' Johnny? You haven't raised an issue. What on are you getting at? sqrt(-1) is just some object that may or may not exist within some field.

 

If you were to say, does sqrt(-1)=1 lead to any contradiction (in any field except one of characteristic two) then yes, because you're saying 1=-1. But just (sqrt(-1)? it is not a well formed question. an object can not lead to contradictions until you try to relate it to other things.[/quote']

 

I disagree. Having to resort to saying sqrt(-1) exists and -1=1 because you are attempting to model real life space with a primitive euclidean coordinate system, or rather just saying -1=1 IS a logical contradiction...

 

What you are trying to do is to redefine logic itself to get around the fact that you have contradicted yourself...

[Dave]

Say what now? That post didn't make any sense whatsoever. He was simply pointing out that Johnny5 hadn't actually stated what his question was properly.

[matt grime]

I disagree. Having to resort to saying sqrt(-1) exists and -1=1 because you are attempting to model real life space with a primitive euclidean coordinate system' date=' or rather just saying -1=1 IS a logical contradiction...[/quote

 

 

I did nothing of the sort.

 

What you are trying to do is to redefine logic itself to get around the fact that you have contradicted yourself...

 

Wow, did you not understand my post. Shall we try another one?

 

Paraphrasing Johnny: does sqrt(2) lead to a contradiction?

 

It is just a meaningless statement vague and open ended - objects do no lead to contradictions; propositions may do so. In this case "the square root of two is rational" would led to a contradiction. You can't get a contradiction if there is notthing to reason from.

[Johnny5]

 

Wow' date=' did you not understand my post. Shall we try another one?

 

Paraphrasing Johnny: does sqrt(2) lead to a contradiction?

 

It is just a meaningless statement vague and open ended - objects do no lead to contradictions; propositions may do so. In this case "the square root of two is rational" would led to a contradiction. You can't get a contradiction if there is notthing to reason from.[/quote']

 

hmmm just hmmm Really i wanted a smiley face with one eyebrow raised, but we don't have that so i chose the dude with the dark glasses.

[zaphod]

the square root of a negative number exists in C by definition.

 

thats pretty much all that really needs to be said about that.

[kriminal99]

If you are talking about imaginary numbers other than to say that they have no relevance and do not exist then one can assume you believe they have some relevance. So it seems we can get a question from his original statement after all... And to answer this question I think it is a contradiction. I realize I misread your original post though, grime...

 

@ zap's post- This is exactly what I meant by mathematicians often trying to redefine logic... You can't give something a definition that contradicts the definitions of the composing ideas.

[AL]

@ zap's post- This is exactly what I meant by mathematicians often trying to redefine logic... You can't give something a definition that contradicts the definitions of the composing ideas.

What do you mean? I assume by "composing ideas" you mean the field axioms of C, but in this case, the definition of i does not contradict any true statement in C. Can you clarify please?

[Johnny5]

If you take away their imaginary number i, they will not know what to do.

[matt grime]

If you are talking about imaginary numbers other than to say that they have no relevance and do not exist then one can assume you believe they have some relevance. So it seems we can get a question from his original statement after all... And to answer this question I think it is a contradiction. I realize I misread your original post though, grime

 

You realize that just using someone's surname like that could be considered rude?

 

Imaginary numbers have no relevance and do not exist?

 

Right, ok, point out a number that does "exist" whateve exist means. I suspect that you believe that reals exist and that R^2 exists (the plane - but if you care to send me one of these in the post I'll be most impressed), and since the complex numbers are simply an algebraic structure on R^2 I have no idea what you're waffling on about.

 

I don't think you understand the mathematical use of the word 'contradiction', or even 'exists' for that matter.

[kriminal99]

You realize that just using someone's surname like that could be considered rude?

 

Imaginary numbers have no relevance and do not exist?

 

Right' date=' ok, point out a number that does "exist" whateve exist means. I suspect that you believe that reals exist and that R^2 exists (the plane - but if you care to send me one of these in the post I'll be most impressed), and since the complex numbers are simply an algebraic structure on R^2 I have no idea what you're waffling on about.

 

I don't think you understand the mathematical use of the word 'contradiction', or even 'exists' for that matter.[/quote']

 

Oh thats your real name?

 

Ok so existence is a bad way of putting it. How about self contradiction? Or relevance to reality?

 

Q^2? Maybe something similar as an arena for using a spatial metaphor to model a relation between two factors, but not as a continuum but rather discreet some sort of discreet "space" depending on the smallest unit of what you are measuring. (Even if it is something like energy) Not as some kind of space like real life space...

 

Real numbers, at any point to which they are defined numerically, are equivalent to some rational value. We will never need an infinite amount of precision so they will never have any relevance as a "number". The only reason we consider them is because of the difficulty in trying to model real life space with axes...

 

Real life space is a continuum, not defined by little perpendicular axes that are infinitely small in all but 2 directions...

[matt grime]

You've stil not pointed out a number (rational or otherwise) that actually "exists" in whatever badyl chosen sense you are talking about. So what that every number used in calculations is rational. That has nothing to do with anything.

 

Stop confusing maths with the real world, cos we, the mathematicians aren't, and never claim to be. So things are contradictions to you if they have no "relevance" (whatever that may mean) to the real world. Well, thanks for the opinion, but that's all it is. In that case why even think about maths almost all of which is in this sense completely irrelevant to the real world?

[zaphod]

 

@ zap's post- This is exactly what I meant by mathematicians often trying to redefine logic... You can't give something a definition that contradicts the definitions of the composing ideas.

 

which logic is being redifined here? i dont understand.

 

if you take a set of objects and perform certain operations on them' date=' sometimes you will realize that the set isnt closed under that operation.

 

for example, when the greeks tried to take the square root of 2, they couldnt fathom the existence of a number not part of the rational numbers. do the so called "irrational" numbers lead to a contradiction? i would hope to think not.

 

so when you've got the idea of the square root of a negative number, the solution falls outside the original set of numbers. so just as math was plunged into irrational numbers to find solutions to square roots of "non-square" numbers, the "imaginary" number was created and the complex number system was used to find solutions to square roots of negative numbers.

 

"irrational" and "imaginary" arent meant to be taken literally.

 

they're perfectly valid number systems which allow calculations to be made.

 

just because you cant have 3[i']i[/i] apples doesnt mean that the number system is a fallacy. i mean you cant have pi apples either (no matter how hard you try).

[Johnny5]

I would like to say something.

 

As algebra developed, so too did the notation.

 

We have now arrived at simple expressions such as:

 

3x^3-2x^2+8 = 0

 

9x-5=0

 

x^2+2x-1=0

 

And so on.

 

Such polynomials have a degree n; (the highest power of x in the expression).

 

The complex numbers emerge right when we consider second degree polynomials. It doesn't matter whether or not the coefficients are rational irrational, or complex, my point is this.

 

From the desire to have n solutions to an nth degree polynomial equation (a few examples listed above) came the set of complex numbers.

 

So, whatever the answer is to my original question, it lies in my previous sentence.

[zaphod]

 

 

From the desire to have n solutions to an nth degree polynomial equation (a few examples listed above) came the set of complex numbers.

 

 

 

if thats the case, then irrational numbers (and hence real numbers) were invented from the "desire" to have a solution to sqrt(2).

 

and i guess natural numbers were invented from the "desire" to have a solution to the question: "how many apples in basket, mr caveman?"

 

whether or not you think you've discovered something about the nature of mathematics, the fact of the matter is, you havent.

 

complex numbers are a valid (and useful, to boot) number system.

[Johnny5]

if thats the case' date=' then irrational numbers (and hence real numbers) were invented from the "desire" to have a solution to sqrt(2).

 

[/quote']

 

That's not false. What you have written above here, is actually true.

 

Not my desire of course, someone elses.

 

I'm not out to topple mathematics or anything, I'm not really sure what I'm out to do, except maybe see what others think about a few things which we never question our professors about, because we anticipate that there is no reasonable answer.