What is the use/need of complex numbers in real life if they are imaginary?
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In my mechanical engineering degree we used them in electrical engineering, and in fluid dynamics. They make some calculations a lot easier.
A fuller explanation here, but not suitable for 5 year olds!
www.theguardian.com/notesandqueries/query/0,5753,-18864,00.html
Complex numbers are often used for audio analysis and quantum physics, because they’re good at describing circular movements and waves.
Also they’re used to calculate rotations in video games and computer graphics. (in this case an even more advanced version called *quaternions* are used, with four total “dimensions” of *real, i, j, k*)
They make calculating and understanding things easier. But you don’t need them, it’s possible to reformat maths/physics to not use them.
Edit:
Sabine’s video on this is a useful insight into the debate on whether imaginary numbers are real or needed.
They are just as “imaginary” as negative numbers are. You can’t have negative sheep. If you put three of them in a pen, it’s entirely preposterous to think that you could take five away from there.
Negative numbers just happen to be very useful for representing amounts which can fluctuate between two states. For example, credit and debit. If you deposit five gold pieces to a bank, your balance says “5” which represents the banker owing that much to you. If you go there and withdraw seven gold pieces, the balance says “-2” and represents you owing that much to the bank. At no point do any sort of “anti-gold pieces” actually appear.
Complex numbers are the same. They’re a very useful tool for representing things which don’t flip between two directions, but cycle through four of them. As a tool, it doesn’t really have day-to-day applications to a layperson, but they’re crucial for solving a wide variety of math problems which, for example, let your cellphone process signals.
They’re not so much “needed” as “helpful.” A lot of engineering problems, (if I recall from university, particularly electrical and fluid) are greatly simplified when using them.
A lot of useful things in real life are imaginary. Money is just a paper, but it greatly simplifies trade, just one example.
Just to add, “imaginary” numbers are just as real as the “real” numbers. Past mathematicians just called it “imaginary number” as a placeholder because they did not know what it was, but unfortunately the name stuck.
They are no more or less imaginary than regular (“real”) numbers, that was just a bad naming choice. All that it means is you can’t mix them with regular numbers. Like you can’t add 3 real + 2 imaginary = 5 something. 3 + 2i must always remain separate components. The real number and the complex number are in different mathematical dimensions.
Because of this property, complex numbers are useful when calculating two properties that are mathematically related, but cannot be substituted for one another – like electricity and magnetism. You can have 5 electricity and 3 magnetism from a wire (which could be represented as 5+3i), but saying you have 8 electromagnetism is invalid. You could also just write the maths with electricity and magnetism as separate numbers, but it hides the fact that when one changes so does the other.
In electricity calculations, it is possible to depict capacitors and coils to have imaginary and negative imaginary resistances. This is called impedance.
A system could have an impedance of 13+4j Ohm which means it is somewhat capacitive. (in electricity we use j instead of i to avoid confusion with current, which is also depicted as i).
A capacitive or inductive system will also modify the relationship between current and potential, which can also be depicted as an imaginary number.
It’s a long time since I did this, but that’s the gist of it. It makes electrical calculations significantly easier by using complex numbers instead of regular numbers.
When you start with solving quadratic equations in high school, you are given, as examples specially selected equations that have solutions among the real numbers. But when you go out in the real world, with equations derived from measurements of real things, that is no longer the case. And more so as equations get more complex – Almost any time you try to do anything, you end up with negative numbers under square root signs. If you were forced to stop there, you wouldn’t be able to find out much about the world.
So instead, we ‘imagine’ that √-1 has a value, just one we don’t know, call it *i*, and keep on going with maths. And when we do, we discover many things.
The device that you are using right now, applies complex numbers all the time. Computer graphics take advantage of [quaternions](https://en.wikipedia.org/wiki/Quaternion) which are an extension of complex numbers. Similarly, those are also used in image processing.
Any device that handles sound, or transmits data over the network (radio, phones, wifi) need complex numbers. Signal processing (Fourier transformation) relies a lot on complex numbers.
I wish we could somehow rename them and not call them imaginary ever again.
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