Do imaginary numbers have more significance in the universe than just a simple mathematical convenience?
The square root of minus one, and hyperbolic functions in general ,seem to be at the (mathematical) heart of virtually all modern physics .
Is it simply a "thought up", mathematical convenience , or does it actually point to something "real" and fundamental about the universe? .
Edit : What I'm getting at here is that without imaginary numbers and hyperbolic functions -- virtually all modern theories in physics and cosmology would be impossible . That is , we had to have invented this branch of mathematics, just to make the universe understandable to us.
Is it therefore just a coincidence that this form of mathematics "works" when looking at what makes the universe tick, or is it because its language is particularly "in tune" with it for some reason?
- Satan ClawsLv 78 years agoFavourite answer
<QUOTE>Do imaginary numbers have more significance in the universe than just a simple mathematical convenience?</QUOTE>
Let's try to establish first what is the importance of real numbers in the universe before we try to determine how to evaluate the importance of complex numbers in the universe.
<QUOTE>What I'm getting at here is that without imaginary numbers and hyperbolic functions -- virtually all modern theories in physics and cosmology would be impossible .</QUOTE>
Correct. You may feel satisfied to know that you're not the first one to wonder about this. Have a look at this when you find the time:
You may also want to read a counter-argument:
<QUOTE>Is it simply a "thought up", mathematical convenience , or does it actually point to something "real" and fundamental about the universe? </QUOTE>
Honestly, I don't know how to answer that question.
You can count the number of people as 1, 2, 3, .... Does it make sense that there are no 2.08 persons? What "law" prevents that? But that at the same time you can say that a certain person has 2.08 times the average weight of a certain sample of persons? And how exactly do you evaluate whether that has some "meaning" -- and how do you describe something as "having a meaning" in this respect?
My point is: you have to take extra care to define exactly what it is you're trying to find out, before attempting to establish an answer. I'm not sure how exactly you'd describe something as being significant. Is it that if it were another way, the world would behave differently? That could be a criterion -- and then you could start playing around with a "mock universe", writing laws for it and see how it'd look like. http://www.youtube.com/watch?v=b240PGCMwV0
Unfortunately, I can't see how we might be close to try to explore that question. I don't have such a high regard of Philosophy; I consider it mostly a sort of mental masturbation, because you're going around in circles. It's so too easy to fall into that trap, as illustrated in this question in a public meeting: http://www.youtube.com/watch?v=CIlNWybVx5c
#t=1m02s It's OK to ask them, but you have to be EXTREMELY careful about wrapping yourself into it and giving yourself the impression of discovering something important.
<QUOTE> Is it therefore just a coincidence that this form of mathematics "works" when looking at what makes the universe tick, or is it because its language is particularly "in tune" with it for some reason?</QUOTE>
In my opinion, we don't yet know enough to answer that question.
And you know what? Personally, I *LOVE* that. It means that we'll have to find out MORE things along the way before we might get close enough to be able to answer that question.
You can easily come up with other stuff; not just with natural numbers, real numbers or complex numbers. For example, Noether's theorem tells you that for each symmetry of a system there ought to be a conserved quantity of motion, and vice-versa. For example, in periodic systems there's usually an associated energy conservation law; for translational symmetry there's an associated linear momentum conservation; for a rotational symmetry there's an associated angular momentum conservation; and so on. WHY the Universe would be like that, I don't know. And that's where Wilson's 1960 paper comes in.
So it's not just the complex numbers. There's a whole lot of stuff that we don't know yet, or at least that I don't (personally) know how much we know.
You can even pick on complex numbers (which have one real part and one imaginary part) and whip up some extension of it. http://en.wikipedia.org/wiki/Multicomplex_number I don't know how you'd evaluate the "significance" of that thing you make up, and how you can apply it to some Physics and get new knowledge or tools out of it; after all, it's something you invented and is useful.
- NinaLv 45 years ago
(1) They do exist, as much as any real number does. The fact, however, that the basic unit is the solution to x^2 + 1 = 0 is mostly just an algebraic coincidence to the reason why they're used a lot in proof or real life (I'll get to that soon). (2) They do exist. (3) They do exist. [get the point, yet?] Imaginary numbers, or more generally Complex numbers, do not algebraically have to be written by using a letter i (or even j, to cover those electrical engineers). We could write i as a special coordinate pair (0,1). If we combine it with the following rules: (a,b) + (c,d) = (a+c, b+d) (a,b) * (c,d) = (ac - bd, ad + bc) Then in fact these are the complex numbers. No 'i' necessary. I didn't even have to talk about some root of a polynomial that never intersects the x-axis. You may say that that's a weird rule for multiplication, but actually it turns out it's quite typical of objects which are periodic in nature (such as anything that orbits another object). Now, I ask you: if it describes reality, how can we responsibly avoid using them in proof? Edit: By the way, if you really don't like the coordinate definition I could give an algebraically equivalent definition using 2x2 matrices of _real numbers_. What my main point is, is that we can make "Complex numbers" which have no mention of the infamous 'i'. Unfortunately, if you don't believe in any of the constructions of the real numbers, I can't help you there. That's a philosophical stumbling block.
- Larry454Lv 78 years ago
Well, I am trying to figure out the difference between "significance in the universe" and "just a simple mathematical convenience." To me, a mathematical convenience that reveals something about how the universe works is pretty significant.
One example that certainly has some significance is in the area of dynamics and vibration analysis. The real and the imaginary portions of an eigenvalue solution to a differential equation describing a dynamic system provide two separate and critical pieces of information:
1. Real parts determine stability
2. Imaginary parts indicate resonance conditions
These are pretty important when you are designing structures like aircraft wings, turbine blades, or flight control surfaces, just to name a few.
In this sense, the complex eigenvalues are indeed in tune with the physical process known as vibration. Perhaps this is because vibration is a sinusoidal response, so describing it as a straight linear motion along a number line is not as accurate as describing it as a circular motion along two axes, one real and the other imaginary - in a complex number system. This can be projected into any axis, but the sinusoidal nature implies that the source is in a circular fashion - thereby driving the description to at least two axes.
- Anonymous7 years ago
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- 8 years ago
Nothing in reality has an imaginary value. Measurable quantities are measurable as real numbers (along with a unit of measure to gives it real world meaning). Its possible to compute an imaginary value from real things, but that imaginary value represents a real measure of something. Pretend its real and change the unit of measure, and you have a real number again.
The fact is, no number tangibly exists. They are ALL abstractions. Some, in a practical sense, are meant to describe real things. Others, are meant to describe other abstract concepts and relationships. All of it is man-made. Numbers didnt lie around the cave floor waiting to picked up. They were invented, imagined, and have no tangibleness.
Here is an interesting thought for you to consider.
You concern yourself with imaginary numbers too much. What about real numbers?
Are negative numbers real? You can have a positive number of apples, but you cannot have a negative number of apples. You can measure a positive distance in one direction, but negative distances are impossible. Negative velocities are impossible. Rather, when we see a negative sign in distances and velocities, we change the direction of the vector. A negative distance is really just a positive distance in the opposite direction.
What about non-integers, for that matter? Anything that exists in reality can be divided into discrete units. The universe is not a continuous thing, it is a discrete thing. Plank units. Everything can be measured in whole units of something else. You can have 1 whole piece of a half-apple, but not 1/2 of an apple. You cant have half an atom. You cant have half a life. Get right down to it, on a fundamental level, we can count in whole integer discrete units. We can count atoms. Plank distances. What is a half of a mol of water? 3.01 x 10²³ atoms of water. A "dozen" is a word, a convenience of language. We can SAY we have a half-dozen. But what we really mean is we have 6. You can say you have a half of an apple but what you really mean is you have a half-apple. To say you have a half of an apple is to talk about the other half that isnt even there and to relate what is to what isnt. It makes little sense. Each half-apple is its own discrete unit and I have two of them. If I said I had two half-apples, are you immediately going to assume that the two halves came from the same whole apple? It is semantics.
Non-integers, irrational numbers, etc. and imaginary numbers, are all just phenomenon of pure, abstract mathematics. Math CAN be used to describe the real world, but it doesnt have to. And, indeed, the real world can only be described approximately, not exactly. So the reality is that it doesnt matter if integers or imaginary number or rational numbers or numbers in general are real or not. What matters is how fruitful our understanding of the world is on account of our descriptions of it.
- NucleusLv 68 years ago
Imaginary numbers to you are real numbers to some.
A reputed university must unconditionally sponsor his further research, if his knowledge of 'Imaginary numbers' is to be revealed, in order to clarify the confusions in the likes of your question addressed.
The above offer would be a win-win deal for knowledge. Is it a big ask?
- Doc MarzLv 68 years ago
Even though imaginary numbers aren't real numbers, their design is to add a real dimension. This is what makes complex numbers work in explaining real effects in the universe. So to answer your question in simple terms, it does point to something real even though it was devised as a mathematical convenience.
- John WLv 78 years ago
Complex numbers are often used to express multiple dimensions such as vectors or the phases of regular cycles. They certainly have a significance, most likely with processes, but not in the unobtanium dilithium crystal sense that you are imagining.
- Anonymous8 years ago
Read up on quantum physics.