You can make some progress by thinking of the imaginary numbers as being a two dimensional vector space over the reals, equipped with a very peculiar multiplication operator that is a combination of the usual stretch with a rotation.
Addition is the usual addition of components, but the multiplication (using polar coordinates) looks like
C = ABexp(T1+T2),
where the T are the phases of A and B.
To multiply two positive reals, since their phases are zero, it’s only a stretch: AB
To multiply two negative reals, since their phases are pi, it’s ABexp(pi+pi). But exp(2pi) is unity, so we have AB again, a positive number.
To find the square root of a positive number D, find a positive real (zero phase) that, when multiplied by itself, gives D. AA, for example.
So, real number arithmetic is a subset of complex number arithmetic, with zero phase. And therefore zero rotation.
Now, find the square root of a negative number, say -4. To do this, we have to step off the real axis for the first time: out into the Argand plane!
-4 is represented as
4exp(pi)
OK?
So we are looking for a number out here on the Argand plane that when multiplied by itself following our rule gives 4exp(pi).
And that number is
2exp(pi/2),
which is two units out on the “imaginary” axis of our Argand plane.
Addition is the usual addition of components, but the multiplication (using polar coordinates) looks like
C = ABexp(T1+T2),
where the T are the phases of A and B.
To multiply two positive reals, since their phases are zero, it’s only a stretch: AB
To multiply two negative reals, since their phases are pi, it’s ABexp(pi+pi). But exp(2pi) is unity, so we have AB again, a positive number.
To find the square root of a positive number D, find a positive real (zero phase) that, when multiplied by itself, gives D. AA, for example.
So, real number arithmetic is a subset of complex number arithmetic, with zero phase. And therefore zero rotation.
Now, find the square root of a negative number, say -4. To do this, we have to step off the real axis for the first time: out into the Argand plane!
-4 is represented as
OK?So we are looking for a number out here on the Argand plane that when multiplied by itself following our rule gives 4exp(pi).
And that number is
which is two units out on the “imaginary” axis of our Argand plane. equals 4exp(pi/2+pi/2)equals 4exp(pi)
QED