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Lets say we have two complex numbers and the product of which is a real number.

The idea of multiplying two imaginary numbers to give a real number is something which I am unable to grasp.

Are the imaginary numbers not that imaginary? Do they have some sort of link to the real world? How can we interpret this answer?

Shah M Hasan
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  • Indeed! Not that imaginary. There was a whole debate over even calling them imaginary. The best interpretation for multiplying complex numbers (IMO) would be the graphical one. Do you know how to interpret the product of complex numbers graphically? – Mason Jul 21 '18 at 17:59
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    The name "imaginary" is a marketing disaster. E.g. see Gauss. – mvw Jul 21 '18 at 17:59
  • You can find many resources on this. Youtube graphical interpretation of multiplying complex numbers is surely to be rewarding. If you already know the graphical interpretation... maybe you can clarify your question. – Mason Jul 21 '18 at 18:01
  • @Mason the one that uses argand diagram? – Shah M Hasan Jul 21 '18 at 18:01
  • We add the angles and multiply the magnitudes. – Mason Jul 21 '18 at 18:01
  • I think this is a duplicate of this question: https://math.stackexchange.com/questions/6916/geometric-interpretation-of-the-multiplication-of-complex-numbers?rq=1... in the comments there is this youtube video by 3blue1brown which I think will be what you want: https://www.youtube.com/watch?v=F_0yfvm0UoU – Mason Jul 21 '18 at 18:04
  • thanks, can you post the link of any debate which you mentioned in the first comment? @Mason – Shah M Hasan Jul 21 '18 at 18:08
  • $1$ is a complex number. –  Jul 21 '18 at 18:43
  • @mvw: It's a semantic disaster too. Some people use it to mean "complex number", some to mean "complex number with imaginary part nonzero", and some to mean some to mean "complex number with real part zero" (some of these people probably require imaginary part nonzero too). And rarely does anyone care to disambiguate. –  Jul 21 '18 at 18:45
  • Well, for every complex number there exists a second complex number such that the product of the two is $1$, a real number. So this happens all the time, and you shouldn't be so surprised. If you want to think of real and complex numbers "concretely", then just say that real numbers do geometry on the real line, and complex numbers do geometry on the real plane. – rschwieb Jul 21 '18 at 19:21
  • Please multiply $(a+bi)(a-bi)$ – Lubin Jul 21 '18 at 19:29
  • What assumptions are you using that does not allow you to understand that $i$ times $i$ is $-1$? – Somos Jul 21 '18 at 20:24
  • @ShahMHasan. Sure. I thought the answer by Joe Gobbini in quora post is helpful: https://www.quora.com/Why-are-imaginary-numbers-called-imaginary-numbers-What-makes-them-%E2%80%9Cimaginary%E2%80%9D. TLDR: Looks like less of a debate and more like a couple of different terms used. Descartes and then Euler's use of the expression "imaginary" helps to solidify these into the language we currently use. – Mason Jul 22 '18 at 02:12

1 Answers1

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Updated:

Are the imaginary numbers not that imaginary? Do they have some sort of link to the real numbers?

Each complex number $z = x + i y$ ($x, y \in \mathbb{R})$ can be interpreted as a point $(x, y)$ in the Gaussian plane.

Pure imaginary numbers $z = i y$ are lying on the $y$-axis of that plane.
Pure real numbers $z = x$ are lying on the $x$-axis of that plane.

Multiplication of complex numbers is convenient in polar coordinates: $$ z_1 z_2 = r_1 e^{i \phi_1} r_2 e^{i \phi_2} = r_1 r_2 e^{i(\phi_1 + \phi_2)} $$ so it can be interpreted as changing the radius and the angle.

For two pure imaginary numbers one gets on of these cases $$ z_1 z_2 = r_1 e^{i (\pm\pi/2)} r_2 e^{i (\pm \pi/2)} = \pm r_1 r_2 $$ which all happen to lie on the $x$-axis, which consist of the real numbers.

mvw
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  • i know about the significance of complex numbers in sciences, what i really wanted to ask was that do the imaginary numbers have some sort of link to real numbers? – Shah M Hasan Jul 21 '18 at 18:11
  • Saying merely that they "simplify mathematical modelling" is an underestimate. Complex numbers are in fact indispensable in Quantum mechanics, for example, without which you would not be able to write your answer. – uniquesolution Jul 21 '18 at 18:14
  • @ShahMHasan You should update your title "physical interpretation" and "real world" if you are not asking to the relationship to physics, but to real numbers. – mvw Jul 21 '18 at 18:28
  • @uniquesolution If one would write everything with $\mathbb{R}^2$ vectors instead, using the proper operations, it would work as well, but be more cumbersome. The same holds for quaternions in robotics and CG. – mvw Jul 21 '18 at 18:42