# Questions about visual/physical representations of imaginary numbers.

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I'm familiar with Euler's identity, but in reality, how do you actually raise something to the power of an imaginary number? Ever since I first encountered imaginary numbers I've been searching for some way to make physical sense out of them. Where the hell does x^2+10 cross the x axis? I don't see any "i" values on the y-axis. wtf? I thought of some way that might work, I'm too tired right now to try it myself right now, maybe someone else can play around with it.

Take an imaginary plane, where coordinates are (iy, ix). Then solve for the equation iy= either (ix)^2+2 or x^2+2, maybe it will make more visual sense, but not too much because where would that situation even be applicable?

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You're right, and you shouldn't. Imaginary and complex numbers actually hold crucial application in the real world. If you haven't been formally introduced to the topic yet, then it certainly can sound confusing and senseless. Before even thinking about Euler's identity, I would suggest starting a thread on imaginary and complex numbers (as not to stray the discussion here).

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$\displaystyle\int_{a}^{b} f(t)\;dt=F(b)-F(a)$

The fundamental theorem of calculus, which relates differentiation and definite integration.

$\Gamma(z)=\displaystyle\int_{0}^{\infty} t^{z-1} e^{-t}\;dt$

The gamma function, though beyond my understanding as a student, seems quite beautiful by its definition via an improper integral. Shifting its argument equates it to the factorial, and both are ubiquitous in math.

$e^{i\tau}=1$

An identity from Euler's formula. Here, $\tau$ denotes the constant ratio of a Euclidean circle's circumference to its radius (isolated from $\pi$ to emphasize its geometric definition). Values of $\cos(x)+i\sin(x)$ parametrize a unit circle in the complex plane. $\tau$, being the circumference-radius ratio, is equivalent to a full circle as an angular measure. Thus, it helps to show the meaning behind complex exponentials, where this ratio "completes a full turn" in the complex unit circle.

$e=\displaystyle\lim_{x\to\pm\infty}\dfrac{^2{(x+1)}}{^2{x}}-\dfrac{^2{x}}{^2{(x-1)}}$

Just think this one's really cool. It's a symmetric limit for a symmetric complex-valued function, that gives $e$, and involves tetration! Also, it's simply attractive from an aesthetic point of view. Just look how elegant it is.

No I've defanitely been introduced to it and I know it's important, especially for 3D modeling when you need to use quaternian numbers for shapes and vectors like in video games. I guess another thing I use to find confusing is how the square root of i contained i. I can't remember what it is, its something lke (2^(1/2)i+i)(2^(1/2)), but then I started thinking about i like the number "1", I mean the square root of 1 is itself, its 1, so the square root of -1 should be some kind of identity of something like "1" but I guess in a different form, and you can model "4" as being a composite of four ones, so I kind of like to think of "i" as some kind of base number identity that can composite many others like the number "1" can, but I still don't understand it's physical meaning. X^2+1 never actually crosses the x axis so how does it do it at some "i" coordinate?

Edited by SamBridge
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x^2+1 never actually crosses the x axis so how does it do it at some "i" coordinate?

The problem is that you're thinking of the situation just in terms of an ordinary "Cartesian co-ordinate system". Of course, the graph of the function $f(x) = x^2 + 1$ doesn't ever touch the y-axis - however this function does have zeroes in the complex plane (they could be represented on an Argand diagram), and in this case the zeroes are $\pm i$.

It is good to think of complex numbers as a way to "extend" the roots of polynomial equations - so that they all have roots, in the complex plane or real plane.

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The problem is that you're thinking of the situation just in terms of an ordinary "Cartesian co-ordinate system". Of course, the graph of the function $f(x) = x^2 + 1$ doesn't ever touch the y-axis - however this function does have zeroes in the complex plane (they could be represented on an Argand diagram), and in this case the zeroes are $\pm i$.

It is good to think of complex numbers as a way to "extend" the roots of polynomial equations - so that they all have roots, in the complex plane or real plane.

The problem isn't that it's on a Cartesian plane because I am completely familiar with not only a complex plane but also quaternian numbers in 3-D mechanics, what the problem is, is that it isn't a number you can count to. You can count to 1, the square root of one, 0, negative one, negative two, ect, but not the square roots of negative numbers. I'm completely familiar with complex and imaginary numbers that's not at all what I'm asking about what I'm asking about is more fundamental like "what causes distance?". What I'm asking about is what the value "i" really is beyond simple analysis and why it has specific properties, there's a whole section of mathematics devoted to investigating why numbers work the way they work and it's called number theory, but unfortunately I can't find what I'm looking for. What I want to know is, what would it look like if I was holding "i" apples?

Edited by SamBridge
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What I'm asking about is what the value "i" really is beyond simple analysis and why it has specific properties, there's a whole section of mathematics devoted to investigating why numbers work the way they work and it's called number theory, but unfortunately I can't find what I'm looking for. What I want to know is, what would it look like if I was holding "i" apples?

You seem to be thinking of this in the wrong way - the last question you asked is rather meaningless as it implies that i is a real, measurable number and of course i is purely imaginary. In physics a quantity that you want to measure damn well better be real (i.e. not complex/imaginary) otherwise you simply will not be able to measure it. For example in quantum mechanics a lot of the time the wavefunction ψ will be a complex equation and so it would be erroneous to think of it as an expression representing a quantity of a mechanical wave which can be measured.

As to your first question of what i actually is - well, in the most simple and elegant way, i is just the square root of negative 1; it is an imaginary unit, however the arithmetic behind it is purely logical and works perfectly as well as it having a plethora of applications to physics in particular (I am a physics student and so I know and understand how important complex analysis is to so many fields of my subject).

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In physics a quantity that you want to measure damn well better be real (i.e. not complex/imaginary) otherwise you simply will not be able to measure it.

I disagree. Impedance is quite a real phenomena and measured and described using complex variables. Potential flows in fluid mechanics are quite real and well described using a complex variable formula.

What I want to know is, what would it look like if I was holding "i" apples?

This, quite simply, is a misuse of the language of mathematics. You might as well ask what it looks like if you were holding 'lightbulb' apples, or if you were holding 'golf club' apples. What I mean is that 'i' is not a descriptor of apples any more than 'lightbulb' or 'golf club' or many other words.

i's properties derive from its definition as the square root of negative one. There are things that can be described having a quantity of i, again impedance or potential flows, but you can't describe apples the same way. It also doesn't make sense to describe apples using words that describe impedance or potential flows, either. E.g. you don't call apples 'radio frequency' or 'inviscid'. Nor do you call something a 'Granny Smith impedance' or a 'golden delicious flow'.

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I disagree. Impedance is quite a real phenomena and measured and described using complex variables. Potential flows in fluid mechanics are quite real and well described using a complex variable formula.

This, quite simply, is a misuse of the language of mathematics. You might as well ask what it looks like if you were holding 'lightbulb' apples, or if you were holding 'golf club' apples. What I mean is that 'i' is not a descriptor of apples any more than 'lightbulb' or 'golf club' or many other words.

i's properties derive from its definition as the square root of negative one. There are things that can be described having a quantity of i, again impedance or potential flows, but you can't describe apples the same way. It also doesn't make sense to describe apples using words that describe impedance or potential flows, either. E.g. you don't call apples 'radio frequency' or 'inviscid'. Nor do you call something a 'Granny Smith impedance' or a 'golden delicious flow'.

You can say the same about "infinity" but you still have an idea of what it looks like, infinity isn't an actual number you can count to but if I had infinite apples it would be a never ending chain that would stretch as far as I can see, I still know what it looks like. The square root of 1 is 1, so I would only have to imagine that "i" has some kind of similar inherent property like that, after all, the square root of negative 4 is i times the square root of 4 which is 2i, but the positive square root of 4 is just 2, there's some kind of hidden reflexive property of i that's similar to taking the square root of 1, I'm starting to think "i" apples would look in some way like 1 apple.

Sure there's axioms of math, but there's still proof that 1+1=2, there has to be something about the proof that i^2=-1 that shows what it really is, and I think it's a property that makes it similar in some way to 1.

Edited by SamBridge
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there has to be something about the proof that i^2=-1 that shows what it really is, and I think it's a property that makes it similar in some way to 1.

If you're going to insist on that, I think it will lead towards incorrect conclusions. Because i is not 1. The only similarity is that i is 1 unit into the complex plane, like 1 is 1 unit into the real plane. Apart from that, the similarity ends.

What you're saying by analogy is that if you have a function of both x and y that 'x is similar in some way to y'. If you have f(x,y), or some coordinate point (x,y), and y=1, you can't really say anything about x. Just like i doesn't really say anything about 1 or any other value on the real plane.

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You can say the same about "infinity" but you still have an idea of what it looks like, infinity isn't an actual number you can count to but if I had infinite apples it would be a never ending chain that would stretch as far as I can see, I still know what it looks like.
You don't really know what infinity looks like - you think you can comprehend its size by applying an object (apples) to it, however (as you said) it is not a real number and one cannot fathom what infinity actually looks like.
...I'm starting to think "i" apples would look in some way like 1 apple.
No, an imaginary unit i of apples doesn't "look" anything like 1 apple as it's just pretty meaningless to label objects such as apples with a non-real quantity. Trying to visualise complex numbers through the use of "apples" is not the correct way to go.
Sure there's axioms of math, but there's still proof that 1+1=2, there has to be something about the proof that i^2=-1 that shows what it really is, and I think it's a property that makes it similar in some way to 1.
This statement requires mathematical justification - simply stating that "you think" i has a property which makes it similar to 1, doesn't really mean anything.

I disagree. Impedance is quite a real phenomena and measured and described using complex variables. Potential flows in fluid mechanics are quite real and well described using a complex variable formula.
Of course one can use complex variables to describe systems (such as quantum systems) - however, as far as I know, when measuring something in a physics experiment the actual quantity (or quantities more likely) that you are measuring has (have) to be real!
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You don't really know what infinity looks like - you think you can comprehend its size by applying an object (apples) to it, however (as you said) it is not a real number and one cannot fathom what infinity actually looks like.

Regardless of whatever you say about it I still have a crystal clear idea of what it looks like, the universe is theoretically infinite in size, we can view the universe. You can view a chain that is infinitely long you can't just view an infinite number of components of it simultaneously. This is where your logic regarding the physical representation of something breaks down. "Beauty" isn't a mathematical number we can count to on a number line, like "infinity" it's more of a concept, but we can still see something beautiful.

Trying to visualise complex numbers through the use of "apples" is not the correct way to go. This statement requires mathematical justification - simply stating that "you think" i has a property which makes it similar to 1, doesn't really mean anything.

But if you have no idea what they look like then you can't say what they don't look like. You have no evidence that a physical quantity of "i" couldn't be similar to the square root of 1. There's logical justification to support that it could, because the square root of 1 is 1, the square root of negative 1 is something also to do with 1, which is why the coefficient in an imaginary number retains the mathematical properties of it's real coefficient, such as the square root of negative 4 being equal to 2i. The square root of 100 is 10* the square root of 1, just like the square root of -100 is 10*i. And then 10 is still the square root of 100 times i^4. There's no "right or wrong" way to "go" at this point, this is more like philosophy, I'm not trying to look smart like you I'm trying to figure out what "i" really is and its physical representation in reality, how mathematics fits into reality is something that's been debated for hundreds of years.. If you're not going to do that you don't have a reason to respond to my posts in this topic. I am well versed in the knowledge of the mathematics of imaginary and real numbers, but what reality is and numbers actually mean to it is something different, and that's because math uses axioms, it is not reality itself. I'm pretty sure it can't be proven that i=1, or that it equals any real number we are familiar with, that's why it's philosophy how it physically fits into the real world which we like to think of as comprised of real numbers, like real numbers of atoms bonded at fixed angles moving at real speeds, ect.

Which reminds me: can this be split off into the philosophy section?

(Also just an off topic sidenote: I'm a senior member when I only have 35 posts? wtf? I've only been here like less than a week.)

Edited by SamBridge
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Which reminds me: can this be split off into the philosophy section?

Definitely second this, and for more than a few reasons.

...that's why it's philosophy how it physically fits into the real world...

No offense, but this explains a lot. I thought your inquiry was like most posted here... practical and specific math questions. I wasn't able to discern that it actually dealt with a subject that, admittedly, touches on the philosophy of mathematics, philosophy in general, and even physics/cosmology.

You know we've gone beyond discrete counting integers and real numbers with the complex plane. And we also have quaternions. What else do we have?... the extended reals, the surreal number system, the hyperreal numbers, surcomplex numbers, tessarines, biquaternions, coquaternions, and the octonions.

Had to look most of these up, and I'm sure abstract algebra can deal us more if we're in for it. But most of these structures cannot be reconcilable, where some are constructed from contradictory definitions. Still, almost of all of them have had their application in physical reality one way or another. So I think it's just a matter of logical consistency and the ability to accurately describe/predict physical processes.

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Definitely second this, and for more than a few reasons.

No offense, but this explains a lot. I thought your inquiry was like most posted here... practical and specific math questions. I wasn't able to discern that it actually dealt with a subject that, admittedly, touches on the philosophy of mathematics, philosophy in general, and even physics/cosmology.

You know we've gone beyond discrete counting integers and real numbers with the complex plane. And we also have quaternions. What else do we have?... the extended reals, the surreal number system, the hyperreal numbers, surcomplex numbers, tessarines, biquaternions, coquaternions, and the octonions.

Had to look most of these up, and I'm sure abstract algebra can deal us more if we're in for it. But most of these structures cannot be reconcilable, where some are constructed from contradictory definitions. Still, almost of all of them have had their application in physical reality one way or another. So I think it's just a matter of logical consistency and the ability to accurately describe/predict physical processes.

Of course they all have their uses, but mathematics merely describes our observed patterns of reality, mathematics itself is not a reduced form of logic, it is it's own composite of systems which have their own axioms.

Looking at "i" a component of a vector still doesn't help nail down the meaning, the only thing I can seem to come up with is that it's related to the number "1" in it's properties. Perhaps we can look into it more by analyzing division. A negative number divided by "i" yields that number time's the coefficient of "-i". Similarly, a positive number divided by the square root of "1" yields that number times the coefficient of 1, and a real number divided by "i" yields that same number accept multiplied by the coefficient -i. I'm seeing a lot of properties that remind me of the number 1, but that could be because of the obscurity of the actual value of "i".

Edited by SamBridge
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Of course they all have their uses, but mathematics merely describes our observed patterns of reality, mathematics itself is not a reduced form of logic, it is it's own composite of systems which have their own axioms.

I think that mathematics is our quantitative model for describing the universe. The complex numbers are an algebraic construction based off the standard axiological system mathematicians have developed. If you say that mathematics is its own logical abstraction, a model so-to-think, then why are trying to look for a physical explanation?

I'm seeing a lot of properties that remind me of the number 1

Well, this seems kind of subjective. One can notice a similar relationship between certain trig functions, e.g. the tangent and cotangent functions. They can have very similar identities with all but a plus or a minus distinguishing the two. They're closely related by definition, so this shouldn't be surprising.

But a strong case can also be made that $i$ and $1$ are quite different. Similarity in this sense is completely subjective, or rather, a strictly relative description, like hot or cold. The only thing we can definitely say is whether or not they are equal, which they're not.

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I think that mathematics is our quantitative model for describing the universe. The complex numbers are an algebraic construction based off the standard axiological system mathematicians have developed. If you say that mathematics is its own logical abstraction, a model so-to-think, then why are trying to look for a physical explanation?

No I'm not saying math isn't logic because that's my own opinion, it was actually proven true, There's statements which can't be put into terms of math and even arithmetic theorems which can't be proven with arithmetic and Kurt Godel's statements and research on it.

But otherwise, math is what we use to descrie patterns that we observe, and it's one of the most highly developed methods of exploring patterns which is why I use it.

But a strong case can also be made that $i$ and $1$ are quite different. Similarity in this sense is completely subjective, or rather, a strictly relative description, like hot or cold. The only thing we can definitely say is whether or not they are equal, which they're not.

Yeah I know it's subjective, but there's no other thing to really put it into sensible terms or to describe it's physical meaning that I can think of, so unless you can come up with something better I'll just have to stick with that.

Edited by SamBridge
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but there's no other thing to really put it into sensible terms or to describe it's physical meaning that I can think of

It is a rotation. It is immensely useful for describing things that vary with time sinusoidally. Which has a nice representation as rotations about the origin on a complex plane.

Again, this has direct application in describing the very real phenomena of phase in circuits. Such as alternating current. And impedance.

Really, anything that time varies on a periodic cycle is a candidate for a complex variable representation. And then, if you need to do operations like add or subtract them, etc., the representation make that particularly easy.

Lastly, I guess I think it is funny that you apparently can't find a single thing that is sensible about them... have you done any independent reading about them? There are many, many texts on complex variables and their uses in physics. For example, Brown's Complex Variables & Application (McGraw Hill, 2008). I personally enjoyed Needham's Visual Complex Analysis (OUP, 1999).

Rather than just naively and blithely resigning yourself to accepting that i is something similar to 1 (which really it isn't), why don't you look through some of the texts on the subject and see if you can't improve that opinion of them?

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It is a rotation. It is immensely useful for describing things that vary with time sinusoidally. Which has a nice representation as rotations about the origin on a complex plane.

Again, this has direct application in describing the very real phenomena of phase in circuits. Such as alternating current. And impedance.

Really, anything that time varies on a periodic cycle is a candidate for a complex variable representation. And then, if you need to do operations like add or subtract them, etc., the representation make that particularly easy.

Lastly, I guess I think it is funny that you apparently can't find a single thing that is sensible about them... have you done any independent reading about them? There are many, many texts on complex variables and their uses in physics. For example, Brown's Complex Variables & Application (McGraw Hill, 2008). I personally enjoyed Needham's Visual Complex Analysis (OUP, 1999).

Rather than just naively and blithely resigning yourself to accepting that i is something similar to 1 (which really it isn't), why don't you look through some of the texts on the subject and see if you can't improve that opinion of them?

As I've said multiple times I'm knowledgeable about complex numbers and I had found in my research prehand that complex numbers were used in electrical circuits, but all your blabbering still doesn't answer what "i" apples looks like. I can picture what a beautiful apple looks like, I can picture part of an infinite number of apples even though neither of those things are things that I can count to on a number line, so what's so complicated about imaginary apples? It isn't sensible, its just useful, I know more about what 5-dimensional objects look like than what imaginary numbers look like, I know how to use them but I have no idea why they work like that, I would think as a math expert that you would be able to use some fancy number theory proof at least as some evidence of them physically correlating to rotation. I've seen them in trigonometric mathematics before, and sine waves can represent circular motion or harmonic oscillation and a variety of any other period phenomena, they can also have imaginary/complex solutions on both Cartesian and polar graphs, but I still don't see the "i" of the tide or of sound waves or ect. Slapping a unit circle on a complex plane doesn't mean "i" represents physical rotation either.

I'm not saying it's proven true or anything, but "i" seems like it has a lot of similarities to the number one, you can even raise it to the power of 4 and have it equal 1, and you can raise 1 to the power of 4 and it will equal 1, "i" is derived from the square root of 1, except that specific "1" happens to be on the opposite side of the number line, and that's it. I've been doing research in my free-time for months even before I came to this website, this is the closest I got, nothing you are saying is anything new to me, although as you mentioned rotation did almost make sense, but when you deal with describing locations on a complex plane with cosx+isiny, the mathematics can get complicated quickly. A rotation really really almost made sense, but there's still that problem of using an imaginary axis in the first place, it's like using a word in its definition or saying x=x+1.

Edited by SamBridge
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but all your blabbering still doesn't answer what "i" apples looks like.

You might as well complain that I also didn't tell you what an honorable apple looks like. Or what a charismatic apple looks like. Just because you can put two words next to each other, doesn't mean it has to make sense. There is no such thing as an honorable apple; and, there is no such thing a i apples. It isn't a meaningful question, so it doesn't have an answer.

i can be thought of as a rotation because of the way points are plotted on the complex plane. Rather than me retyping a lot of stuff, why don't you take a peek at http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/ ?

Or, as I said above, check out a text on them. In particular since you seem so fixated on visualizing it... check out that text I recommended on visual complex analysis.

Also, can I ask you to tone the apparent attitude down a little? I would like to think that your use of the words like "blabbering" and "fancy math expert such as yourself" aren't meant to be aggressive, but it sure sounds that way. If you don't care for my answers or have a problem with me, you can PM me, report me to a mod, or simply not reply.

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There is an honorable apple and a smart apple and an industrious apple, I can picture things in my head that fit all those criteria. Those terms are subjective of course but I easily can, and in fact I was trying to use imaginary numbers to investigate the art of juxtapoz and the meaning beyond the abstract seeming art.

Rotation almost makes sense, but I still don't see what "i" apples looks like from it. Before I was at that "i" is similar to one which there is even more evidence for looking at the very complex plane in the website you posted, I would have liked to think of imaginary numbers as extrapolations of extra dimensions, a whole world that is intangible but that is responsible for shaping things, this is where I ran into rotation in my research. However, it's still doesn't make sense that you can just insert an imaginary axis into any random place of reality and get negative 1. Lets say I have a straight stick. If I decide to rotate from my relative position of which I measure it being at 90 degrees from 2pi, and I rotate it 180 degrees, where's the "i"?

Edited by SamBridge
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There is an honorable apple and a smart apple and an industrious apple, I can picture things in my head that fit all those criteria.

Ok, well, I guess you have a greater imagination than I. I am picturing an honorable apple that works for The Innocence Project going around saving people who have been wrongfully convicted... and then I think it is some reality TV show pitch that is going to air right after Here Comes Honey Boo Boo. It should have some corny pun for a title... Like 'The Good Bite' or something...

Anywho... I guess I can't really help any more here.

Edited by Bignose
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Rotation almost makes sense, but I still don't see what "i" apples looks like from it

Okay. This is my personal view, and it holds no objective merit. But to me, it makes sense. I simply think that mathematics is a quantitative model when applied to physical phenomena. It does not physically define, epitomize, or fundamentally cause whatever it is it's describing. But it's quantitative and it's exact, and it's almost always based off some intuitive structure.

Using the natural numbers to count ordinary objects makes sense, and it works. There is an established and intuitive relationship between the two. However, using non-integral reals and negative numbers aren't practical. Even less for imaginary or complex numbers. This isn't a case where you can simply extend the math to any generalization you want. If we could, then why why stop at the complex numbers? How many apples is "j" or "k" apples in the numerous hypercomplex algebras?

So in my view of the subject, "i apples" is meaningless. 1) You can't just extend the application to some arbitrary generalization which is otherwise "pure" in nature. 2) Math doesn't physically epitomize what it describes. It models.

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!

Moderator Note

I've split this conversation from another thread, found here. Please try to keep this one on topic.

SamBridge, you need to keep your attitude in check. Being impolite doesn't help the discussion.

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Ok, well, I guess you have a greater imagination than I.

Good, I should, I've spend a lot of time exercising it to understand visual extra-dimensional shapes and abstract art as well as art that just doesn't make visual sense like juxtapoz.

Anywho... I guess I can't really help any more here.

You don't really need to state that, it is already apparent you prefer cold mathematics to philosophy.

Okay. This is my personal view, and it holds no objective merit. But to me, it makes sense. I simply think that mathematics is a quantitative model when applied to physical phenomena. It does not physically define, epitomize, or fundamentally cause whatever it is it's describing. But it's quantitative and it's exact, and it's almost always based off some intuitive structure.

But it's quantitative and exact by it's own standards of defining quantitative and exact using it's own axoims. I'm not saying it's a cause, but if mathematics was itself a reduced form of logic, then we wouldn't need all these various mathematics systems, we should just be able to use 1 logic to deduce what is logically happening, but we can't, so we know math isn't logic, we assign whatever meaning we want to various symbols.

So in my view of the subject, "i apples" is meaningless. 1) You can't just extend the application to some arbitrary generalization which is otherwise "pure" in nature. 2) Math doesn't physically epitomize what it describes. It models.

But as according to you math "models" reality, "i" should model something in reality.

Edited by SamBridge
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Good, I should, I've spend a lot of time exercising it to understand visual extra-dimensional shapes and abstract art as well as art that just doesn't make visual sense like juxtapoz.

OK. I gotta ask, then, oh imaginative one: what is an honorable apple?

it is already apparent you prefer cold mathematics to philosophy.

I prefer whatever gives me good predictions. So, whatever mathematical tools I need in order to make good predictions that agree with reality is what I prefer.

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OK. I gotta ask, then, oh imaginative one: what is an honorable apple?

It can't be descried using only English words, you would have to see it for yourself. The best way I can describe it, an honorable apple looks like it has a slightly scarred and randomly rugged surface and patches of discoloration, it looks as though it has been through a lot, and accomplished what it was meant to, that's about all I can put into terms of words.

I prefer whatever gives me good predictions. So, whatever mathematical tools I need in order to make good predictions that agree with reality is what I prefer.

Predict what "i" apples looks like.

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What about Thinking of i as rotation? Not a physical object, but a physical transform.

On the imaginary plane 1*i=1i or a 90 degree transform,1i*i=-1 another 90,-1*i=-1i and one more multiply and you're back to 1 for a full 360.

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