Half the battle in coming to grips with complex numbers is not letting
the look of the expressions intimidate one. All of a sudden, one finds onself on
what looks like a Cartesian plane, only the ordinate is i - the square root of minus 1 -
and the expression of the point contains a plus sign rather than a mere colon: (a + bi).
What is going on here?
The Complex plane looks like the Cartesian plane, and expressions on it are liable to
two different approaches. Here, a and b are real numbers, but because b
ia affixed to i, it is called imaginary. Nonetheless, if one needs to find r, the
length of the line to the point, one proceeds as always and merely ignores the i.
Thus, (3 + 4i) gets evaluated to square root of ( 9 + 16) ie (25)^.5, which is equal to 5.
For the second approach, one might want to use trigonometric functions; one then needs
to find the angle which the line to the point creates. Again, for length 5, and with sides 3
and 4, the angle, theta, will be that of .927 rads (53 degrees).
Where the fact of i makes a difference is when one wants to calculate with it. For example,
to multiply (3 + 4i) by its conjugate (3 - 4i) yields 25 ( and not -7, which is what
(3 + 4)(3 - 4) would have given)!!
Our little plus sign is a stern reminder...
There are many sites which go into al this, but the one below - intended for French candidates
tot he Bac S - is a charmer:
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