What is the use/need of complex numbers in real life if they are imaginary?
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Let me try to put it in simple terms. You have a straight path. Forward is one direction that you call normal or positive. To go backwards, you call it negative and you basically turn 180º and go the opposite directio.
If we’re talking numbers, if you add 5 meters you go forward, and if you add negative 5 what you’re doing is adding 5 but 180º rotated, so negative. You are walking 5 meters but backwards. So you can think of the – sign as short for 180º degrees rotated.
This is consistent. If you apply – twice it is 180+180 = 360=~0 so its back to the same direction. That would be subtracting negative 5 -> 10- (-5)=10+5=15
Ok, now where it is interesting is i=square root of -1 means i*i = -1. If you do it 4 times, (i*i) times (i*i)= – times – = + You just did a 360. Funny that. Replace i for 90 and you got 90+90+90+90=360=~0
So you can say i is a 90º rotation just as you can say – is a 180º rotation, and it rotates stuff 90º. Instead of going back and forward, you’re going sideways.
This makes it funny in that you can think of – (minus) as ii (i times i) or 180 as 2×90
Complex numbers are complex because now you can either think of it as describing stuff on a plane, or rotated. They work really well to describe stuff that is rotating or cycling.
You can even write them as an absolute value with a rotation. 5 rotated 90 is 5*i, 5 rotated 180 is minus 5.
Are they imaginary? Like all other numbers, they are symbols with which you describe something -> and that something may be real, or another mathematician’s dream.
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