Re: The State-of-the-Art in Mathematics, but didn't I solve this? (Smart1234)

[G.E.Ivey]

On 19 Dec 2004, S. Enterprize Company wrote:
>>>The problem here is not the choice of values of sqrt(-1) but the concept i
>>>itself. It is not well-defined because the mapping sqrt is well-defined only
>>>on perfect square such as 1, 4, etc.
>>>
>>>E. E Escultura
>>
>>Ok can it be represented like this?
>>
>>n=1 to oo
>>with cartesian coordinates x,y,z
>>Use only integer powers of n
>>otherwise it's an indeterminate
>>
>>[(-1)^n/2] x
>>[(-1)^n/2] y
>>[(-1)^n/2] z
>>
>>
>>For x coordinate,
>>[(-1)^n/2] x
>>
>>at n=1 --> indeterminate
>>at n=2 --> -1
>>at n=3 --> indeterminate
>>at n=4 --> 1
>>at n=5 ---> indeterminate
>>at n=6 ---> -1
>>at n=7 ---> indeterminate
>>at n=8 ---> 1
>>etc...
>>
>> In cases where you have sqrt(-2), etc..,
>>for n=1 --> oo
>>Integer Powers of n Only
>>
>>sqrt (-2) = sqrt(2) [(-1)^n/2)]
>>
>> = (1.41421...) [(-1)^1/2)]
>> = (1.4142...) ( indeterminate)
>>
>>And,
>>
>>(sqrt (-2))^2 = (1.41421...)^2 [(-1)^2/2)]
>> = (2)(-1)
>> = -2
>>And,
>>
>>(sqrt(-2))^3 = (1.41421...)^3 [(-1)^3/2)]
>> = (2.8284...) (indeterminate)
>>
>>And,
>>
>>(sqrt(-2))^4 = 3.9998...
>>
>>And so on...
>>
>> Does this map sqrt correctly with negative sqrt's?
>>
>
>
> Why don't you reply? Why doesn't anyone reply? What's wrong you don't like to
>give me credit for anything?
>
>
>
>
Credit for what? you haven't said anything worth paying attention to. The answer to "S. Enterprize Company"'s original statement is just that he is completely wrong about everything. Mathematicians DON'T define i as "the square root of -1" for exactly the reason he states: every number has two square roots and since the complex numbers are not an ordered field it is impossible to distinguish between "+i" and "-i".

What mathematics DO is to define the complex numbers as the set of PAIRS of real numbers (x,y) with additon defined by (a,b)+ (c,d)= (a+c, b+d) and multiplication defined by (a,b)*(c,d)= (a*c-b*d,a*d+b*c). That way, it happens that (0,1)*(0,1)= (0,-1)*(0,-1)= (-1, 0) which can be corresponded to -1. i is specifically defined as (0,1).

[S. Enterprize Company]

First of all, I don't agree with you when you imply that if I don't know one
thing, then I don't know anything, or if I appear to be wrong about one thing,
then I am wrong about everything. Let's look at this mathematically.

Let x= 1 mistake or something you think I said wrong.
n = 2 to oo
Let y = lim SUM A_k = no mistakes

You can see that I could be right 99.999... % of the time with only one
thing mistake. This would show that YOU are mistaken about everything except
one thing. So who is the most correct? I am.

Now going back to this topic of imaginary numbers. I already know the
standard or conventional way imaginary numbers are handled. But it was pointed
out to me by someone that "i" is ill-defined and there are problems mapping it.
If you look at the meaning of i,

sqrt (-1) = i

there is no such thing unless you pair it like you say then relate it to real
numbers.

Well since then I offered another proposal which was to use a non-standard
approach to this, classifing it similarly to surreal, superreal and hyperreal
numbers, except I called it " a surreal positive or negative sign". In other
words,

a = a certain degree of the sign itself
b = another certain degree of the sign itself

and the surreal notation would be like this,

a|b where { | } = 0

so,

(.5_-) | ( -- )

(.5_-) < ( -- )

This states that a certain degree of the sign is used in this case and the next
think after it would be the total negative sign. Using this method of analysis,
you can have decimal degrees of negativeness. Like for example, instead of

( -1 )^.5 = (1_-)

you could have,

(-1)^.5123 = (1_-.5123)

where the degree of the negativeness, which is a sub-level non-standard
analysis of negative could be represented like this.

(.5123_- ) | ( -- )

In this case there would be an ill-defined state of "i" to start with.

Then I showed ways this sub-level sign could be handled with addition,
multiplication and powers. It seems to work logically in my opinion with
nothing ill-defined.

Just check out the other posts I made about this, a new non-standand proposal
in math. at the sub-level locations that could exist within the sign itself.

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