1. Throughout the semester, we were not allowed to take square roots of negative numbers. The
reason
why this was so is because we never introduced the set of complex numbers, usually denoted by
C.
Numbers in this set are of the form a + ib, where a and b are real numbers. For now, you can
think of
i as a symbol. i has the property that i2 = ??1, which is equivalent to i =
p
??1. For example, one can
show that (a + ib)2 = a2 ?? b2 + 2iab and (a + ib)(a ?? ib) = a2 + b2 (work it out).
Let a + ib be a complex number. The complex conjugate is the number a ?? ib. Division by
complex
numbers doesn't behave easy. The number
a + ib
c + id
is NOT in the form A + iB because there is an i in
the denominator. However, we can always turn a number of this form into a complex number of
the
form A + iB. Show how to do this. Here's a hint: multiply both the numerator and denominator
of
a + ib
c + id
by the complex conjugate of the denominator. This amounts to multiplying the entire fraction
by 1 because you multiplied the top and bottom by the same thing. After some manipulation, you
will get a number of the form A + iB.
Solution
(a+ib)x(c-id)/(c+id)x(c-id) =(ac+bd + i(bc-ad))/(c^2 +d^2)= A+iB (ac+bd)/(c^2 + d^2) =A i(bc-
ad)/(c^2 + d^2) = iB