Why do we need the set of complex numbers to solve?

[astrololo]

I was wondering, why is the set of complex numbers needed to solve problems that the set of reals doesn't permit to ? I mean, in relation to the fundamental theorem of algebra, that is.

 

[Geofleur]

Just to clarify, are you asking why we bother to enlarge the domain of admissible numbers, instead of just saying "This equation has no solutions"?

 

[astrololo]

Just to clarify, are you asking why we bother to enlarge the domain of admissible numbers, instead of just saying "This equation has no solutions"?

Yeah, I'm asking why by using complexs we can obtain solutions while if we use only the reals we can't.

 

[Geofleur]

Your question sounds a little different to me than my attempted re-phrasing, more like "What leads to the need to enlarge the domain of admissible numbers in the first place?", not so much "Given that need, why bother to do it?"

I suppose the simplest answer is that, without complex numbers, there is no meaning of taking the square root of a negative number, so the equation $z^2 = -1$ would just have no solutions. There may be a deeper answer but, if so, I don't know what it is!

 

[Ray Vickson]

Yeah, I'm asking why by using complexs we can obtain solutions while if we use only the reals we can't.

Historically, complex numbers were used first to obtain REAL solutions to real equations that could not have been obtained without them; that is, it was crucial to use complex numbers during intermediate steps on the way towards the final real solution. (I realize that is dealing with a different question than the precise one you asked, but I personally, think it is very interesting on its own.)

 

[HallsofIvy]

To expand on what Ray Vickson and Geofleur said, they are talking about Cardano's solution to the cubic equation. If a and b are any two numbers, then (a+ b)^3= a^3+ 3a^2b+ 3ab^2+ b^3 and 3ab(a+ b)= 3a^2b+ 3ab^2. Subtracting, (a+ b)^3- 3ab(a+ b)= a^3+ b^3. If we let a+ b= x, 3ab= m and a^3+ b^3= n that is the same as x^3- mx= n. That is, given two number, a and b, x= a+ b satisfies the (reduced) cubic equation x^3- mx= n.

Now, suppose we are given m and n. Can we find a and b thus solving that cubic equation? Yes, we can. From m= 3ab we have b= \frac{m}{3a}. Putting that into a^3+ b^3= n, we get a^3+ \frac{m^3}{3^3a^3}= n. Multiplying by a^3, (a^3)^2+ \left(\frac{m}{3}\right)^3= na^3 or (a^3)^3- n(a^3)+ \left(\frac{m}{3}\right)^3= 0, a quadratic equation in a^3.

By the quadratic formula,
a^3= \frac{n\pm \sqrt{n^2- 4\left(\frac{m}{3}\right)^3}}{2}= \frac{n}{2}\pm \sqrt{\left(\frac{n}{2}\right)^2- \left(\frac{m}{3}\right)^3}.

Now, of course, there are an enormous number of applications for the complex numbers, but that was the original impetus for their definition.

The point is that there exist cubic equations, having all real roots such that \left(\frac{n}{2}\right)^2- \left(\frac{m}{3}\right)^3 is negative. That is, in using that formula to find real roots to the equation it is necessary take the square root of a negative number. Of course, it happens that, in completing the formula the "imaginary parts" cancel but it was still necessary to create the imaginary numbers in order to use that formula.

 

[epenguin]

To say this further, Arab mathematicians I think if not earlier recognised that many quadratic equations like
x² + 1 = 0, or
x² - x + 1 = 0 needed a square root of -1 in a solution, and this really seemed to make no sense so these solutions were called not real in the ordinary everyday, not the modern mathematical technical sense of real. They could just be dismissed for that reason, they were not a real solution to any problem. But then in surely one of the most brilliant mathematical steps ever, it was discovered in Renaissance Italy that they could be used for solving something quite concrete and real, as explained in posts above. This is the first example, another I have seen is to work out in now many different ways can you give change for a dollar?

These are minor things one could live without but on the applied side they open the mathematical treatment of e.g.oscillations and vibrations and circuits to engineers and physicists in a truly eye-opening and convenient way. And quantum mechanics is thought to need complex numbers inherently, not just for convenience of calculations. And there is much else.

One landmark theorem is the 'fundamental theorem of algebra' (which oddly is not algebraic) which says that, as per previous examples, every algebraic equation with real coefficients has a solution in the field of complex numbers (which includes the real numbers as special case). (From that you can deduce that if it is nth degree it has n solutions.) Then if the coefficients are complex numbers, real or non-real, they still have just complex-number solutions. Because of their 'nonrealness' It was hoped that their uses like above would be temporary and a way to solve the problems like the cubic without them would be found. I have read that this has been proved impossible. (I do not know how advanced the proof is.)

It is found that the real number system is essentially incomplete, and with complex numbers you see "behind the scene". The things most magic and mind-blowing in math seem mostly to come from complex numbers. Some mathematicians (I am not one) might agree with my impression that without much exaggeration, math is divided into two parts, glorified accountancy with no really big surprises on the one hand, and stuff depending on complex numbers on the other.

 

[jssamp]

Personally I feel the choice of wording is unfortunate. Complex numbers are no more unreal than integers. Because they are called imaginary doesn't make Euler's any less fundamental. It's sad that somebody had to use real and imaginary to describe two sets of one fundamental tool of knowledge and so lead so many of us to think half of the world is "not real" and "too complex" to understand.

 

[aikismos]

Personally I feel the choice of wording is unfortunate. Complex numbers are no more unreal than integers. Because they are called imaginary doesn't make Euler's any less fundamental. It's sad that somebody had to use real and imaginary to describe two sets of one fundamental tool of knowledge and so lead so many of us to think half of the world is "not real" and "too complex" to understand.

I think I'd have to take issue with the notion that imaginary numbers are just as "real" as real numbers, since given it's readily accepted that it's not possible using integers to to multiply a number by itself and obtain a negative result. Integers are intuitive in the sense that the negative has an interpretation of direction relative to a point in a geometrical interpretation. Even the products of integers can have a geometric interpretation on the real number plane in regards to the slope of the main diagonal swept out by the area of the vectors that represent them. I don't know that the same can be said for the products of complex numbers on the Argand plane.

 

[jssamp]

It's not possible using integers. The bias that led to the name is inherent in that statement. Why are integers your measuring stick for "realness". Is pi not a real value. what about the distance across my square room corner to corner, is that not a real distance? It isn't rational. Imaginary numbers have a very real visual interpretation on the number plane for me. I visualize complex exponentials in the phasor domain. If you saw it you would recognize it but you might call it a sine curve.

 

[jssamp]

To say this further, Arab mathematicians I think if not earlier recognised that many quadratic equations like
x² + 1 = 0, or
x² - x + 1 = 0 needed a square root of -1 in a solution, and this really seemed to make no sense so these solutions were called not real in the ordinary everyday, not the modern mathematical technical sense of real. They could just be dismissed for that reason, they were not a real solution to any problem. But then in surely one of the most brilliant mathematical steps ever, it was discovered in Renaissance Italy that they could be used for solving something quite concrete and real, as explained in posts above. This is the first example, another I have seen is to work out in now many different ways can you give change for a dollar?

These are minor things one could live without but on the applied side they open the mathematical treatment of e.g.oscillations and vibrations and circuits to engineers and physicists in a truly eye-opening and convenient way. And quantum mechanics is thought to need complex numbers inherently, not just for convenience of calculations. And there is much else.

One landmark theorem is the 'fundamental theorem of algebra' (which oddly is not algebraic) which says that, as per previous examples, every algebraic equation with real coefficients has a solution in the field of complex numbers (which includes the real numbers as special case). (From that you can deduce that if it is nth degree it has n solutions.) Then if the coefficients are complex numbers, real or non-real, they still have just complex-number solutions. Because of their 'nonrealness' It was hoped that their uses like above would be temporary and a way to solve the problems like the cubic without them would be found. I have read that this has been proved impossible. (I do not know how advanced the proof is.)

It is found that the real number system is essentially incomplete, and with complex numbers you see "behind the scene". The things most magic and mind-blowing in math seem mostly to come from complex numbers. Some mathematicians (I am not one) might agree with my impression that without much exaggeration, math is divided into two parts, glorified accountancy with no really big surprises on the one hand, and stuff depending on complex numbers on the other.

Good answer. I can't think of any interesting technical endeavor today that would be possible without complex numbers. Take communications and digital signals processing. Our electronic lives rely on the Fourier transforms and how fun would that be without using complex exponentials in
the integration.

 

[aikismos]

)

It's not possible using integers. The bias that led to the name is inherent in that statement. Why are integers your measuring stick for "realness". Is pi not a real value. what about the distance across my square room corner to corner, is that not a real distance? It isn't rational.

I think that the measuring stick for "realness" (a term you introduced, I didn't) is that it is derived from Peano's axioms, extended with limits, and isn't contradictory to those systems. The fact that Rene Descartes coined the term imaginary is perfectly suited, because that an even root of a negative real number has an answer is neither intuitive nor is it useful in the graphical domain as an extension of the Cartesian plane. I appealed to integers because you mentioned them in your post, and because integers arise as an necessary extension of the natural numbers when the operation of subtraction is introduced. In a way, complex numbers draw a parallel to this, as the definition which extends the reals to the complex numbers fulfills a similar role and is no doubt useful in fluid dynamics, electrical engineering, and other sciences. However, the point I was trying to make is that while integers and rationals and reals have a way of being interpreted graphically on the Cartesian plane, $i$ (as far as I know) has no similar role. How can it, as it exists outside the partially ordered plane? (This necessitates creating the complex plane mapping this new set onto one of the axes.)

Let's talk about the diagonal of your room. From one corner to it's kitty corner, has a length, and if you were to measure it with a ruler, you could accept a rational or non-terminating, non-repeating value (if you wanted to use mathematical formula and use the Pythagorean theorem), but what you CAN'T get for a length is an imaginary value. That's because in the theorem, the hypotenuse is the root of the squares of two reals (which regardless of their signs in the context of vectors) must be a positive value. Hence your room can never have an imaginary value. THAT'S why complex numbers are not intuitive like the naturals, integers, rationals, or even to some extent the reals (where the epsilon-delta definition doesn't eliminate the wonder of infinite regress but at least gives formal criteria to evaluate any process seeking to find a more precise value). I think the OP is trying to understand, WHY were the reals extended? Certainly, prior posters have covered the notion that as a more complex formal system (https://en.wikipedia.org/wiki/Formal_system), the complex numbers essentially through the addition of axioms has practical use in proof. I think that's the crux of what it is being asked... the OP is trying to integrate the progression into his/her ontology which probably has progressed like this:

Where do numbers come from? Counting. (The naturals)
Why does 0 exist? Because sometimes when we take away everything, nothing is left to count. (The wholes)
Why do negatives numbers exist? Because sometimes when we want to take away more than exists, we need to keep track of what we didn't get to take away but still might like to (The integers)
Why do rational numbers exist? Because sometimes we like to divide rectangles up into squares (The rationals)
What number times itself gives us 2? We find we can't find a rational number answer, though it must exist to make arithmetic work. (The irrationals)

Remember that the Pythagorean theorem doesn't allow for imaginaries. (The hypotenuse is the sum of the squares of the legs.) So along comes someone who asks, well if the root of 2 exists, what about the root of -1? BAM! In the 1700's they really started to explore the UTILITY of assuming it does exist (even though it doesn't by the rules of arithmetic) and find that not only can they make it work despite the fact the it doesn't exist, but it can answer all sorts of math questions. I think the OP is trying to get a sense of what those initial uses are. I would suggest reading up on the roots of unity too.

Useful? Yes. Philosophically intriguing? Absolutely. Real? Nope. Intuitive? How can assuming something that you know doesn't exist exists in an even more abstract form possibly be intuitive?

EDIT: I like this phrase in (https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra):

"Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed."

This is the essence. New sets of numbers are about creating closure! The FTA closes yet another formal system on the road of maddening complexity!

 

[jssamp]

Excellent response! This is why I love this forum. Always intelligent discussion to be had. Glad I had time today to visit, aikismos.

 

[sshai45]

@aikismos: I'd want to say that with regards to complex numbers and "reality" thereof, a difficulty with your "length" example is that it would also "defeat" negative numbers as well, since you cannot have negative lengths, either. Yet most people have less trouble with negative numbers than with complex numbers. The question remains: why?

The reason I think is because that things which you can measure as having a negative value are easily experienceable, e.g. we can all understand what a negative number of degrees on a Fahrenheit or Celsius thermometer means, or what it means if a bank transaction has a negative value (which is, historically, how negatives came about, in China and India, I believe), or what "negative growth rate" means (it's shrinking), etc. But examples of a quantity which we can measure as having a complex value are rarer and less-connected to everyday experience, or perhaps, are just not usually described that way. For example, everyone knows of AC current, but most lack any electrical engineering experience, thus describing an AC signal with a complex amplitude and the meaning of that is less clear. And even fewer would have experience with the quantum wave function, which was only recently "measured" directly.

Negative numbers also have a simple intuitive observation: they represent an opposite, and so it only makes sense to measure a negative quantity when there is an opposite direction to the one we are measuring. Complex numbers are a little more weird: to mark off an "imaginary" amount of something it has to be in some sense "orthogonal" to both the thing and its opposite. Like the AC waves I mentioned: negative amplitude just means the wave is flipped relative to some conventional reference frame. Imaginary amplitude means it is phased half-way between those, neither opposite flip nor straight. In particular, the magnitude of the amplitude (always a nonnegative real number) is not changed, but the phase is changed, which is not an "obvious" parameter yet is the natural "orthogonal" in that situation. I submit that it is hard to think of what the "orthogonal" would be in other situations. With regards to coordinates, say, we measure lengths plus or minus when we are given a direction. You could also measure complex coordinates as well: one complex coordinate is a point in a 2-dimensional plane of measurement, and we need to specify a convention for both axes to make it work. Though you could just as well use a pair of real coordinates. Yet there's nothing wrong with treating that as one, complex, "number". It's just that the way it is taught doesn't make it seem as such. But you can, in fact, even describe all points on the Earth's surface using a complex coordinate system with nice mathematical properties (the Riemann sphere). When you get into topology and abstract algebra, you find that you can treat it as a unified space, not two spaces (e.g. instead of as \mathbb{R} \times \mathbb{R} with a funny multiplication, you can take it as, say, a quotient ring of a polynomial ring). But we don't have to go that far. A more elementary treatment is to try "base-2i" representation... Then you can actually write complexes as a single string of digits with no "I"s, making them really feel like singular entities given our usual way of relating to numbers as strings of digits.

Finally, I think a lot of the confusion results from that unfortunate name of "imaginary". We hear that and on some level believe it. Personally, I think it should be scrapped. Call it "transverse" or "lateral" numbers instead (maybe +I is "obverse" and "-I" is "reverse"). Complex numbers can still be called that, since it has no word "imaginary" in it.

With regards to "reality" of numbers, I'd say that numbers are "real" in the same sense that words are "real", and "imaginary" in the same sense that words are "imaginary". They are mental constructs we have that allow us to make sense of reality. As I read somewhere, or maybe I even thought of it myself but inspired by something else, you cannot stub your toe on a 1, so clearly, 1 is not "real" in a material sense. All numbers are invented by our minds. All numbers are "imaginary" in this sense.

 

[PeroK]

I think I'd have to take issue with the notion that imaginary numbers are just as "real" as real numbers, since given it's readily accepted that it's not possible using integers to to multiply a number by itself and obtain a negative result. Integers are intuitive in the sense that the negative has an interpretation of direction relative to a point in a geometrical interpretation. Even the products of integers can have a geometric interpretation on the real number plane in regards to the slope of the main diagonal swept out by the area of the vectors that represent them. I don't know that the same can be said for the products of complex numbers on the Argand plane.

If complex numbers are in a sense not real, then what do you make of the realness or imaginary-ness of matrices, for example?

 

[Geofleur]

I'd like to elaborate a bit on the point that complex numbers do have a nice graphical interpretation. Set $a + ib = (a,b)$, a point in the Argand plane. Multiplying by $i$ rotates the position vector for this point counterclockwise by 90$^{\circ}$. More generally, using the polar forms, multiplying two complex numbers gives

$z_1z_2 = r_1e^{i\theta_1}r_2e^{i\theta_2}= r_1r_2e^{i(\theta_1+\theta_2)}$,

So $z_1$ stretches $z_2$ by amount $r_1$ and rotates it by an amount $\theta_1$. There's a delightful book that shows the unfolding of complex analysis from this geometrical perspective, Needham's Visual Complex Analysis.

 

[aikismos]

Good answer. I can't think of any interesting technical endeavor today that would be possible without complex numbers. Take communications and digital signals processing. Our electronic lives rely on the Fourier transforms and how fun would that be without using complex exponentials in
the integration.

Thanks! I think like many people here, I'm just here to rub shoulders with a subset of the population to learn a thing or two through dialogue spiced up with a bit of argumentation. I work in the business world in an IT capacity these days, so I don't get to rub elbows with thinkers too often. I tip my hat to everyone who spends moments of their free time engaging in conversation about sometimes convoluted logical musings. :D A hearty thanks for your pushback.

 

[aikismos]

@aikismos: I'd want to say that with regards to complex numbers and "reality" thereof, a difficulty with your "length" example is that it would also "defeat" negative numbers as well, since you cannot have negative lengths, either.

I'm going to pushback on this because it's possible in the formal system for introducing the notion of a vector. Moving from the NE to SW corner, for instance, is the opposite of treading from SW to NE. In this regard, negativity as a concept can be used to differentiate models for motion in time-space. I'm no physicist, but all of Minkowski space (See https://en.wikipedia.org/wiki/Minkowski_space) is built on this as being useful in mathematical physics. My very limited understanding is that integers are what give us the ability to have relativity, because without them, we would have an absolute origin (in the mathematical sense) in space-time and therefore an absolute frame of reference. I could be wrong, and if so, someone correct me.

@aikismos:Yet most people have less trouble with negative numbers than with complex numbers. The question remains: why?

I think it's a matter of complexity wherein students who reach complex analysis don't really have a mastery of the reals or their mapping to geometric structures. Let's be honest that the majority of people who study math in secondary ed never really understand anything at all beyond the fundamentals of arithmetic, and those who do venture into undergraduate math ultimately leave without really understanding anything about the nature of computation as a physical system or formal system. I'm going to even go one step further an say that much of the teaching that occurs at the graduate level is dominated by people who, though excel in the manipulation of formal systems, often fall prey to the Platonic delusion that numbers are somehow independent of computational system. In this regard, the nature of complex analysis (CA) presents a challenge, because it falls back to the point made earlier that imaginary numbers seem to contradict intuitive geometry. Many students who survive undergraduate math often walk away with a sense of bewilderment at the variety of formal systems used and may miss the generalization of mathematical systems, which are essentially explored, not for the pedantry of axiomatic systems, but for the utility of modeling. I personally view complex analysis as one of the watershed moments precisely because philosophically, it allows the generalization and therefore differentiation of fields. The Fundamental Theorem of Algebra (FTA) and complex numbers allow us to have two different algebras map to the same geometry, that of the Euclidean plane. I think complex analysis versus real analysis is analogical to Eucidean to Lobachevskian geometry, and forces a thinker to accept that any mathematical system is an edifice that is built with no claim to absolute truth. Where as non-Euclidean geometry forces us to accept that no geometry is absolute in modeling space, complex analysis by the way of the FTA forces us to accept that no algebra is absolute in dividing that space up by mapping an algebra onto a geometry, if you will. For students who don't understand that geometries and algebras are just formal systems which we use to organize our thoughts about external time-space, they get stuck on the idea of how can something which is imaginary (in the normal Cartesian sense of space-time) be so damned useful? In this sense, CA by way of the FTA is just as important as the of zero, operation, inverse, identity, etc. because it creates an algebra which is not a subset of the reals (like Boolean algebra), but outside with scalars which have no geometrically intuitive use from a Cartesian perspective. In this sense, using complex number to find roots in algebra is an example of adding axioms to a system to extend the reach of provable truths in a Goedelian sense, if that makes sense to you. What was unprovable in an algebra of real numbers because somewhat trivial (relatively speaking :D) in an algebra with imaginary numbers!

EDIT: Let me say as a former high school teacher, that the NCTM did a good job of pushing the idea that there are four cognitive domains in math (a notion being backed by cognitive science): verbal, graphical, numerical, and symbolic domains. In this context, it would be simplest to say that complex analysis forces us to accept there is a bigger system of numerical and symbolic thinking than the reals which can compete with the reals to describe graphical thinking.

@aikismos:Negative numbers... Then you can actually write complexes as a single string of digits with no "I"s, making them really feel like singular entities given our usual way of relating to numbers as strings of digits.

Thanks! This is the sort of thing that I'm clueless about not being an EE. I'll cogitate upon these ideas.

@aikismos:With regards to "reality" of numbers, I'd say that numbers are "real" in the same sense that words are "real", and "imaginary" in the same sense that words are "imaginary". They are mental constructs we have that allow us to make sense of reality. As I read somewhere, or maybe I even thought of it myself but inspired by something else, you cannot stub your toe on a 1, so clearly, 1 is not "real" in a material sense. All numbers are invented by our minds. All numbers are "imaginary" in this sense.

Agreed absolutement! Here you and I strongly agree about the utility of CA deriving from the fact that it's just an abstract system for modeling IN a physical system. Sadly, though, there are many bright minds in physics, economics, and math which seem to miss this point such as Seth Lloyd, Steve Landsburg, and other mathematical physicists who seem to think that imaginary numbers are some non-scientific Platonic ideal which have mystical implications in life, a point that I lament nearly daily.

 

[aikismos]

If complex numbers are in a sense not real, then what do you make of the realness or imaginary-ness of matrices, for example?

I'd say that matrices with reals are in a sense "real", and those with complex elements aren't.

 

[aikismos]

I'd like to elaborate a bit on the point that complex numbers do have a nice graphical interpretation. Set $a + ib = (a,b)$, a point in the Argand plane. Multiplying by $i$ rotates the position vector for this point counterclockwise by 90$^{\circ}$. More generally, using the polar forms, multiplying two complex numbers gives

$z_1z_2 = r_1e^{i\theta_1}r_2e^{i\theta_2}= r_1r_2e^{i(\theta_1+\theta_2)}$,

So $z_1$ stretches $z_2$ by amount $r_1$ and rotates it by an amount $\theta_1$. There's a delightful book that shows the unfolding of complex analysis from this geometrical perspective, Needham's Visual Complex Analysis.

Fantastic! I totally am going to hunt that down. Only on the Internet can I find a EE and a fluid dynamicist to help me increase my ignorance. I'm familiar with the idea you're stating here, because Lakoff and Nunez have an appendix (https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From) which posits that this same technique is the whole point of Euler's $e^{\pi i} + 1 = 0$ in that it describes simple geometrical transformations on the Argand plane. (How else would you make sense numerically of raising an non-terminating, non-repeating decimal number to the product of another and the root of a number that doesn't exist and having it equal to the inverse of counting to one?) But to push back a little, could those same transformations be accomplished on the Cartesian plane with reals? I would proffer that transformation you offer is only a second example of what had already been accomplished in coordinate geometry, and so while complex analysis and the Fundamental Theorem of Algebra offer a new algebraic system, they don't really create anything new from an ontological standpoint from mapping an algebra onto a geometry.

 

[Geofleur]

I often have this same kind of argument with myself about complex numbers. On the one hand, considered as a vector space, $\mathbb{C}$ (with $\mathbb{R}$ as scalars) is linearly isomorphic to $\mathbb{R}^2$. On the other hand, considered as an algebra, multiplication of complex numbers seems to follow very naturally from the definition of $i$ and the normal rules of arithmetic, whereas the "multiplication" of points in $\mathbb{R}^2$ that one can define to achieve the same results does not at all seem natural.

 

[Edwin McCravy]

The term "imaginary" is a misnomer. As has been pointed out, x+yi is simply the ordered pair of real numbers, or the vector, or the point (x,y). The operations are addition, (x,y)+(w,z) = (x+w,y+z) and multiplication (x,y)(w,z) = (xw-yz,yw+xz) . It's easy to remember the multiplication operation if you know the double angle formulas for cosine and sine. That's because for (cos(θ),sin(θ)) (cos(Φ),sin(Φ)) on the unit circle, their product is (cos(θ+Φ),sin(θ+Φ)).

 

[zinq]

As has been mentioned above — if you believe in the real numbers and just want to add to them the ability to solve the quadratic equation

x² + 1 = 0​

and so we toss in the imaginary units ±i, then amazingly, the resulting number system is algebraically closed: Every polynomial (of degree ≥ 1) with coefficients in it has a root in this number system.

Instead suppose you're interested in number systems that are based on Euclidean space Rⁿ. We already have a nice way to add the points of Rⁿ: vector addition. Can we also find a way to multiply the points that satisfies the same nice axioms that the real numbers already do? (We indicate multiplication by adjacency, and ||x|| denotes the distance of a point from the origin; addition is the usual vector addition in Rⁿ.)These axioms for multiplication are as follows.

a) commutativity: For all x, y in Rⁿ

xy = yx​

b) associativity: For all x, y, z in Rⁿ

x(yz) = (xy)z​

c) distributivity: For all x, y, z in Rⁿ

x(y + z) = xy + xz​

d) composition: For all x, y in Rⁿ

||xy|| = ||x|| ||y||​

Theorem: The only dimensions n for which Rⁿ has a multiplication satisfying a), b), c), and d) are n = 1 (the real numbers) and n = 2 (the complex numbers). There are no other ways to define such multiplicative systems on Rⁿ for any n.

Note: If we decide to omit a) (commutativity), there is just one more n that works: n = 4, and the only multiplication that works on R⁴ is that of the quaternions. If we also decide to also omit axiom b) (associativity), then there is just one more n that works: n = 8, and the only multiplication that works on R⁸ is that of the octonions (a.k.a. Cayley numbers).

​

 

[aikismos]

The term "imaginary" is a misnomer. As has been pointed out, x+yi is simply the ordered pair of real numbers, or the vector, or the point (x,y). The operations are addition, (x,y)+(w,z) = (x+w,y+z) and multiplication (x,y)(w,z) = (xw-yz,yw+xz) . It's easy to remember the multiplication operation if you know the double angle formulas for cosine and sine. That's because for (cos(θ),sin(θ)) (cos(Φ),sin(Φ)) on the unit circle, their product is (cos(θ+Φ),sin(θ+Φ)).

Tell me, @Edwin McCravy, which integer multiplied by itself has a product of -1? :D

 

[symbolipoint]

Tell me, @Edwin McCravy, which integer multiplied by itself has a product of -1? :D

The imaginary unit, i, is not an integer. It is an "imaginary" number.

 

[aikismos]

The imaginary unit, i, is not an integer. It is an "imaginary" number.

Precisely! Which is exactly why it's not a misnomer as submitted by McCravy.

 

[symbolipoint]

Precisely! Which is exactly why it's not a misnomer as submitted by McCravy.

I looked, but I guess I did not look closely enough.

 

[Mark44]

The imaginary unit, i, is not an integer. It is an "imaginary" number.

However, i is a Gaussian integer. I'm not saying that i is an integer in the usual sense, though.

 

[sophiecentaur]

Since the earliest humans started to count, the gamut of 'familiar' numbers has been widening and widening. We quite take for granted that the 'threeness' of three cows corresponds with the 'threeness' of three oranges but using "three" to describe a group of three objects - sometimes looking the same and sometimes different (e.g. three animals) - is incredibly sophisticated. It would seem that collective descriptions of things was object-specific - e.g. a pair of shoes, a brave of pheasants and a couple of pints. Just using addition of integers was a massive step and we are, perhaps too familiar with it to appreciate just how clever it is. Moving on to rational numbers, negative numbers, irrational numbers etc etc has involved some massive intellectual leaps. In many ways, I would have said that the concept of a negative number (of cows?) is a pretty 'imaginary' idea, too.
So yes - giving a particular set of numbers the term 'real' is really a bit arbitrary. The word 'real' would perhaps have been better, replaced with 'familiar' but soon afterwards, another group of entities would have become 'familiar'. The etimology of Science and Maths leaves a lot to be desired and is a rich ground for argument.

 

[Geofleur]

So one person's "imaginary" is another's bread and butter :-)

 

[Ssnow]

''The shortest path between two truths in the real domain passes through the complex domain.'' J.Hadamard ;-)

 

[Orodruin]

I would go so far as to say that no mathematical construction is "real" to some extent. Mathematical constructions are abstract concepts which may or may not be useful when we describe what we observe. In this sense the real numbers are closer to us because they have a larger number of applications in situations which are familiar to us.

 

[Geofleur]

I suppose some could argue that the mathematical world exists independently of our concrete, everyday one, to be perceived only by "the mind's eye". Penrose talks about the infinite structure of the Mandelbrot set as an example - it's there whether anyone ever studies it or not. The same would then go for abstract structures and the logical relations between them. A less strong statement may be that at least the logical relations are real in the sense of being objectively verifiable.

 

[mathwonk]

he suckered you in with the question about matrices, i.e. the 2x2 matrices with real entries of form first row [a -b], second row [b a], are precisely isomorphic to the complex numbers, i.e. this matrix represents a+bi. So the matrix of form [a 0], [0 a] corresponds to the real number a, and then the matrix [0 -1], [1 0], when you square it gives [-1 0], [0 -1], i.e. the real number -1. so if you believe in real entry matrices you also have to believe in complex numbers.

 

[robphy]

Possibly interesting reading...
complex numbers as a certain-type of 2x2 matrix with real elements...
http://math.stackexchange.com/questions/570709/complex-numbers-and-2x2-matrices
http://www.sosmath.com/matrix/complex/complex.html

[oooh... mathwonk beat me by two minutes]

 

[lavinia]

What is intuitive seems to refer to what can be visually seen or imagined and circular motion, (rotation through an angle), is intuitive as is changing the length of a measuring rod that is pointed in some direction. These two operations, change of length and rotation through an angle, are the two components of complex multiplication. If one thinks of i as rotation through 90 degrees, i is just as intuitive as 1 which is rotation though zero degrees or -1 which is rotation through 180 degrees.

So if real numbers are intuitive or "real" so are the complex numbers.

What is less intuitive - at least to me - are the real numbers themselves. For instance,one can think of rational lengths as the total of all possible divisions of integer lengths (some number of measuring sticks laid side by side)using ruler and compass constructions. After a bit of thought one can see how one might go about making one that you are given, that is you could define a mathematical procedure for making it. This involves a high level of mathematical abstraction. It is not intuitive but rather deductive or at least requires deduction before a visual picture can be achieved if even possible.

If one extends the idea of ruler and compass constructions to the plane, one gets some irrational numbers and again with some thought one can imagine how one might go about constructing such a number e.g. the square root of 2. But again we are in a world of abstraction and mathematical procedures. What does it even mean to say that one can intuit the number sqrt( 2 + sqrt( 2 + sqrt(2))/27?

Gauss showed that all linear lengths that can be constructed with a ruler and a compass are iterated square roots but this does not come close to all of the real numbers. There are even lengths of regular polygons that can not be constructed in this way e.g. the length of the side of a regular septagon. Is this length intuitive?

Mathematicians conceived of the real numbers in various ways but not using algorithms or intuitions. Rather they came up with the idea of the metric completion of the rational numbers. But one can not intuit these numbers because there are uncountably infinitely many of them. There is no procedure even to generate any except a paltry countable subset.

There is a subfield of the real numbers for which there do exist algorithms, that is if a computer were allowed to run forever it would be able to express the number in terms of limits of rational numbers. These are called computable numbers. For instance the square root of 2 and π are both computable. One might say that in principle a number is "real" or worse "intuitive" if it is computable. There are only countably many computable numbers so this excludes almost all real numbers.

 

[sophiecentaur]

What is intuitive seems to refer to what can be visually seen or imagined and circular motion, (rotation through an angle), is intuitive as is changing the length of a measuring rod that is pointed in some direction.

It all depends upon who you are dealing with here. I think you should look at this.

 

[mathwonk]

lavinia of course is right. The difficult concept is the real numbers. the complex numbers are an almost trivial extension. but to go from rationals to reals, that is a tour de force. With reference to another thread asking for the greatest mathematicians, I think this raises the name of the seldom mentioned genius Eudoxus very high.

 

[xAxis]

lavinia of course is right. The difficult concept is the real numbers. the complex numbers are an almost trivial extension. but to go from rationals to reals, that is a tour de force. ...

I just can't see this. I would think real numbers were "intuitive" at least from the time of Zeno and his motion paradoxes. For Greeks mainly geometry was the mathematics. I would think once a Greek mathematician draw the number line with his ruler, some idea of continuum would (or should have been) there. Pythagoras knew that sqrt of 2 was not rational, and yet he could place that number on the line. It was easy for them to "see" that 4/3 is bigger than 5/4, which is bigger than 6/5 an so on. On the other hand they measured weights, volumes and other things and expressed them with the same numbers. So I think that first intuition about numbers was something like "Bigness."
Sure it's easy to visualise complex numbers by varying length and angle, but had not complex numbers been already invented, could you think of them by just looking at varying length and angle?
And let me be the first one in this insightful thread to say that I like the name 'imaginary' for imaginary numbers. I couldn't think of better name today, let alone in time they were invented.
Sophie great insight into the cause of great variety of terms for a group of objects. This morning in my backyard I saw a murder. (And later a glaring) :)

 

[mathwonk]

well look at it this way. a complex number is just two real numbers (a,b) with multiplication defined by (a,b).(c,d) = (ac-bd, ad+bc). That's it. You will notice that (0,1).(0,1) = (-1,0). So if you regard the subset of real numbers as the pairs of form (a,0), then (0,1) is the square root of (-1,0).

But to describe a real number it takes an infinite sequence of rational numbers. And it has to be a "Cauchy sequence", so a sequence {xn} of rartional numbers defines a real number if and only if for every e>0 there is a positive integer K such that for every n,m > K, we have |xn-xm| < e.

Then two Cauchy sequences {xn}, {yn} define the same real number if and only if for every e>0, there exists a positive integer K such that whenever n,m > K we have |xn-ym| < e.

So a real number is an equivalence class of Cauchy sequences of rationals. Then you have to define the ordering on them and prove the basic "least upper bound" property for them, that for every non empty, bounded above, set of reals, there exists a unique least upper bound. All this is very tedious and lengthy.

Isn't that more complicated? I.e. going from rationals to reals is very sophisticated, even if it was understood by the ancients to some extent, but going from reals to complexes is pretty simple, once you get over the psychological barrier of not wanting to believe in them. And exhibiting a complex square root of -1 is easy, we just wrote it down. Admittedly the fundamental property of complex numbers is not so easy, that every equation of positive degree has a complex root. But defining them is much easier than defining reals.

 

[FactChecker]

I'd like to elaborate a bit on the point that complex numbers do have a nice graphical interpretation. Set $a + ib = (a,b)$, a point in the Argand plane. Multiplying by $i$ rotates the position vector for this point counterclockwise by 90$^{\circ}$. More generally, using the polar forms, multiplying two complex numbers gives

$z_1z_2 = r_1e^{i\theta_1}r_2e^{i\theta_2}= r_1r_2e^{i(\theta_1+\theta_2)}$,

So $z_1$ stretches $z_2$ by amount $r_1$ and rotates it by an amount $\theta_1$. There's a delightful book that shows the unfolding of complex analysis from this geometrical perspective, Needham's Visual Complex Analysis.

I second the recommendation of Needham's book. The definition of multiplication that gives you rotation in the complex plane is very important. The usual XY plane does not have a basic rotation operation unless matrices are introduced.

 

[Allin]

The set of complex numbers should be taught at the primary /elementary school. The radical of integer -1, is an operator. Taken 4 times as a factor: the product is integer +1.

 

[chad kimsey]

If you just touch the fundamentals of mathematics, as well as any subject really, you just get to see the ice above the water. When in reality, the majority of the iceberg is under the water and you can't see it yet.

I don't know how far you have studied, but this same question stumped me all the way through high school and up to a point in college. Even at differential equations, I know complex numbers are necessary but why are they "imaginary"? And, what good is it really? So you solve a cubic equation, who cares right? Why would you ever need this crap??

The answer to that question is deeper at the fundamental understanding, the root of how things work and why. Going as far as engineering, creating things from scratch, modeling any world situation with blocks of differential equations impossible to solve by hand...understanding that level is awesome. However, it still never truly hit me until physical chemistry my senior year as a chemical engineer. This was by far the most outrageous, difficult class I ever had (and I'm really good with math/engineering way of thinking). Guess what popped up? Complex numbers. They don't make sense unless you go into quantum mechanics, where they are essential to understanding anything in the atomic world. Touching on the Schrödinger Equation, movement and behavior of electrons, and describing MO theory all use complex sets. It still blows my mind how "imaginary" numbers were created to describe things that can't normally happen fit perfectly in the world of the atom.

And one other thing bothers me, I try not to think about it but I can't help myself...is why is there two separate sets of rules, equations, physics, whatever you want to call it, that describe the nano and macro worlds? If everything is made of atoms, then why does it behave according to completely different laws? Why do properties and models fall apart, and exactly where does that happen? Gather enough mass, and you have gravity warping the space-time around the body. All that mass is a collection of endless atoms...but in the atomic world there is no gravitational force. Whats that all about? I love science, the more I learn the more I want to know

 

[Allin]

In any equation of one unknown value, of the first degree, the implicite coefficient 1 is identifiable to: (square root of -1) exponent 0. The first fudamental theorem of algebra should be superficially shown to the pupils at the last quadrimester of primary studies. In our world, rondness and circular movements are omnipresent and in diverse ways, thus the operator "square root of -1" is usefull and most practical for measuring and evaluating distance, center of mass, instant accelation, energy-productive impact, collapse critic point, etc. Generally speaking, wherever the infinitesimal (afterwards called "differential & integral") calculus is concerned, and its offspring mathematical subjects as ODEs (ordinary differential equations), PDEs, integral équations, etc, the complex set of numbers _imaginary cousins of the set of real numbers_, this operator is ubiquous i.e. most necessary in the calculations /computations. In a 3D cartesian coordinates system, the course of the extremity of an arrow of wrist watch, moreover the course of the lone electron tourning around the core of an atom of hydrogene, moreover the random-like traveling of an abandonned debris of satellite moving marygoround our planet Earth, could be computed and drawn with formulas containing the operator "square root of -1". ODEs are so difficult to compute /solve that even the most powerfull computers cannot succeed, just can make successive approximations (thence the reniewed methods of making approximations, called "numerical analysis", compulsory in the baccalaureate in engineering' (almost nothing to do with infinitesimal calculus analysis that is in the curriculum of major-level fundamentals mathemathics). The studies of engineering in physics, includes 3 to 6 credits in numerical analysis /methods.