By greatly simplifying otherwise complicated problems that don't originally involve imaginary numbers. A HUGE shortcut in some problems. Never heard of phasors? Never heard of expanding a problem to the complex plane? Thought not.
I just recently started AP AB Cal, but if I'm not mistaken, it would also be much harder, or even impossible, for us to make electronics. Some calculations to make that cicrutry come from imaginary numbers.
Not impossible, but much harder indeed. Not only the notation is greatly simplified with the use of "i" but also it is much easier to work with complex impedances instead of differential equations to solve circuits.
Ok I'll Try and explain this,
the square root of -1 is i yes
but the square root of 4 is 2
2i means 2 times the square root of -1
so if you square 2i you get -4
It's how to explain using logic.
Why imaginary?
Look at a graph.
If you go westwards 4 and southwards 4 you are at (-4,-4).
Then you are asked to square root both sides
On technicality you would square root -4 both
That would give you a wrong answer cause square rooting a negative number.
So they give you the 2i for both, although it would be wrong
You would know that though for it being imaginary
So you would see both sides lengths are actually 4
Even though it says -4 on the graphing point.
So you would square root your 4's for a length of 2
Which would give you the point of (-2,-2)
And Thank You Pre-Cal and My Teacher for me to learn this.