UPDATE: My answer to this problem. Which, at the end of the day, isn't really an answer at all.

As part of our explorations of "why this math" for aspects of physics, I pose an obvious and seemingly simple question: which type of numbers is required to describe the world around us?

We use different types of numbers in mathematics. For example, we have the integers, the rational numbers, the real numbers, and the complex numbers. Furthermore, there is a distinct hierarchy among number types - the hierarchy for the four types above is:

\(integers \subset rational ~numbers \subset real ~numbers \subset complex~ numbers\).

There are additional number types such as the hypercomplex numbers (which are fascinating if you've never run across them). Physics, however, should only need to use a certain type of number to describe the universe, and which type should be dictated by nature. Which number type is **required** for our current physical theories, and why?

A couple comments for this question:

Arguing that one number type is simply more convenient to describe phenomena is not a valid argument, as we are looking for what is required.

On its surface this seems like a very straightforward question, but I warn everyone that it is actually rather subtle.

There are different answers to this question, depending on what you believe about measurement and/or the fundamental structure of our world. Hence I hope everyone will provide a number of interesting viewpoints.

Finally, as always - only if enough interest and discussion is shown by the members of our community will I post my own answer. I and your peers on Brilliant want to hear everyone's thoughts!

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## Comments

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TopNewestI like the book of Paul J. Nahin, "An Imaginary Tale: The Story of i" (you can see on Amazon) :)

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First of all focussiong on what numbers are? guyz they are nothing except something used as a magnitude to describe teh physical quantities if the universe.ANd according to Model delpendent realism it occurs in our brain.Numbers are only in our brain and are create dfrom logic and i dont know the numbers which will decribe the universe but i know is the last and frist number of this number system- the numbers are -0 and Infinity...and guyzz thay are same

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I think that complex numbers are necessary, because they make a complete system. This means all equations can be described. This is important to mathematics (I don't know about QM and that stuff). And for all those people who say "everything can be constructed by integers", that doesn't matter. For example, some people said that fractions are simply integers over integers. Yes, but they aren't INTEGERS, they're RATIONAL NUMBERS. If you need to construct something to use it, face it, you need it. Now real numbers. I'm sure we all love calculating out infinite fraction sequences (not really), but for all purposes, it's better to use the irrational numbers. Complex numbers are similarly useful, even if they can't describe actual amounts. If you only count amounts, then sure, natural numbers are fine. But the universe is obviously more complex than that. And like shown dozens of times, physical equations use complex numbers. So I think that complex numbers are necessary, if not in a too obvious way, to our physical system.

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Interesting, i find this kind of discussion fascinating, and would really like to see what Davids own answer is... One method of reasoning is that all rational numbers can be expressed using integers - simply use fractions.... real numbers also tend to be able to be expressed using integers - taylor expansions etc, although i do not know if it would be possible to express all real numbers in this way... maybe not.... but the ones necessary for current physical theories, quite possibly. Complex numbers such as 'i' can just be expressed as 'root(-1)', using integers and common notation.

The possible flaw in this method is that although you can express these using integers, overall what you have expressed isnt actually an integer - if you replace i with 'root(-1)' surely you're still using complex numbers - I guess it just depends on how you like to think about things.

If you ignore this flaw and continue using this reductionist method, you can actually dispose of "numbers" altogether, and instead use symbols and notation - sort of like 'typographical number theory', it is probably possible to express current physical theories using pure logic. It would be horrific - but I think its possible.

On the topic of beyond current physical theories, if we assume that the universe is finite, and that in a certain region there can only be a certain finite amount of information, I think that this information can be expressed/approximated in lots of different ways, to varying degrees of accuracy - different constructs are required for different approximations. I think if there was an ultimate fundamental "theory of the universe", it is likely to be simple in nature, in that it shouldn't require as a necessity certain man-made constructs (which I believe numbers essentially are), but would have complex implications - think fractals and chaos theory, however I also feel that such a theory would also be impossible... because surely any such system must be either incomplete or inconsistent - according to Godel. If this is applicable to 'the universe' or what implication it would have if it was is beyond me.

Whichever which-way you look at it, the "fundamental structure of our world" seems to be so far beyond human understanding that all we can do is approximate using crude models based on artificial constructs which seem to agree with our observations, which are also crude.

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Fascinating view.

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Seems like this answer has not come up: real numbers. Complex numbers, can actually be represented as an ordered pair, and just make the operations just like the ones in complex numbers. I do not have much time to build this up but I believe it is possible. In my opinion, complex numbers is just a tool for convenience. Other stuff can just replace it. It is obvious that rational numbers are needed, otherwise how are you going to evaluate \(P=\frac{F}{A}\) (just an example)? So we have left why we need irrational numbers. Again I think this is obvious, as many constants are irrational and we need them. \(\pi\) is significant enough. Well I think this is not the answer, just my opinion, so I hope someone out there can tell me my misconceptions, mistakes, gaps or whatsoever.

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I have another question.... Are quaternions real?

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I'd like to make another pitch for an answer that seems mostly neglected on this thread, which is the answer of INTEGERS!

One of the first issues with this answer is obviously the irrational numbers. However, I think that there is an easy way around this. Consider that no measurement is exact, that any measurement has a finite amount of significant digits in any measurement. Then, any measurement we could ever take has a finite number of decimal points, and can then be expressed as a fraction of two integers. In other words, it should never be necessary to use every infinite digit of pi to describe the circumference of a circle, or any other physical quantity. Only in pure math would you require these irrational numbers.

I am not so far in physics to be able to speak confidently about the issues of these claims in quantum mechanics, nor the role or necessity of imaginary numbers in quantum mechanics, so I will leave that alone. But perhaps there is a similar argument to be made there.

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I like it! Any applied science needs only enough digits to have the calculations turn out accurately

enough.Log in to reply

I think complex numbers should be used , though we can only use the reals and the tasks where complex numbers were used should be assciated with pairs of real numbers.

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Hey all. Sorry I haven't been able to monitor this discussion as closely as I'd like. We got hit by lightning over the weekend and it took out our internet connection. However, it's a great discussion. I'm not going to reply to everyone's comments as I'll post my overall take on this question on Wednesday (after I consult Mariam B's tarot cards). The comments on QM are right in a certain way of looking at it, and the comments about deriving complex numbers from simpler number types are also right. I'll try and distill these viewpoints into my own response.

There is a physics question besides QM involved too, that no one has incorporated yet. I'll toss this question into the ring as well:

Is there a fundamental limit as to how much information one can squeeze into a certain region of space and time? If yes, then our universe, which is finite in extent, could be built out of a large but finite number of these small regions. How would this change the required number system?

Note, this goes a bit beyond the limitation "current physical theories", which was a deliberate choice.

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I like this question!

Some top answers have focused on the fact that complex numbers are required because basic equations describing quantum mechanics rely on complex numbers. Others have countered that only natural numbers are required because from them we can construct a system that mimics the behavior of, e.g., complex numbers using only the naturals.

As others have pointed out the Natural numbers purists are kind of cheating - if you construct a system that behaves like complex numbers it doesn't matter that you have avoided using the symbol 'i', you are still using complex numbers. Why not go even further and claim that we don't need numbers at all, just the concept of and axioms of a set (from which we can reconstruct all math)... On the other hand, I don't think that those on the complex numbers side of the debate go far enough.

Let's examine the meaning of the phrase "a number system". A number system is a collection of items (which we will call numbers) and relations and operations (different ways to compare or combine these numbers).

Numbers are useful because we can talk about them in the abstract, replace an actual number with an unknown variable, and we know how the operations will be evaluated when we assign any specific value to the variable. The problem with this abstraction is that sometimes you can write down unsolvable questions. For example if your number system is the natural numbers, you can write down 4+x=3, but now there is no number that makes the equation true unless we introduce integers. Similarly if we want to solve all types of equations we must expand our number set to include rations, reals, and complex numbers to solve polynomial equations. The thing is, as soon as we have introduced 0, 1, the operations of addition and multiplication, and, crucially, the idea that we want to be able to 'undo' those operations. That is, we want to be able to 'do algebra'. Then we have magically already created all the complex numbers.

What about vectors? surely in a multi-dimensional world we need vectors to describe things. Vectors appear to just be a list of numbers, but they have different operations. For example, Maxwell's laws require the use of cross product. We need matrices to deal with vectors. And while the matrix is just an array of numbers, and the matrix operations can be described as a sequence of steps using basic number operations, the overall algebra of matrices has a different structure to that of 'numbers'. So I believe that we have to include matrices on our list of required number systems for the same reason that we have to include complex numbers.

It is a false cop-out to claim that because we can describe a more complex system using the symbols of a simpler system, we don't really need the complex system at all.

Are there other number systems needed? Surely. I don't pretend to know any details about cutting edge particle physics theory, but I know that the ways that these particles interact can be quite complicated, and cannot be modeled within the algebra of basic numbers.

On the very cutting edge of what might not even be real physics after all, string theorists make predictions about reality using even more exotic algebraic structures.

At the root, numbers are an abstraction that allow us to makes predictions about the universe. For any complicated way that objects in the universe can interact, we need a number system that incorporates and models that kind of interaction. As our knowledge of physics in incomplete, I'm sure that there is no final answer to the types of number systems that we need to describe the world around us.

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I find that many of the people here that argue for the coarser left side of David M.'s inclusion chain above argue something like this: "You don't need complex numbers because you can construct those from real numbers by algebraic closure" or like this: "You don't need real numbers because you can construct those from the rationals using Dedekind cuts or Cauchy sequences", If that is the case, then you're still using those numbers, you've just created them anew. Just call a spade a spade and admit it. Also, I think a few interesting options are left out: the algebraic numbers, the Gaussian integers and \(\mathbb Q(i)\)

For the current quantum theory, I believe we need a continuum of values (or, at least, something dense in the continiuum) that multiply without changing modulus. Thus we need values from all over the complex unit circle. Also, all rational numbers should be present, since ratios are a thing. So, at least all complex numbers with rational polar coordinates (with argument a rational multiple of \(\pi\)), extended to a field (Again, since ratios are a thing, I think we need to have a field). That would be \(\mathbb Q(i)\) extended with all possible values of \(\sin (q\,\pi),\; q\in \mathbb Q\). Some of these might be trancendental. I can't tell this late at night.

Do we really need \(\pi\) or \(e\)? How about all the other trancendental numbers? I don't know. I am not certain enough to give an answer, but I have told you what I believe to be a minimum of numbers needed.

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Mursalin gave an answer which I think everybody is liking. But honestly speaking I don't think that it is right. If you ask what type of numbers is required to describe universe around us then my answer would be

Natural numbers!Yes.. It looks very stupid but let me explain. First of let me clear to you that it is not true that it is impossible to imagine quantum mechanics without complex numbers. This is a very famous misconception. Let us ask.. if nobody would have thought about complex numbers before the advent of quantum mechanics, wouldn't scientists have solved puzzles of blackbody radiation or photoelectric effect? Actually quantum mechanics can be constructed without using complex numbers, we just need to write equations for 2 variables instead of 1 with proper constraints for them. Complex number has two such quantities contained in it (magnitude and phase) and so if we use complex number then we need to write only single equation. Thusmathematicallyquantum mechanics or LCR circuit equations become easier if we use complex numbers.Now what about 0,negative integers,rational numbers and irrational numbers? It is well known that all these quantities can be constructed mathematically. I think everybody knows about constructions of 0 and negative integers: they are constructed as solutions to certain equations. Last of all irrational numbers are constructed using well known procedures of Dedekind cuts or Nested intervals (you may read wikipedia articles about these).

Now the next question: suppose we don't construct any of these and only use natural numbers. Then is it possible to write say theory of general relativity or quantum mechanics? And answer is

YES!Only problem would be that it would be too complicated since each variable would need lot of description! May be Kroneker had realized this long back and said "God made the natural numbers; all else is the work of man."People who don't like my answer may also want to read about "Preintuitionism" on wikipedia.

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Though members of this thread have successfully acknowledged that the entities that represent rationals and irrationals are completely built from natural numbers (e.g. equivalence classes of ordered pairs for fractions and sequences for irrationals), one significant component of the construction of real numbers has been ignored: the operations on these new structures, addition and multiplication.

At each stage of construction, one must redefine addition and multiplication for ordered pairs, sequences, etc. For example, if the ordered pairs (a,b) and (c,d) are fractions, we have the rule, which we must

definetheir sum by the rule (a,b) + (c,d) = (ad + bc, bd). So even if we only use ordered pairs and sequences of ordered pairs of natural numbers, we have to provide new axioms that define the structures of the real numbers. So even employing this construction, one is still using a system isomorphic to the real numbers, equipped with the entire structure of the continuum. Thus if physics uses any idea of the continuum, it uses real numbers, no matter how they are disguisedLog in to reply

I disagree on one point,can all irrational numbers be constructed from natural numbers?How would you construct pi or e?

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Well, one possible way is this:

\(\pi=4(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}- \cdots\)

\(e=1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+ \cdots\)

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But it has got really interesting.I read that there are many irrational numbers which can't be explicitly represented(by some root,ratio,or series).Do these numbers have any significance in physics?

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The second one is a direct implementation of the Taylor Expansion of \(e^x\).

I'm trying to express them just from their definition and without using any idea of where they might be on the number line. But I think we're deviating away from the topic a little bit....

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Read about Dedekind cut on wikipedia or in any book on analysis.

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What you've said is absolutely right! But that wasn't what I was going for. See the EDIT part of my comment.

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I think all we need is real numbers.other things are all structures constructed from real numbers.A complex number is only an ordered pair of real numbers. An n-tuple of real numbers constitute an n-dimensional vector,a matrix of real numbers is an n cross n tensor,and so on.If necessary,we can also add another dimension and get a 3 dimensional analogue of a tensor,which will have n cross n cross n elements.So I think faith in 'reality' should be restored :)

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[Deleted for irrelevance- Peter]

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!!!!???

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We are interested in your 'situation'....please tell about it

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Internet preachers are everywhere! Unbelievable!

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What this has to do with the original question?? LOL

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I think COMPLEX NUMBERS is required to describe the whole world around us.

Reason being , Complex numbers exists in

Conjugate pairs. In the same way , In physics Forces exists in conjugate pairs (Fundamental theory of classical mechanics : newtons 3rd law). We can't move a inch without forces .SO, forces are there in every fundamental structure of the world and exists in pairs (analogous to complex numbers) .Secondly, In physics we use vectors everywhere which is analogous to complex numbers.Thirdly , We can't compare complex numbers. We can't say whether 3 + i or 4 - i greater. Can we? In the same way in physics , We can't compare different measuring units in physics. For example , we can't compare ' Kilo grams and pascals ' etc. etc. ( somewhat analogous )

(I think my 3rd reason makes no sense , But I am just trying to prove it )

Fourthly , Maths in the mother of all sciences. It is required in every walk of life and physics. In physics we use mathematical equations instantly. There is always a tendency to get a purely complex number as a root of the equation. And there are infinitely many pure complex numbers and infinitely many real numbers (or integers etc) .SO the probability of getting a pure complex number is equal to the probability of getting a real number. So , we can say , complex are used in almost every equations of maths (as integers⊂rational numbers⊂real numbers⊂complex numbers. )

Combining above four conditions , I can say that COMPLEX NUMBERS is required to describe the whole world around us.

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I was quite impressed by Mursalin's answer to the question. I know some basics about quantum mechanics, so I would try to make the reason here:- The Schrodinger's equation gives the probability of finding an electron in a given amount of space. It is the square of the wavefunction, and the wavefunction itself consists of a complex number. In fact, even the Newtonian mechanics can be derived from the Schrodinger's equation after assigning some values and parameters to the equation. So this clearly shows that this equation is a much more basic way of describing the universe than the newton's laws itself. So the answer comes here, everything in this universe is described by its wavefunction, and this wavefuction is an imaginary quantity. Hence complex numbers are the more basic and essential number type required for describing the universe around us. I don't know anything about hypercomplex numbers, but as per my knowledge complex numbers may satisfy the requirements.

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hmmm

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I think that the whole process of classifying number types has begun from simple systems such as integers to intricate ones like complex numbers or hypercomplex numbers (quaternions, tessarines, coquaternions, etc. ). Now one can question what would be the need for newer (or more complex) number systems and the answer to that lies in the fact that these new number systems provide the appropriate mathematical formalism to realize ideas and theories. Now why are they appropriate and the answer to that would be if any other number system be used to formalize a theory it's be insufficient to describe the nuances of that theory. This brings us to question as to why advanced mathematics like lie algebra, or group theory, etc. is needed to do advanced physics. One would not be able to get far along if one continues to abandon complex numbers to do Quantum Mechanics ( as someone rightly pointed out). Coming back to number systems, the process with which new number systems were developed tells us an important fact as we made progress with the advent of complex numbers and hypercomplex numbers, our understanding of the universe improved. Does this not hint towards the fact that there is a possibility of an all encompassing number system which will qualify as the singular type of system needed for all physical theories. Definitely this number system can't be integers, real numbers, or even complex numbers because these prove to be parts of a bigger better system for theories. Taking a crude example, we can always use complex numbers instead of real numbers or integers, but let's say does it serve any purpose to use complex numbers in simple arithmetic. The point here is these are all subsets of what is required. Striving for a better, bigger, "all encompassing" and universal number system is what would be required for our current physical theories. And as complex numbers is in a sense a subset of hypercomplex numbers. So "currently" my answer would be hypercomplex numbers. And yeah, i could be totally wrong but this is what I feel should be. Hope I have conveyed my idea properly.

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All number systems are essentially a mathematical construct to explain the space, e.g. 1-D space can be described by a real number, 2-D space by a complex number or an ordered pair of two real numbers, 3-D space by vectors, and 4-D by quaternions, etc.

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Really, all we ever need is \(1\), \(-1\), addition, and multiplication. Because with these few tools and a little algebra, you can construct all other numbers. However, I did realize that transcendental numbers aren't the easiest to construct with this system, so we should also include helpful numbers like \(\pi\), \(e\), or anything else like that. (Here is a more detailed explanation of this construction system and transcendental numbers) Whereas real and complex numbers are convenient for most physical applications, they are not necessary. Furthermore, if you consider \(1\) and \(-1\) as "real," then undoubtably, all these other numbers are real.

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You asked us what number type is

requiredfor our current physical theories. Well, I can name certain theories where complex numbers are a necessity. Quantum Mechanics is such an example. It seems that you can't even approach Quantum Mechanics without introducing complex numbers.In QM the probability of something happening is the square of the magnitude (absolute value) of the “probability amplitude”. This probability amplitude can be complex valued. For example if the probability amplitude is\(\frac{i}{\sqrt2}\), then the probability is \(|{\frac{i}{\sqrt2}}|^2=\frac{1}{2}\).In fact, the Schrödinger equation, which is arguably the backbone of Quantum Mechanics, has an '\(i\)' right at the beginning. You can't escape it! The equation of a single particle looks something like this:

\(i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},\,t)=-\frac{\hbar^2}{2m}\nabla^2\Psi(\mathbf{r},\,t)+V(\mathbf{r})\Psi(\mathbf{r},\,t)\) [It took me a really

longtime to render this in LaTeX!].I don't know much about Quantum Mechanics and most of what I wrote here is copied and pasted from things I found on the internet. So, I will stop talking about Quantum Mechanics now.

A classmate of mine once asked me, "Why are we studying about complex numbers anyway? They don't exist after all! So why make a fuss about things that don't even exist?"

In reply, I asked him another question, "What is the physical significance of multiplying something by \(-1\)? Multiplication by \(-1\)

rotatessomething by \(180\) degrees. If something is heading towards north at a velocity of \(1 ms^{-1}\). This means that thing is heading towards south at a velocity of \(-1ms^{-1}\).What if we could find a number that would rotate something by \(90\) degrees? Assume that we have such a number. What would happen if we multiplied something by this number

twice? It would rotate that thing by \(90+90=180\) degrees. This is remarkable! Because that's very thing multiplying by \(-1\) does. In other words,\(something\times new\) \(number \times new\) \(number = something \times (-1)\)

Or in other words, \(new\) \(number^2=-1\).

And this is how the imaginary unit \(i\) is defined. Just because you can't count \(i\) chickens doesn't mean complex numbers are any less real. By this definition, even negative numbers don't exist (you can't actually count \(-5\) chickens)."

My answer was able to convince my classmate.

Another point: we know that the distance between two points in Euclidean space is \(\sqrt{\bigtriangleup x^2+\bigtriangleup y^2+\bigtriangleup z^2}\) [this is just the Pythagorean theorem].

When relativity came along, we realized that space and time were very closely related. And the distance [this is also known as the space-time interval] between two points (events) in

space-timeis:\(\sqrt{\bigtriangleup x^2+\bigtriangleup y^2+\bigtriangleup z^2 -c^2t^2}\).

What does that have to do with complex numbers? Watch again: the formula is

\(\sqrt{\bigtriangleup x^2+\bigtriangleup y^2+\bigtriangleup z^2 +(ict)^2}\).

So \(i\) also creeps up here! So physical theories need numbers and sometimes those numbers happen to be complex. But as you have said, there are different answer to this question and a lot of people will come up with different viewpoints. According to me, in order to understand and describe

[I'm putting a lot of stress on this] in this universe, complex numbers are necessary.everythingLook at the size of this comment! I think I'll stop now.

EDIT: there have been a couple of comments recently that say that complex numbers are not actually

necessaryfor describing the universe. They are absolutely right!But this raises another question:

Do we even need numbers to describe the universe around us?

Any physical theory is a model of the universe. Numbers are a tool to describe a theory. They are not a property of the theory itself. Numbers have

propertiesand we use thosepropertiesas atoolto try to describe atheorythat in turn describes theuniverseto some extent. We can have theories that don't use numbers at all!There are certain things that we observe in the universe and we try to capture those things with numbers and end up using those numbers in theories. Complex numbers exist in nature in the same way other numbers exist. They have certain properties that we can experience, perceive and observe. I tried to demonstrate that with the example of rotation. I tried to illustrate current physical theories that use the properties of complex numbers to describe natural phenomenon without getting too philosophical about it. The universe doesn't care if we use numbers to understand it. Numbers are merely a tool for

us.I understand that this has gotten a little bit more philosophical than I would have wanted. So, I'm stopping here.

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Although I agree with Mursalin and have nothing to add about QM, I do object to his observation that \(i\) also creeps up in special relativity when considering the interval ("distance") between two points in spacetime. Unfortunately, properly explaining why this interpretation is not suitable, requires quite a lengthy explanation.

First off, vectors with complex components are used in QM for instance. The norm squared of such a vector is (considered to be) equal to the inproduct with its

conjugaterather than itself, so even with complex components you still get something strictly positive.The four-vector as used in special (and general) relativity is not a proper vector in the mathematical sense, or at least not in the normal four-dimensional vector space. The scalar product of two four-vectors is defined as

\( \mathbf{A} \cdot \mathbf{B} = -A_0 B_0 + A_1 B_1 + A_2 B_2 + A_3 B_3 = \sum_{\mu,\nu=0}^{3} \eta_{\mu\nu} A^\mu B^\nu \),

where \( \eta \) is the Minkowski (or flat-space) metric tensor with \( \eta_{00} = -1, \eta_{11} = \eta_{22} = \eta_{33} = 1 \) and all other components zero. It should be noted that sometimes the scalar product (and hence \( \eta \)) is defined with the exact opposite sign; there is no ironclad convention for this. Also, typically the summation sign is omitted; summation is implicit whenever an index is repeated (that

isironclad).At this point I cannot resist but demonstrate what makes four-vectors and their special scalar product so useful. The fundamental principle (or postulate, if you like) behind (special) relativity is that it does not matter what (inertial) frame of reference you use to describe something. This is nicely reflected in this scalar product: the scalar product of two four-vectors is

invariantwith respect to a change of reference frame. That goes for the norm of the spacetime four-vector \( (ct, x, y, z) \),\( \eta_{\mu\nu} x^\mu x^\nu = -c^2 t^2 + x^2 + y^2 + z^2 \),

which gives you the spacetime interval. But it also works for other four-vectors such as the energy-momentum four-vector \( (E/c, p_x, p_y, p_z) \), for which we have

\( \eta_{\mu\nu} p^\mu p^\nu = -E^2/c^2 + p_x^2 + p_y^2 + p_z^2 = -m^2 c^2 \),

where \( m \) is the invariant restmass. You might recognize the above formula for the special case \( \vec{p} = 0 \), but this is the more general form.

Going back to the definition of a scalar product, it may seem that using \( \eta \) is just overly complicated; why introduce a 16-component tensor for a sum of four products? And why not use imaginary components to deal with the minus sign? I suppose the most convincing answer (which is what I have been working towards) is that, when you go to general relativity, the scalar product generalizes to

\( \mathbf{A} \cdot \mathbf{B} = g_{\mu\nu} A^\mu B^\nu \),

where \( g \) is still real-valued but may now have nonzero off-diagonal components.

Finally! Sorry for the long post.

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I agree with you. It's like algebra, that one use to describe a system, and same system can be described by different algebras, but it doesn't mean that algebraic model exists physically. As we have different tools like matrices system, similarly complex numbers and it's properties are used to describe systems, it doesn't have to do with physical existence.

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I think complex numbers must be the numbers used for describing the universe around us.

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Can you give a reason?

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yes sir ,in real life situations like free fall under all the effects of nature like drag, viscosity, etc..the energy dissipated by the falling object is in the form of complex number magnitudes..

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I think it should be complex numbers. Though I have not seen them(

purely complex numbers) being used so often, but I know one field where it is used. My teacher had showed me using it in solving parallelLCRcircuits. I don't know whether I am going as per the requirements of the discussion.Log in to reply

Aah, so this is a good example of ease of description. Is it necessary to have complex numbers for LCR circuits, or is it merely a convenient description because it makes the math easier?

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Well I think most of the things can be simplified out using one or more approaches. It depends on the effectiveness(

I mean accuracy) of the technique being employed.Log in to reply

It makes the math easier I guess. I remember that while learning LCR circuits, I would end up with a second order differential equation and solving them isn't included in High School mathematics. Instead, we were introduced to phasors and use vectors(?) to solve the problems.

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From what I can tell, in classical wave analysis, complex numbers are introduced as a way to simplify representations of wave equations, using Euler's formula. In the example of LCR circuits, one sets up the differential equation describing the system; an oversimplified method of seeing how complex numbers relate to this is by showing that the solution of the differential equation may take the form of \(e^{i(wt-c)}\), where w is angular frequency and c is phase shift. This is convenient, as it is represents the wave equations involving sine and cosine, so we know intuitively that we're on the right track like this. When you plug it all back into the original differential equation and solve with initial conditions and such, we find that, indeed, all the imaginary terms from the \(e^{ix}\) term disappear by the time we arrive at the final solution. Thus, it becomes apparent that the complex numbers were only introduced to help solve the equation, and are devoid of physical meaning. Quantum mechanics is a different story, however...

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