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In mathematics, how does something like complex numbers apply to the real world? Why do complex numbers exist? How can we comprehend addition of complex numbers? For example, addition of natural numbers can be understood as putting together two apples and two oranges makes four fruits. How can we apply this thinking to complex numbers?

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That doesn't explain real number addition, only natural number addition. What are real numbers? Do they "exist" in any sense? –  Thomas Andrews Dec 25 '13 at 3:24
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The algebra of complex numbers can be thought of as an algebra for rotations and scalings. For every complex number, we can find an angle and length (i.e., put it in polar coordinates). The multiplication of complex numbers adds the angles and multiplies the lengths. –  nomen Dec 25 '13 at 3:30
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But one of the mysteries is that this idea of scaling and rotations and the idea of translation (addition) work together as well as they do. @nomen –  Thomas Andrews Dec 25 '13 at 4:52
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@okarin: it is because the exponential function is periodic on circles centered at $(0,0)$, as the angle varies. In other words, the exponential function is the sum of a sine function and a cosine function, and complex algebra corresponds to combining waves. en.wikipedia.org/wiki/Euler's_formula –  nomen Dec 25 '13 at 5:15
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How can we comprehend addition of complex numbers? We can understand it the same we understand natural addition. We have 2 tomatoes and 3 potatoes, add them up and we get... 3 potatoes and 2 tomatoes. Potatoes are vegetables, Tomatoes are fruits but they both sorta feel like vegetables and both are generally things we eat as part of a healthy diet but since they are of different types you can't technically add them together under one specification (here we're assuming that such a specification does not exist). Same with complex numbers real and imaginary are different types of numbers. –  Nick Dec 25 '13 at 6:02
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Numbers count $(\mathbb{N})$ and measure $(\mathbb{R})$. Yet complex $(\mathbb{C})$ or imaginary $(i\,\mathbb{R})$ numbers do neither. So what good are they anyway ? $($Is this what you're asking ?$)$ Well, let's just say that engineering as we know it would be a whole lot more difficult to either understand or apply without their help, as would mechanics and computer graphics, or even modern physics, for that matter. Many of the practical problems that arise in these various fields often require solving contour integrals, which in many instances simply cannot be done without the use of complex integration. Their trigonometric applications range from geographic location and cartographic projections to signal processing and other branches of electrical engineering. Basically, all radio or acoustic signals, as well as electricity itself, are nothing else sinusoidal waveforms (and those that aren't can easily be decomposed into such), and whose study would become very tedious really fast, were it not for Euler's relationship, $e^{ix}=\sin x+i\cdot\cos x$.

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TLDR –  Lucian Dec 25 '13 at 6:37
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"how does something like complex numbers apply to the real world?" Type $$\rm circuits\ and\ complex\ numbers$$ into Google, and you will find that computations of currents in electrical circuits are done using complex numbers.

"Why do complex numbers exist?" The equation $x^3-4x+1=0$ has three real solutions, but it is impossible to express them (in terms of arithmetical operations and square roots and cube roots) without using complex numbers. So complex numbers come up naturally in finding real solutions of real equations. Key search phrase: casus irreducibilis.

"How can we comprehend addition of complex numbers?" What is there to comprehend? If we want to add $2+3\sqrt{-1}$ to $4+5\sqrt{-1}$, we do $2+4=6$ and $3+5=8$ to get the answer $6+8\sqrt{-1}$. Just like adding 2 apples and 3 unicorns to 4 apples and 5 unicorns, getting 6 apples and 8 unicorns.

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There are lots of ways to develop intuition with complex numbers and they've been mentioned above, so I'll try to say something different. I don't think it matters whether or not complex numbers "exist", it simply matters that they are very useful and therefore worth studying. In fact, its hard to say whether any numbers really exist. For example, what is the number $5$? There isn't a tree somewhere in Iceland where $5$'s grow and are shipped all over the world for commercial use. Instead, we have a concept of $5$ that has been universally agreed upon in some sense and can be represented by $5$ of something, for example the following represents $5$ in some sense: $* * * * *$.

My point is, to me natural numbers don't even "exist" in any meaningful sense. They just turn out to be concepts that seem to occur in nature and we've come up with a useful way to characterize them. You could even argue that complex numbers occur "in nature" in the sense that we can use them to describe certain physical laws. However, that is less satisfying to me, since there are plenty of mathematical constructions with no physical interpretation. Instead, I like to think that complex numbers are simply a mathematical construction that turn out to be very useful in solving different types of problems.

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