system
Given the following post, respond as if you're commenting on Reddit:
human
Is there a real world analogy for imaginary numbers?

Just like how you can't have -1 oranges, but we can analogise at as you owing 1 orange, are there any real world analogies for imaginary numbers?
gpt
For me the most intuitive is in terms of 2d space.

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In the context of the number line, let's assume "1 metre" is the distance between 1 and 0 and let's arbitrarily assign "East" to be the direction in which positive numbers go away from zero. Think of any real number as being that far away from your origin (zero) due east. (So for example, the number 10 is 10 metres east of zero.)

Negative numbers can be thought of as distances due west. So -1 is 1 metre west of zero.

So now, on this line going west to east, each real number corresponds to a point on the line at some distance from a point we call 0. 

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Addition (the operator) corresponds to walking on this line due east. So 2+3 (=0+2+3) is equivalent to you starting at 0, walking two metres east and then walking another 3 metres east to end up at 5 (the point 5 metres east of zero).  
Now, subtraction which is just addition of a negative number, corresponds to walking along this line due west. so 2-3 (=0+2+(-3)) would be equivalent to walking 2 metres east and then 3 metres west to end up at -1 (1 metre west of 0).

Now the imaginary number i corresponds to the point 1 metre North of zero. This number cannot be reached by any combination of walking east or west, so it is not part of the group of real numbers, which all exist on the line through zero going west to east.

Addition of any number by i corresponds to walking north by 1 metre.

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Where this begins to get really interesting and where it ties in with everything else is when we look at multiplication.

Now, multiplication by a positive real number n corresponds to stretching space by a scaling factor of that number (i.e. what was 1 metre before becomes n metres now in all directions). So 2\*3 corresponds to where the old 2 (2 metres due east) ends up after stretching space such that what was 1 metre earlier (in any direction) now becomes 3 metres (in the same direction). So the point which was 2 metres due east from zero earlier now ends up 6 metres due east yielding 6.   
Multiplying by 1 does nothing \[space is stretched so what was 1 metre earlier is 1 metre now, so nothing changes\].

    
Multiplying by i corresponds to rotating space around zero (keeping all distances constant) by 90 degrees so that what was 1 metre due East earlier is now 1 metre due North. This concept fits in very nicely with what we know about i. 

1\*i corresponds to finding where the point 1 (1 metre from zero due east) ends up after rotating space by 90 degrees. This is the point i (1 metre North of zero) => 1 times i is i.  
i\*i corresponds to where the old i (1 metre due north) ends up after rotating space by 90 degrees. (This point will now end up 1 metre west of zero). => i times i is -1 !

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Now within this framework, it is easy to see what complex numbers correspond to and what various arithmetic operators do.  
Any complex number is expressed as a real part (a) and an imaginary part (ib), which basically means that to get to it, you need to walk "a" metres East \[if a is negative, this corresponds walking some distance West\] and then "b" metres North.  
Adding complex numbers involves adding the real parts (figuring out what the net amount along the East-West line you need to walk) and the complex parts (figuring out what the net amount along the North-South line you need to walk) to get to the new point.  
Multiplying by a negative real number involves stretching space by a scaling factor of the absolute value of the number and then rotating space by 180 degrees.  
Multiplying by an imaginary number ib involves stretching space by scaling factor b and rotating space by 90 degrees.