1.1. Definition and Simple Properties 3
A complex number of the form x + i0, with x ∈ R will be denoted simply as x.
This identifies R as a subset of C. Similarly, a complex number of the form 0 + iy
with y real will be denoted simply as iy. The numbers of this form are traditionally
called the imaginary numbers.
Note that, from the above definition, if x, y ∈ R, then
yi = (y + i0)(0 + i) = 0 + iy = iy
and so x + iy and x + yi are the same complex number. Which form is used to
describe this number is usually dictated by which looks best typographically. When
specific numbers replace x and y, the latter seems to look best. Thus, we usually
write 2 + 3i rather than 2 + i3.
Example 1.1.2. If z1 = 5 + 2i and z2 = 3 − 4i, find z1 + z2 and z1z2.
z1 + z2 = 5 + 3 + (2 − 4)i = 8 − 2i,
z1z2 = (5 · 3 − 2 · (−4)) + (5 · (−4) + 3 · 2)i = 23 − 14i.
Example 1.1.3. Show that the quadratic equation (1.1.1) has solutions which are
Solution: If b2 − 4ac ≥ 0, the quadratic formula (1.1.2) tells us the solutions
b2 − 4ac
b2 − 4ac
On the other hand, if b2 − 4ac 0, then 4ac − b2 is positive and has real square
roots. By squaring both sides, and using i2 = −1, it is easy to see that
± b2 − 4ac = ±i 4ac − b2.
This suggests that the solutions to the quadratic equation in this case are the two
4ac − b2
4ac − b2
That these two numbers are, indeed, solutions to the quadratic equation may be
verified by directly substituting them in for x in (1.1.1). We leave this as an exercise
Field Properties. In our definition of the product of two complex numbers,
we were guided by the desire to have the usual rules of arithmetic hold – that is,
the commutative and associative laws for addition and for multiplication and the
distributive law. Did we succeed? These are some of the properties of a field. Do
these laws actually hold in C with the operations as defined above? The following
theorem says they do.
Theorem 1.1.4. If z1,z2,z3 are complex numbers, then
(a) z1 + z2 = z2 + z1 – commutative law of addition;
(b) (z1 + z2) + z3 = z1 + (z2 + z3) – associative law of addition;
(c) z1z2 = z2z1 – commutative law of multiplication;
(d) (z1z2)z3 = z1(z2z3) – associative law of multiplication;