Mathematician: Imaginary numbers arise quite naturally when you start asking certain basic mathematical questions. Probably the best example is the following: Once we know how to multiply and add, we might ask ourselves “are there any numbers x that satisfy the equation 
?” The answer is yes, 
and 
both satisfy it. If we try some more equations of this form, we might at some point ask the similar question, are there any numbers x that satisfy the equation 
? None of the ordinary (real) numbers satisfy this property. If we are feeling lazy we can stop there and say that this equation can’t be satisfied. But, if we are feeling creative, we might assume for a moment that there is a (special) number that satisfies this equation, (we’ll call it the number 
), and then see if we can derive its properties. By definition, of course, our number satisfies 
, but we can ask other questions about this bizarre number, such as “what is 
for any positive integer n?” Some simple calculations show us that 
is either equal to , 
, 
, or 
depending on the value of n. We will now say that is an “imaginary number”, as is any multiple of , such as 
and 
. We can now also introduce what we are going to call “complex numbers”, which are formed by the sum of a real and an imaginary number, such as 
, 
, 
and 
. It is not too hard to show that complex numbers yield a consistent theory, in the sense that when you add them, subtract them, multiply them, or divide them, you get a complex number back as the result of the operation. Furthermore, it turns out that all polynomial equations (which includes the equations 
and 
that we considered before, as well as others like 
) have at least one complex number solution (notice that real numbers are also complex numbers because, for example, we can write 
as 
). That means that no other weird types of numbers need to be introduced in order to find solutions to these equations, so complex numbers are “enough” to find every polynomial equation solution.
Okay, so maybe imaginary and complex numbers make sense to introduce and lead to a reasonable theory, but how could they possibly be useful? After all, in the real world we have real numbers of things (e.g. 3 frogs) and real amounts (16.2 dollars), not imaginary numbers of things. As it turns out, complex numbers are fantastically, staggeringly useful. This is especially true once we allow ourselves to start plugging complex numbers into functions (like 
, 
, 
, etc.) , and see what output they produce. For example, imaginary numbers give us a useful and surprising link between the exponential function and the sine and cosine functions, in Euler’s beautiful formula:
We see this funny object, 
surface again in an essential way in the subject of Fourier Analysis, which finds applications in the study of heat flow, signal processing, music filtering, data compression, image processing, and many other areas (in fact, one of the most referenced applied math papers of all time relates to how to quickly approximate a sum involving on a computer in order to do a discrete version of Fourier Analysis). Complex numbers pop up again in the theory of Taylor Series, one of the most used mathematical tools ever invented. One way to think about this is that Taylor Series provide a method for locally approximating a function using polynomials, and the region in which such approximation succeeds depends in a crucial way on the function’s behavior when it’s thought of a function of complex rather than real numbers. Complex numbers make an important appearance yet again in the theory of matrices. When we have a matrix (which is like a grid of numbers) that has only real numbers, and we want to split it apart into multiple matrices (via an eigen vector decomposition, let’s say) we discover that in general the matrices we divide it into may have complex (non-real) entries! What’s more, questions about prime numbers, solutions to problems in electrical engineering and the differential equations in quantum mechanics are deeply connected to complex numbers. The list of ways in which complex numbers appear (in a fundamental way) throughout mathematics is truly enormous.
Fine, so imaginary and complex numbers arise naturally and are extremely useful, but do they really exist? Well, this raises deeper (or should I say, different) questions, like whether numbers in general exist. Sure, numbers (complex or otherwise) exist as concepts in our brains, but no numbers ever appear in the physical world. We can see 3 cats, or the symbol “3” scrawled on a sign, but we will never see the actual number 3. Imaginary numbers, like real numbers, are simply ideas without any physical existence. They are both very useful (though with real numbers, it is much more obvious why that is so). But it is hard to see how one would (convincingly) argue that real numbers actually exist while imaginary numbers do not.
Some commentators have taken my argument above to imply that I think numbers (including imaginary numbers) actually exist (perhaps in some platonic sense). However, as was pointed out, this hinges very much on your definition of “exist”. Although the idea of numbers is very natural (you might almost say “obvious”), and probably would have been invented by almost any highly intelligent beings that may have happened to inhabit our planet, they are indeed still simply just an idea that we created. They lack physical existence, but “exist” in our minds as much as any ideas do.
Nice clear explanation! I would love to hear the Mathematician explain why Euler’s formula happens to be true… why should there be such a simple relationship between those important constants? It baffles me.
Also, I think the question of “whether numbers in general exist” isn’t actually all that deep. It just comes down to what you mean by “exist.” When I say something exists I basically mean that it is physically present in the world. (So then obviously “3” does not exist in that sense.) Is this more than a semantic question?
The Physicist wrote an article on this blog about Euler’s equation, which proves it. It doesn’t necessarily make it clear intuitively why the equation is true though. This is one of those very important and fundamental theorems, like the Central Limit Theorem, for which it is hard to come up with a basic, intuitive, satisfying explanation.
I absolutely agree that the existence of numbers really just hinges on what you mean by “exist”. That being said, I think that some people feel that real numbers exist or are truly “real” whereas imaginary numbers do not really exist. I don’t agree with this perspective, which is why I tried to take a moment to address it.
The discussion about existence and being real seems to go towards the direction of Plato’s theory of forms…
It’s nice to see that you’re trying to argue that imaginary numbers arise naturally, Mathematician. I also think about how to explain this sometimes, and if I were to do it, the part that differs from your way is I’d start all the way from the top of the chain:
Natural numbers arose naturally from, well, nature, in some sense
Integers arise naturally from natural numbers
Rational numbers arise naturally from integers
Irrational/real numbers arise naturally from rational numbers
…Finally imaginary numbers arise from reals, and since it is extended from the reals in a natural way much like the previous steps (and which you’ve explained), it could potentially be even more convincing.
The reply by Mathematician is anything but clear. Frankly, it is nonsense.
Complex numbers don’t exist. About the best explanation is that a vector (a,b) is also represented as a + bi. But just as (a,b) is not a number, neither is a + bi.
To define i^2 = -1 is mathematically incorrect. The reason for this is that the operation of “squaring” is only valid for real magnitudes. Sqrt(-1) is not a real magnitude. In fact, it is not even a magnitude.
In truth, complex number theory is not required at all in mathematics. All the results involving complex numbers are possible because of the trigonometric properties of sine and cosine.
The Euler equation is a load of rubbish, that is, e^(i * pi) = -1.
To wit, if one assumes this is true, then ln -1 = i * pi where ln is the natural logarithm. As you can immediately tell, this is nonsense.
Julia? Still a nice clear explanation? If you believe this rot, then you might believe anything. I suggest you think for yourself. Good luck!
Fractals. You use them in fractals. Those amazing things.
“The Euler equation is a load of rubbish, that is, e^(i * pi) = -1.
To wit, if one assumes this is true, then ln -1 = i * pi where ln is the natural logarithm. As you can immediately tell, this is nonsense.”
Actually John….
Now I know I can immediately tell that ln -1= i*pi.
…does the “magnitude” -1 exist? (“Magnitude” as used in John’s post). I have a very hard time conceiving of “anti miles” or “backwards time”. But, since I think we all agree that it’s cool to use negative numbers in working with time or distance I propose that we can be equally cool using imaginary numbers … if they get us somewhere mathematically, which they do. So, IMHO, the point isn’t the “magnitudes” of imaginary numbers per se, but rather the demonstrated fact that they can be integrated into our concept of real numbers … which gives us reliable results in pursuit of knowledge that concretely serves us. The fact that I’m still waiting for the magic day when it all hits me in one gloriously perfect “ah ha” moment is not reasonable cause to conclude that imaginary numbers are bunk.
Nope!
Magnitudes are specifically defined to be non-negative numbers because, while the freedom of complex numbers is useful, the confinement of positive reals is also useful.
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Very nice explanation of complex number!
As you give example of real number 3, “3 cats”, would you give example of any imaginary number?
@Andy
We usually think of numbers in terms of counting things (which is why the natural numbers are sometimes called “the counting numbers”), but of course you can’t have “i” things.
Using the example of angles and phases, you’ll sometimes hear electrical engineers talking about the voltage being i times current. What they mean in this case is that the (sine wave representing) voltage in an AC circuit is lagging behind the current by 90°. Turns out that it’s easier to do the math entirely within the framework of complex numbers, so translating back and forth is often just not done.
What is O divided by O? On my calculator, it always says error?
I can’t help myself here… arguing with the Euler function / natural base for exponentiation is futile.
@John, I am curious how you explain the relationships between the trigonometric functions and Pi without the Euler function… or radians for that matter. if the Euler function was wrong, if imaginary numbers were unnecessary, can you prove ALL the properties of Pi and trigonometric and hyperbolic functions?
furthermore, your arguments about magnitude are misplaced – magnitudes can indeed be only positive. however your assumption that “squaring is only valid for positive magnitudes” is baseless (forgive the pun).
if i plot y = x^2 + 1, it clearly does not have a real-numbered solution for y=0. however it does still have two root solutions, because it is a polynomial of second rank.
y = ax^2 + bx + c = 0
a = 1; b = 0; c = 1;
x = (-b +- sqrt(b^2 – 4ac) / (2a)
x = +- sqrt(-4) / 2
x = +- sqrt(-1) * sqrt(4) / 2
x = +- sqrt(-1) * 2 / 2
x = sqrt(-1) or x = -sqrt(-1)
(equivalently) x = i or x = -i
and to show this is the case:
(i)^2 + 1 = sqrt(-1)^2 + 1 = -1 + 1 = 0
(-i)^2 + 1 = (-1 * sqrt(-1))^2 + 1 = (-1)^2 * sqrt(-1)^2 + 1 = 1 * -1 + 1 = 0
therefore i could say that:
x^2 + 1 = ( x – sqrt(-1) )( x + sqrt(-1) )
the way i like to think of complex numbers is that they are an algebraic vector quantity. they have magnitude, they have direction, and the real numbers (like imaginary) are special cases. in fact you would be hard-pressed to find anything in this universe that is one-dimensional. singularities? illusions due to poor coordinate choice. big bang? possible illusion / artifact we are observing due to incompleteness of relativity and QM making consistent, if not correct predictions. quantum numbers? integers or half-integers.
arguably every quantity in physics is at least dual valued (even probability amplitudes are considered as the squares of magnitude, time and space are not separate ‘things’, energy is broken up into rest mass and momentum, etc), and there is no reason to think numbers themselves are an exception, even if we cannot fully appreciate or extrapolate the meaning of such things to our daily-world macro scales.
I was thinking about the imaginary numbers and I can’t agree with the way they are defined. We have positive numbers and write + before them. For negative number we write -. It is just a property of the number that it can be positive or negative (or zero ), but never both. In the same way I think i has to denote a property of the number just as r. Saying i*i=-1 is as saying +*+=+ or -*-=+. It is true but we multiply the numbers and +, -, i and r are only properties of these numbers.
In real numbers we know that positive*positive=positive. Positive*negative=negative. Negative*negative=positive. With the imaginary numbers we have: positive*positive=negative; negative*negative=negative; positive*negative=positive. We also know that real number*real number=real number; imaginary number*imaginary number=real number; real number*imaginary number=imaginary number. It is just like + and – but at different level. At the next level we’ll want imaginary*imaginary=imaginary and so on, we’ll add two more dimensions, two more letters to denote the new properties and will have 8 types of numbers. Now they are four: positive real, positive imaginary, negative real and negative imginary (and zero of course). Complex numbers are just a combination of real and imaginary numbers.
Looking for imaginary numbers in the real world we have to look for numbers satisfying: positive*positive=negative etc.
There was a time when irrational numbers where not accepted as numbers, until it was proved that they exist (as the length of hypotenuse of an isosceles right triangle with both arms of unit length, or there exists a length equal to sqrt 2, ,equivalently there exists a point on the number line for sqrt 2).do imaginary exists in the sense that there exists a length equal to i(sqrt -1)? Because the proof that sqrt 2 exists lies in showing that such a length exists, on the same lines can u show that i (sqrt-1) exists.
imaginary numbers are two dimensional number, what could be the two dimension in real life, it is the magnitude and time, magnitude is one dimensional and time is another dimensional, magnitude is different for each time, for example for a sine wave magnitude at time 0 is 0 where as time at 2*pi*(f=1)*(t=1/4) = pi/2 (2*pi is the circumference of a circle, frequency is one rotation per second, time 1/4 indicates quarter second) magnitude at this time is +1. what if we want to know what could be the magnitude some time at 1/2 second which is future compared to 1/4 second, in order to know the future value of the sine wave at 1/2 second(remember we now at 1/4 second) we have no clue, we need to wait for another 1/4 second and see the magnitude at that exact time of 1/2 second) what if we could start the wave 1/4 second earlier we can see the value of 1/2 second at 1/4 second it self, also if we want to know the magnitude at 0 sec, when we are at 1/4 second we need to delay the wave to 1/4 seconds to see the value at 0 second, to know the future and past magnitude of sine wave at current time we need to shift the wave correspondingly to future or to past, this phase shift is achieved by imaginary numbers or to say for any value of phase shift we need to use complex numbers (real + imajinary), imaginary values does not exist because they correspond to past or future, since past or future does not exist in present, so as imaginary numbers do not exist in current time but they exist either in past or in future to current time. -1 = i * i, this could be explained like if we want to have a magnitude of -1 at 1/2 second sqrt(-1) indicates what is the magnitude at 1/4 second and sqrt(-1) * sqrt(-1) indicates what is the magnitude at 0th second. all these concepts only works in sine and cosine perfectly, since it works in sine and cosine perfectly and any time domain signal can be represented as sum of sines and cosines using Fourier transform, it works for all two dimensional signals.
@Stan, Nice comment there.
Funny enough, If we had some number system/groups over “positive no * positive no= negative no.” could we see new phenomenons, like explanation why Matter dominates while we see little anti-matter. (https://en.wikipedia.org/wiki/CP_violation & https://en.wikipedia.org/wiki/Baryon_asymmetry)??
Imaginary, real, integers, whatever…doesn’t matter…it’s all conceptual. A family can’t have 2.5 children, you can never divide a pizza into 3 equal slices, etc…you have to use (haha) your imagination to use numbers. Producing “real world” results using numbers is conceptualization too. Does 1+1=2? Not in physical reality. We have to imagine that the two objects we are adding together are exactly the same (never the case) in order to make the equation work.
Our imagination allows us to use constructs (such as mathematics) to define objective reality. Cool huh?
Would it be a stretch to say that the imaginary number really is just a complex variable? I do think it exists just not in the same way other numbers do where they have a definite value.