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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.

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.

Originally Posted by Universal Mind

There is a difference. The rule regarding radicand products was established one way for a very long time, but it had to be given an addition when imaginary numbers were introduced to the system. The rule was generalized to the entire set for ages, though even roots of negative numbers were assumed not to exist. Then the new rule involved an exception.

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 non-negative 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.

You are being flat out dishonest. Read the last post where I addressed Xei. Counter the specific points I made in it. Even if I were fumbling over the specific points creationists do, it would not warrant your immature rudeness, 31 year old.

I challenge you to resolve the paradox involved in the "proof" I posted in the thread titled "1 = -1?" I think it is probably a reductio ad absurdum of imaginary numbers, but I welcome your resolution.

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.

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 x-axis and then walk 4 meters in the positive direction parallel to the y-axis, 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 pseudo-proof. 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 non-negative 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 non-negative 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 pseudo-proof 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?

Originally Posted by Universal Mind

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 non-negative real numbers? Or was it accepted as a universal principle that applies to all numbers?

Of course it would have been frequently stated as a rule for all numbers there not yet being any knowledge concerning the potential to define complex numbers. However that doesn't change anything. The simple fact is that it is still the sort of thing that needs to be proved. Because the proof goes through on non negative reals but not on complex numbers, your identity doesn't hold for complex numbers like it does for reals.

Let me ask you a question.

Do you or do you not acknowledge that the multiplicativity of the square root depends on the domain from which we are extracting roots?

Originally Posted by PhilosopherStoned

Of course it would have been frequently stated as a rule for all numbers there not yet being any knowledge concerning the potential to define complex numbers. However that doesn't change anything. The simple fact is that it is still the sort of thing that needs to be proved. Because the proof goes through on non negative reals but not on complex numbers, your identity doesn't hold for complex numbers like it does for reals.

I cannot prove that it applies to complex numbers. It doesn't. My "proof" shows that. For proving that it does not pertain to complex numbers, what I typed is literally a proof. We are in full agreement there. I am just saying that an exception had to be made up because complex numbers were made up. A much lower intellectual level version of the same thing is paranormal enthusiasts coming up with wild claims about shit walking through walls and levitating. If you tell them that the already established laws of physics don't allow such things, they say, "Oh, but these are exceptions. They are special." So, you can never stump such a person with laws of physics. That is the game mathematicians played, though with a lot more showing of intelligence.

Originally Posted by Universal Mind

I cannot prove that it applies to complex numbers. It doesn't. My "proof" shows that. For proving that it does not pertain to complex numbers, what I typed is literally a proof. We are in full agreement there.

OK. That's all I wanted to know. It's worth pointing out that what you typed nothing more than a proof by contraction that the square root is not multiplicative on the complex numbers.

I am just saying that an exception had to be made up because complex numbers were made up. A much lower intellectual level version of the same thing is paranormal enthusiasts coming up with wild claims about shit walking through walls and levitating. If you tell them that the already established laws of physics don't allow such things, they say, "Oh, but these are exceptions. They are special." So, you can never stump such a person with laws of physics. That is the game mathematicians played, though with a lot more showing of intelligence.

But this isn't the game that mathematicians played and it's not even a fitting analogy. There is no exception.

An exception to a rule P(x) on X would be some x for which P(x) was false. But the rule P to which you're trying to claim an exception to doesn't apply to the X which you're trying to use it on! It never did!

This is like saying all animals are macroscopic because, at one time, only macroscopic animals were known.

I think that is cleared up by what I said here...

Originally Posted by Universal Mind

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?

The idea that Seth the spirit walked through a wall and then levitated would be claimed to be "where the rule doesn't apply," and I would call it a conveniently made up exception. The laws of physics as we currentlyh accept them disprove Seth. The laws of physics with the exceptions spiritualists believe in do not disprove the existence of Seth.

No because what you said there isn't coherent.

You cannot do something that you're not allowed to do with complex numbers and then claim that that disproves the existence of complex numbers.

At one point, all animals were considered to be macroscopic. This was a rule regardless of if it was written down or not. Are you saying that allowing microscopic animals is an exception to this rule?

Originally Posted by PhilosopherStoned

No because what you said there isn't coherent.

You cannot do something that you're not allowed to do with complex numbers and then claim that that disproves the existence of complex numbers.

I am not saying it disproves the existence of complex numbers according to the currently accepted rule. From the standpoint of acceptance of the old rule, it does. If the old rule is "x can't happen" and a proof shows x happening, acceptance of the old rule involves denouncement of what led to x happening. Then if what led to x happening is proven to be a reality, the denouncement is shown not to be valid. I don't think complex numbers have been proven to have true basis in reality, though tkdyo and khh might possibly change my mind.

Originally Posted by PhilosopherStoned

At one point, all animals were considered to be macroscopic. This was a rule regardless of if it was written down or not. Are you saying that allowing microscopic animals is an exception to this rule?

Yes, they are. They are an exception to the old rule, but they have been proven to exist. The new rule is legitimate.

Originally Posted by Universal Mind

I am not saying it disproves the existence of complex numbers according to the currently accepted rule. From the standpoint of acceptance of the old rule, it does.

If the old rule says that the square root function is multiplicative on every thing in sight the square root of which we might want to take then you are stepping beyond the bounds of mathematics in accepting it for the simple reason that it cannot be proven and trivial counterexamples can be given. Hence nobody is interested in that standpoint because it is not internally consistent and therefore has nothing to do with mathematics.

You may as well be interested in the standpoint that the earth is flat and consequently the solar system doesn't exist.

Why? Because complex numbers exist?

Hey, check this out. I just came up with some things. They are called fantasy numbers. They have reciprocals that equal 2 over themselves, by definition. It turns out that when you multiply them by their reciprocals, the product is not 1. So much for that rule about reciprocals being multiplicative inverses. These blow that rule out of the water. Just multiply the coordinates of a point by one of them, and you will see that fantasy numbers have a basis in reality. The sum of a fantasy number and a real number (Well, they are both technically real. The inventors of them just didn't realize it at first.) is called a complicated number. Draw the complicated plane and see for yourself. You can spin mugs on it.

Originally Posted by Universal Mind

Why? Because complex numbers exist?

I don't know what this is responding to.

Hey, check this out. I just came up with some things. They are called fantasy numbers. They have reciprocals that equal 2 over themselves, by definition. It turns out that when you multiply them by their reciprocals, the product is not 1. So much for that rule about reciprocals being multiplicative inverses. These blow that rule out of the water. Just multiply the coordinates of a point by one of them, and you will see that fantasy numbers have a basis in reality. The sum of a fantasy number and a real number (Well, they are both technically real. The inventors of them just didn't realize it at first.) is called a complicated number. Draw the complicated plane and see for yourself. You can spin mugs on it.

Hmmm. Let's see. So the "numbers" are the image of the map @: R^* -> R^*, x |-> 2/x? Note in particular that that's still real. In fact, Consider any non-zero real number c. Then we want to solve for @x = c, i.e. c = 2/x. Multiply both sides by x and divide by c to get x = 2/c. Hence not only are they a subset of the non-zero reals but they are precisely identical to them.

As for you calling this a reciprocal, that's kind of stupid but whatever. It is not the reciprocal to which the rule that the reciprocal is the multiplicative inverse applies so you've made "exceptions" for nothing. You are using the word reciprocal in one way for your definition and in another way for the result you're attempting to construct an exception for. It's two different words.

Finally, as for the "complicated plane", note that for any real y and any "fantasy number 2/x, we have y + 2/x being real as well and so there is no need for a complicated plane. So you can't use them to represent rotations in a plane. You need two dimensions for that.

This whole thing (like your whole argument) collapses in on itself.

Originally Posted by PhilosopherStoned

I don't know what this is responding to.

Your entire post.

Originally Posted by PhilosopherStoned

Hmmm. Let's see. So the "numbers" are the image of the map @: R^* -> R^*, x |-> 2/x? Note in particular that that's still real. In fact, Consider any non-zero real number c. Then we want to solve for @x = c, i.e. c = 2/x. Multiply both sides by x and divide by c to get x = 2/c. Hence not only are they a subset of the non-zero reals but they are precisely identical to them.

Real in what sense? Not involving the square roots of negative numbers? If that's your definition of "real," I agree, but you are missing the point. So I am being realistic in saying there are actual numbers that have reciprocals which are 2 over them? Name such a number, not just a letter that represents it. They don't really exist.

(back to the satire) If x equals c, then the reciprocal of c is 2/x. However, we use the fantasy unit f. f is defined as the reciprocal of 2/f, that number being f/2. So f = f/2 =/= f/1. The Reflexive Property of Equality is old school and proven to not be universal, as I just showed.

Originally Posted by PhilosopherStoned

I
As for you calling this a reciprocal, that's kind of stupid but whatever. It is not the reciprocal to which the rule that the reciprocal is the multiplicative inverse applies so you've made "exceptions" for nothing. You are using the word reciprocal in one way for your definition and in another way for the result you're attempting to construct an exception for. It's two different words.

(end satire) Uh, it doesn't apply precisely because I have made an exception. Every number's multiplicative inverse is its reciprocal. Also, I am using only one definition of reciprocal.

Originally Posted by PhilosopherStoned

Finally, as for the "complicated plane", note that for any real y and any "fantasy number 2/x, we have y + 2/x being real as well and so there is no need for a complicated plane. So you can't use them to represent rotations in a plane. You need two dimensions for that.

This whole thing (like your whole argument) collapses in on itself.

(back to satire) No, f is the fantasy unit. It is not real. It is fantasy, but it exists. The word "fantasy" was misunderstood by the people who came up with it. They did not know that the Flying Spaghetti Monster, I mean fantasy unit represents an actual part of reality.

OK UM. I see you don't actually care about learning how math works.

You should seriously consider getting a new job. You teaching a math class is like letting a creationist teach a biology class. You clearly have no qualms about just making shit up and pretending that you're right. Frankly the thought of it makes my skin crawl.

Consider being a novelist or something. Peace sucka.

Pseudo-IntellectualHazed, I think you should give up trying to debate. You are just too intellectually dishonest and dodgy to get anywhere in this game. However, I invented the version of you that is intellectually honest and always responsive. His name is "La La Unit," represented by L. He is therefore part of reality, and he is even useful in this satire. If you don't believe me, multiply L by mug.

You're a great guy, though.

By the way, I am a writer. I didn't say I am teaching now.

Originally Posted by Universal Mind

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.

If that's what you meant then I don't see the relevance, considering complex numbers can also be written in wood or ink or whatever. By the way, that matrix you said can be 'made out of wood' is the same thing as i, it squares to the negative of the identity matrix.

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 x-axis and then walk 4 meters in the positive direction parallel to the y-axis, 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.

This mistake seems to suggest to me that you've never even taken a preliminary glance at the mathematics of complex numbers. How can you talk about something if you don't know what it is? If you had taken such a preliminary glance you would know that what you're talking about is norms, and that norms are always real; there is no such thing as a complex norm. The norm of 3 + i is not a complex number, but rather sqrt5, just like the norm of (3, 1) is not a coordinate, but sqrt5; that is the distance of that point on the complex plane from the origin.

3 + i is not used for talking about distance, I've explained this so many times now. That is the wrong physical basis for an abstraction, just as discrete objects is the wrong physical basis for real numbers. This really isn't a subtle point, so I think you must be deliberately ignoring it, which is a shame because it's at the heart of the entire issue.

Yet again, all you have to do is try to apply your own arguments critically to things that you accept, and it falls apart. Would it make sense to say you've reached a point (3, 1) feet from the centre? Does it make sense to say your height is (3, 1)? No. Does that say anything about the existence of real numbers? No, you've just shown that distances is the wrong basis of abstraction for points on the plane.

I already explained at the beginning that the real coordinate plane is the same thing as the complex plane, it just has a multiplication operation defined on it. (0, 1) is i. That's exactly what i means. Saying the complex plane doesn't exist is the same thing as saying the real plane doesn't exist. Either that or you don't believe that rotations of coordinates is 'actual'. What's the justification for that?

Originally Posted by Universal Mind

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.

For simplicity we mostly just write down the amplitude of the signal. Also we use j instead of i, since i is most commonly the current. So Ae^jθsin(ωt) is the complete signal. This breaks down to A(cos(θ) +j∙sin(θ))sin(ωt) as per Eulers formula. In a system you select something (most commonly the input) to have θ = 0.
edit: Wait... I'm not entirely sure that's correct... I guess I'll have to research it to make sure.

Originally Posted by Universal Mind

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.

Ok, consider this circuit diagram

Say you want to know the current and voltage over every element, given that u(t) = A∙sin(ωt). Solving this is significantly easier with complex numbers. Now... I haven't actually done in a while, but I'll post the solution later today if I have time.

Ill try to put this in a more accessible way. Im not sure if you have taken any basic circuit theory, but there are 3 main components. resistors, capacitors and inductors. Lets imagine DC first, since DC is all real. On a graph, of current over time, it would just be a straight line. A resistor only changes the altitude of this line, its still straight and unaltered. A capacitor starts off like a break in the wire, and no current goes through, but over time builds up until it becomes like a wire with minimal resistance, and an inductor does the opposite, starts like a wire, and adds more and more resistance. So capacitors and inductors make exponential graphs.

When you take these elements in to AC current, something interesting happens. A resistor, still does nothing and you just get a sine wave. However, capacitors and inductors create a phase shift in the graph of some theta. This is important, because if you are going to represent the power, current, voltage, w/e as a real number to measure and utilize you have to account for that angle. But where did that angle come from?

We use imaginary numbers to account for this phase shift. Ill let khh find the formulas since he is on the case already, but in layman's terms we can say a signals power is R+iZ where R is the real power (or voltage or what have you) you get at the output of the circuit and Z is the power that is eaten by the capacitor/inductor and thus causes the phase shift. This power lost is important because it effects the phase shift of the signal itself. If you were to combine the real and imaginary numbers by vector addition you would get the true power in the system. It gets even more complex dealing with transformers and such, but its still based on this. We separate them in to real and imaginary because you can only utilize the real power, the imaginary power cannot be used, but it still effects the signal.

edit: I forgot to include something. the other reason we use i is because the phase shift of a single capacitor or inductor is always 90 degrees to a resistor (aka real power). Thus, you cannot simply add the power in a capacitor to the power in a resistor, you must add them using vector addition to get the correct total power in the system. and when dealing with vectors, 90 degrees is equal to i, but i^2 equals 1 so it doesnt keep the total power of the system from being real.

A capacitor can be though of as a resistor with a negative imaginary impedance ("resistance"), and a coil can be though of as a resistor with a positive imaginary impedance as per the equations below.
Z_L = jωL
Z_C = 1/(jωC)
This wikipedia article explains pretty well

This means the R and L can be viewed as a resistor with impedance
Z_L = R + jωL
And assuming G is the resistance, G and C can be viewed as a resistor with impedance
Z_C = 1/((1/G) + (1/(jωC)) = (C²Gω²)/(C²ω² + G²) - j(CG²ω)/(C²ω² + G²)
(The last transformation is a bit tedious, which is why I prefer working with numbers)

The current flowing through L and R is
i_L(t) = u(t)/(Z_L + Z_C)
The voltage over G and C is
u_C(t) = u(t) (Z_C)/(Z_L + Z_C)

From here the voltage over and current through every element can be easily found by Ohm's law
U = Z∙I

To find the complex power through an element, we can use the formula
S = U∙I
Which can be transformed with Ohm's law into
S = I²∙Z
S = U²/Z
Only the real part of the power, P = Re(S), can be utilized.

With input like u(t) = A∙sin(2π∙f∙t), you'll end up with equation that look like this for current, voltage and amplitude
X(t) = (x+yj)A = B∙e^jθ∙A
Which you know is a sine with frequency f and amplitude A∙B, that's θ behind the input.

Wow, this has become a veritable TSUNAMI of mathematical intrigue.

You may continue.

Originally Posted by Xei

If that's what you meant then I don't see the relevance, considering complex numbers can also be written in wood or ink or whatever. By the way, that matrix you said can be 'made out of wood' is the same thing as i, it squares to the negative of the identity matrix.

The difference, and the point I was indirectly illustrating, is that a number is a principle that goes beyond the mere symbol of it but a matrix is just a physical construction. It is a man made construction to which mathematicians apply made up rules. In that way, it is like a board game. Any rule you say it has is just what some other human declared the rule.

Originally Posted by Xei

This mistake seems to suggest to me that you've never even taken a preliminary glance at the mathematics of complex numbers. How can you talk about something if you don't know what it is? If you had taken such a preliminary glance you would know that what you're talking about is norms, and that norms are always real; there is no such thing as a complex norm. The norm of 3 + i is not a complex number, but rather sqrt5, just like the norm of (3, 1) is not a coordinate, but sqrt5; that is the distance of that point on the complex plane from the origin.

3 + i is not used for talking about distance, I've explained this so many times now. That is the wrong physical basis for an abstraction, just as discrete objects is the wrong physical basis for real numbers. This really isn't a subtle point, so I think you must be deliberately ignoring it, which is a shame because it's at the heart of the entire issue.

I reiterated something very relevant because it also applies to my bigger point that you just quoted. A complex plane in the physical realm is nonsensical, period. It is two dimensional in concept. It is a plane in concept. It does not work out to anything sensical when you try to physically create one (not just a representation on paper or a computer, etc.) in the actual universe. You just have a representation that does not fit reality. If you walk the distance from (0, 0) to (0, i), you have not really traveled i units. As you said, that cannot happen because imaginary numbers cannot denote distance. So what you have is just a sheet with a bunch of bullshit written on it. You seem strangely close to agreeing with that. Your point about norms helps illustrate my point about realism.

Originally Posted by Xei

Yet again, all you have to do is try to apply your own arguments critically to things that you accept, and it falls apart. Would it make sense to say you've reached a point (3, 1) feet from the centre? Does it make sense to say your height is (3, 1)? No. Does that say anything about the existence of real numbers? No, you've just shown that distances is the wrong basis of abstraction for points on the plane.

(3, 1) is not a number. It is a point. 3 is a number, and 1 is a number. (3, 1) is an ordered pair of numbers. You can't type your full address and call it a distance either. It is not a number, though it has numbers in it.

Originally Posted by Xei

I already explained at the beginning that the real coordinate plane is the same thing as the complex plane, it just has a multiplication operation defined on it. (0, 1) is i. That's exactly what i means. Saying the complex plane doesn't exist is the same thing as saying the real plane doesn't exist. Either that or you don't believe that rotations of coordinates is 'actual'. What's the justification for that?

Rotations are actual, but if they happen on the complex plane, they have mathematical fiction labels. You could just as well rotate a mug on a Star Wars blanket and call the new position R2D2.


tkdyo, that helps me understand things better, but I have a long way to go before I can fully analyze it. It sounds like perhaps complex numbers are just representations used because they work as solutions to equations but represent measures that could be legitimately denoted by real numbers? You said i is really 90 degrees and that i^2 = 1. It looks like that is not the common imaginary unit and that the meaning behind the symbol is the measure of an angle that can be and is, in other maths and sciences, represented by real numbers such as 90 and pi/2. I want to study it some more because it really is fascinating.


khh, you did not explicitly use actual examples of imaginary numbers being used for convenience to represent physical entities and phenomena that are otherwise represented by real numbers, but from what little I do know about what you discussed (which is more than what it was before I read your posts), it seems that my take on tkdyo's post may apply to yours. Despite my extreme skepticism and cynicism, I actually want to believe that imaginary numbers have a much stronger basis in reality than I have thought.

I was reading about imaginary and complex numbers on some web sites last night, and something really interesting I came across is the claim by a physicist that imaginary numbers seem like fiction to our senses that are limited to four dimensions but that with heightened enough perception, we could perceive ten dimensions and actually "see" the realities of complex numbers in the way that we can look at 2 trees and see the reality of 2. I think that is really worth considering. I have an Art Bell Show fascination view on that right now, but I love the idea and am going to look into it a lot more, as I am with what you and tkdyo have brought up.

Originally Posted by tkdyo

A capacitor starts off like a break in the wire, and no current goes through, but over time builds up until it becomes like a wire with minimal resistance, and an inductor does the opposite, starts like a wire, and adds more and more resistance. So capacitors and inductors make exponential graphs.

I think you've reversed the inductors and capacitors. As I recall, in "enough time" an inductor will act as an ideal conductor and a capacitor will act as an open circuit.

Originally Posted by Universal Mind

tkdyo, that helps me understand things better, but I have a long way to go before I can fully analyze it. It sounds like perhaps complex numbers are just representations used because they work as solutions to equations but represent measures that could be legitimately denoted by real numbers? You said i is really 90 degrees and that i^2 = 1. It looks like that is not the common imaginary unit and that the meaning behind the symbol is the measure of an angle that can be and is, in other maths and sciences, represented by real numbers such as 90 and pi/2. I want to study it some more because it really is fascinating.

I believe what tkdyo said (or meant) is that a signal that's purely imaginary would be 90 degrees behind or in front of (depending on the sign) a signal that's purely real. But i*i = -1, which equals 180 degrees, as
-sin(2π∙f∙t) = sin(2π∙f∙t + π)

Originally Posted by Universal Mind

khh, you did not explicitly use actual examples of imaginary numbers being used for convenience to represent physical entities and phenomena that are otherwise represented by real numbers, but from what little I do know about what you discussed (which is more than what it was before I read your posts), it seems that my take on tkdyo's post may apply to yours. Despite my extreme skepticism and cynicism, I actually want to believe that imaginary numbers have a much stronger basis in reality than I have thought.

I'd say that the representation of inductance and capacitance as the imaginary part of impedance is an explicit use, and (as long as you're not exceeding the operational zone for which the components are designed) the math and the actual measured signals do agree. In fact I don't know how to analyse a system containing both inductors and capacitors without the use of complex numbers.

Also all parts of the complex power S = |S|e^jφ = P + jQ for a component can be measured directly from a signal if you have sinus input. There is a wikipedia article on this too, but it's not so good. This is a particularily good illustration I believe, because it can be directly observed that the real part of the power, P, is usable by the component while the imaginary part of the power, Q, is not.

I guess you could argue that while this is a convenient way of writing it, the same could be achieved using xy- and polar-coordinates. My guess is that it would be possible (I don't know for certain, but it would make sense for what I've shown at least), but in that case I'd ask what makes that a more "real" representation of the phase-shift of a sine than imaginary numbers.

mm, Im sorry, I did have a couple typos in there, but khh cleaned them up rather nicely, thank you khh.

UM, that sounds very interesting about the multiple dimensions and definitely sounds like its worth the time to look up. I remember we did use i in a basic relativity class I took, but Im in no position to try and teach anything about that as it was only one class lol.