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For explaining and showing positive integers in real life, we can say something like: 1 = An apple; 2 = two apples, etc.

For explaining and showing the number 0 in real life, we can say something like: No money in your bank account means you have 0 dollars.

For explaining and showing negative integers in real life, we can say something like: -1 = You have to give back 1 apple to someone; Or: you are 1 dollars in debt to someone.

For explaining and showing fractions or rational numbers in real life, we can say something like: You have an apple and you cut it in 2; Each slice is 1/2 of the whole apple before you cut it in two.

How can we explain, show and represent complex numbers in real life using basic, simple, apparent in real life and preferably visual examples on the same level of easiness to understand as the other examples I have given in this question?

THIS IS NOT A HOMEWORK QUESTION! I just want to understand complex numbers and not memorize some a + ib! And in order for me to understand them, I need VERY simple, REAL LIFE examples and explanations! Nothing abstract and/or advanced like Electrical Engineering concepts! Simple, easy to understand and apparent in real life only please.

student signup
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  • Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on [meta], or in [chat]. Comments continuing discussion may be removed. – Shaun Feb 11 '25 at 11:33
  • See https://math.stackexchange.com/questions/285520/where-exactly-are-complex-numbers-used-in-the-real-world. – Integreek Feb 11 '25 at 17:42
  • @RakshithPL Have you fully read my question mate? I have explicitly said that I want VERY simple and basic examples; As basic and simple as saying "1 = An apple" and noting advanced like electrical engineering concepts! The question you referred me to and its answers are not simple. Honestly, I should just delete my question and account all together. – student signup Feb 12 '25 at 03:34
  • @studentsignup sorry, my intention wasn't to discourage you! Your question is a really good one and it should be reopened soon(I've voted, there are 3 reopen votes right now so 2 more are needed) and will be well-received. Please don't delete your question or account. I referred that question since I thought that some answers like the second most upvoted answer and some that talked about the relation between exp and trig functions were fairly easy to understand. Sorry if it was too overwhelming for you. Have you just started learning complex numbers? – Integreek Feb 12 '25 at 05:11
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    @RakshithPL Thank you for your kind words and I apologize for my sudden outburst and rudeness; I'm just frustrated for my question being closed for dumb reasons. No, I've learned (more like memorized) them but never really understood what they are and what they represent; I refuse to just memorize and not understand and that's why I posted this question. – student signup Feb 13 '25 at 03:19

2 Answers2

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The easiest way to visualize complex numbers is by imagining them as rotations. If $(x,y)$ is a point in the plane, then you can represent that as $x+iy$. If you want to rotate that point around the origin $(0,0)$ by an angle, say, $45$ degrees, you find the sine and cosine of 45 degrees, which are both equal to $\frac{1}{\sqrt{2}}$ and then if you multiply $(x+iy)$ by $\frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}}$, this will replace $(x+iy)$ by a new complex number $a+ib$. The new coordinates are the location of the point after rotation.

Nicolas Bourbaki
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    This is a very good example but i still need more detail: Is i the same as the Z axis in a 3D coordinate? Is i a new axis that crosses the Cartesian coordinate system in the point (x=0, y=0)? How does x + iy show a point in the Cartesian coordinate system? – student signup Feb 10 '25 at 06:56
  • Actually viewing the trig identitys $\sin(a+b)= \sin a\cos b + \sin b\cos a$ and $\cos (a+b)= \cos a\cos b -\sin a\sin b$ work *beatifully as complex number. Any non-origin point in a plain $(u, w)$ can be translation to and (angle, distance) where $d=$ the distance of the point to the origin ($\sqrt{x^2+y^2}$) and $a$ is the angle from the x-axis,the origin, ant the point. Then we have $(u,w)=(d\cos a, d\sin a)$. If we view this as a complex number $u+wi = d\cos a+di\sin a $ and if we have complex multiplication as $(u+wi)(j+ki)=uj + uki + wji + wki^2=(uj-wk)+ (uk+wj)i$ then..... – fleablood Feb 17 '25 at 05:37
  • If you have two numbers $d \cos \theta+d\sin \theta i$ and $r\cos \eta+r\sin\eta i$ then multiplying these out we get $dr (\cos\theta \cos\eta-\sin \theta\sin \eta) + dr(\cos\theta \sin \eta + \cos \eta\sin \theta)i= dr(\cos(\theta +\eta)+\sin(\theta + \eta)i)$. This tells us multiplying complex numbers turn int simply adding the angles of rotation and multiplying the real positive values of their distances. Kind of awesome, really. – fleablood Feb 17 '25 at 05:44
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Please read in a textbook of Electrical engineering (often the symbol used $j=i$ for the imaginary part) about some particular examples of complex impedance, the following is copied from the internet.

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Narasimham
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    Thank you for your reply; As I've said, these examples of complex numbers in electrical engineering are too advanced and complicated for me to understand; I need a much more simple example, on the same level of examples that i have given in my question. – student signup Feb 10 '25 at 06:38