Originally Posted by Xei
Read it again. Still no idea what it means. Mainly because you introduced new, undefined words like 'principle'. What does
0 1
1 0
look like when it's made out of wood?
A principle is basically a rule concerning the functioning of a thing or class of things. The matrix you typed would look basically like what you posted if made of wood, except it would be made out of wood. The coloration may be different, but the matrix could be painted black. You might want to add to it by carving tall parentheses and putting them on the sides, and they could be painted black also.
Originally Posted by Xei
Once again you only think this is an argument because you're assuming the consequent.
The real number line is 'based on' the existence of real numbers. This isn't a problem, so long as we can prove that real numbers exist. The complex plane is based on the existence of complex numbers. This isn't a problem, so long as we can prove that complex numbers exist.
Yes, and the complex plane cannot be used to do that. How tall and wide a building is can prove the existence of real numbers.
Originally Posted by Xei
We establish that real numbers exist because we can find a physical situation to which they exactly correspond (your definition of existence), for example, displacements along a continuous line in space.
We establish that complex numbers exist because we can find a physical situation to which they exactly correspond, for example, rotations of objects.
Again, the complex plane is nonsense. Think of it this way: You stand on a big flat surface and accurately paint a real coordinate plane on it. If you then stand stand at (0, 0) and walk 3 meters in the positive direction to the right along the xaxis and then walk 4 meters in the positive direction parallel to the yaxis, you will in fact be 5 meters from the origin. All real, actual, true, measurable distances in the universe. Now imagine similarly constructing a complex plane. Explain a similar situation involving walking on it. Would it make sense to say that you could arrive at a point 3 + i feet from the center? Would that be an actual distance? If so, the next time somebody asks your height, say what it is in complex numbers.
Originally Posted by khh
A better example, I think, would be the amplitude and phase of a sine. In electronics when you're working with AC, voltage and current is almost always written on the form Ae^{iθ}.
That was Xei's statement you quoted. I don't know jack about electronics. Could you give me an AC For Dummies explanation of how that equation translates to reality? A times 2.71828 to a power that is the product of sqrt 1 and an angle measure... I want to research this myself, but I wanted to see if you could give me a quick summary. It looks like we might be exploring an interesting avenue here.
Originally Posted by tkdyo
This was my thought as well khh. In electrical engineering we use complex numbers A LOT to signify real world things. Especially when talking about power and energy lost.
That sounds fascinating. Considering the general issue I have taken with complex numbers, can you summarize how that works in the universe? I don't know enough about what you are saying to rule it out.
Originally Posted by Olysseus
The problem I still have is that the "contradiction":
1 = i*i = sqrt(1)*sqrt(1) = sqrt(1*1) = sqrt(1) = 1
can be reduced to: 1 = sqrt (1) = 1
The imaginary numbers in the the first equation are, to my mind, an unnecessary bit of sophistry to make the whole problem look more complex (crap...I just made a really bad pun.). But surely no one would claim that the second equation is really a contradiction or that it really "proves" 1=1. Yet it is effectively the same line of reasoning as the first without any imaginary values.
It seems to me, and I welcome thoughts if you disagree, that to explain why the second equation is not a contradiction, we would say, "negative one can be equal to the square root of one and the square root of one can possibly equal one or negative one. Thus negative one is equal to something that can possibly equal negative one or positive one, so negative one is only equal to one or the other." The context of what we started with does matter. But once we have conceded this, it applies to the first equation with imaginaries as well.
Am I missing something? Can you explain 1 = sqrt(1) = 1 otherwise? So I am not convinced of the premise that the principal root is all we consider if no variable is involved. I could be wrong, but I thought that anytime a root was brought into both sides of an equation, whether there was a variable or not, then both +/ roots had to be considered. The only time we default to the principal square root is when we are asked to actually calculate the value of a given root. In other words, I thought the concept of principal square root was really just a convention. I should have raised this question earlier because I may have led you astray by not directly questioning your premise. If I could find a reference that definitely said you only take the +/ root when dealing with a variable, then I would have to concede your point that my line of reasoning is wrong. I'll look into my math books and see if I can find something like that.
Until then I'm still unconvinced that these sorts of paradoxes alone really tell us anything about the reality versus fictionality of imaginary numbers.
The first proof involves only principal square roots. The radical sign with an understood index of 2 indicates principal square root. Also, the +/ does not apply there. If you have 81 = 81 and get the principal square root of each side, you have 9 = 9. If you put a minus sign before each radical, you have  9 =  9. If you get the general square roots of each side, you have +/ 9 = +/ 9, which would mean that 9 = 9 and  9 =  9. You could never legitimately infer that 9 =  9. As for  1 = sqrt 1 = 1, it depends on what is meant by "sqrt 1." I use it to mean "principal sqrt, but that is just internet lingo I have hardly used in my life. I type radicals in my job related writings. If you use "sqrt" to mean "anything that multiplies by itself to get the number," then your equation would not be accurate at 1 = sqrt 1. You would need to indicate that it is just one of the two square roots. = and "could be" are two different things. Your equation suggests that 1 is the only square root. That is why I see it as something different from the issue raised in my pseudoproof. I will think about it some more, though.
Does that clear up anything?
Originally Posted by PhilosopherStoned
No there is no difference. In fact it was given no "addition". The correct statement was never "taking square roots is multiplicative on all numbers" it was "... on nonnegative reals".
Suppose that, as before, P(x) holds for all x in X and that Y extends X. Let us suppose that P(y) also holds for all P and all y in Y. Let P(x) be "x is an element of X." Applying it on some element of y that does not lay in X yields P(y) = "y is an element of X". But by assumption, it isn't.
So if we're allowed to just extend a statement like that without proof, we can proof that if all x are y then all y are x. This is creationist stuff.
Hence we can't do that. The formal argument is formally wrong. But the bolded part of your argument is an instance of this formal argument. "It applied to the real numbers so it should apply to the complex numbers or they don't exist".
To claim that the square root is multiplicative on the complex numbers, you'd have to prove it.
Ok dude. As I said, you didn't make any points, specific or otherwise. There is no paradox and there is no challenge. I'm not inclined to waste much more of my time discussing this with you but, as Xei said on that thread, sqrt(a)sqrt(b) = sqrt(ab) is the same identity as sqrt(a)/sqrt(c) = sqrt(a/c) and this is obtainable by substituting b = 1/c (valid everywhere c =/=0) to get
sqrt(a)sqrt(b) = sqrt(ab)
sqrt(a)sqrt(1/c) = sqrt(a*1/c)
sqrt(a)*1/sqrt(c) = sqrt(a/c)
sqrt(a)/sqrt(c) = sqrt(a/c)
Hence if it's not valid in one form it's not really valid in the other, now is it? So it's the same problem as on this one, you're not allowed to just make up rules and assert that they hold without proving it.
Now read my posts to see where I have beaten dead horses on those topics. Answer a simple question for me. Before any mathematician ever mentioned anything about imaginary or complex numbers, was the rule concerning the bottom proof stated to be limited to nonnegative real numbers? Or was it accepted as a universal principle that applies to all numbers? Maybe you can open your mind enough to see what I am saying if you will tackle that. You have dodged it so far, and it is my key issue. Try not to be outlandish enough to claim that I have not asked a question.
Also, read my posts in the other thread more carefully. I said that the pseudoproof is a reductio ad absurdum of imaginary numbers short of the declaration of "exception." When you answer my question, you may understand that point better. The point you are illustrating is essentially what immediately pertains to the first rule they teach in a standard logic class. I am in complete agreement with it. So again, "Exception," is the only defense to the reductio ad absurdum, and the exception was conveniently made to accompany the crock of shit and battle the type of reasoning used in the reductio ad absurdum. I call it a reductio ad absurdum, and you call it an exception that came long after the establishment of the rule. My view of that is, "How convenient."
I will put it in a way that you may agree with. It is a reductio ad absurdum that works according to the long established rule and not to the exception that came later. How about that?


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