Here’s how to understand negative numbers that makes understanding complex numbers even easier.
First, realize that numbers are not objects, they are foremost operators. You can’t have “one” or “two”, you always have “one mouse” or “two cats”; that is, you apply your operator to your base object, in basically what amounts to a form of multiplication. When you multiply numbers, you compose the operators; that is, you do one of the operators, then take that result and do the other. When you add them, you add their effects together and figure out what operator would give that effect. Notice that in general you can’t multiply “three houses” and “four cars” unless you have defined what it means to multiply by a house or a car, or at very least that specific operation. Similarly, if you try to add them, you’ll just get “three houses plus four cars” unless you’ve specifically defined an operator that does exactly that. Also, while you can add “one cat” and “two cats”, you can’t multiply them unless you know what “cat” times “cat” is.
Now above a number is defined as an operation on an object. Consider instead we define it as an operation on a segment of the number line; “4 cubits” means go forward a cubit (whatever that happens to be) 4 times. Now, define the operation “-” as “turn around”, and define “-X” as “-” applied to “X”, so that “-6 furlongs” means “turn around, then go a furlong 6 times”.
Now what happens when we multiply two of these operators? If we multiply “3 times 4”, we pick a distance, say angstrom, then go 4 angstroms forward. Now we reset ourselves, take “4 angstroms forward” as our base distance, and move that distance forward 3 times, clearly ended up 12 angstroms forward. If we multiply “-2 times 3” angstroms, we first measure 3 angstroms forward, then reset ourselves facing forwards, turn around to face backwards and go that distance twice, leaving us with the distance 6 angstroms backwards. For “2 times -3”, we first measure 3 angstroms backwards, then reset ourselves, face forwards, and step that distance twice. But since we measured that step as going backwards, each step is “3 angstroms backwards” leaving us at “6 angstroms backwards”.
Now the payoff: for “-4 times -5” angstroms, we first turn around and go 5 angstroms, then go back to the beginning, face forwards, but then turn around to face backwards as we’re multiplying by a negative, and now we step backwards as well since the distance we’re going is “5 angstroms backwards”. Thus we step 20 angstroms forward.
What about adding them? Now instead of applying a new operator to an old result, we stand at the distance of the first result and walk a distance specified by the second result. “5 angstroms” plus “-4 angstroms” means “walk 5 angstroms forward, turn around, walk 4 angstroms backwards”, leaving you at “1 angstrom forward”.
Now, just define i as “turn to the right”, and you’ll know all about the complex numbers. Or rather, a different version of them, because usually people define it as “turn to the left”, but it really doesn’t matter.
