Lecture Notes in Differential Equations - Bruce E. Shapiro

161
The motivation for equation 18.74 is the following. Suppose z = (0, 1).
Then by 18.74,
z 2 = (0, 1) × (0, 1) = (−1, 0) (18.79)
We use the special symbol i to represent the complex number i = (0, 1).
Then we can write any complex number z = (a, b) as
z = (a, b) = (a, 0) + (b, 0) = a(1, 0) + b(0, 1) (18.80)
Since i = (0, 1) multiplication by (1, 0) is identical to multiplication by 1
we have
z = (a, b) = a + bi (18.81)
and hence from 18.79
i 2 = −1 (18.82)
The common notation is to represent complex numbers as z = a + bi where
a, b ∈ R, where i represents the square root of −1. It can easily be shown
that the set of complex numbers defined in this way have all of the properties
of a Field.
Theorem 18.8. Properties of Complex Numbers
1. Closure. The set of complex numbers is closed under addition and
multiplication.
2. Commutivity. For all complex number w, z,
}
w + z = z + w
wz = zw
3. Associativity. For all complex numbers u, v, w,
(u + v) + w = u + (v + w)
(uv)w = u(vw)
}
(18.83)
(18.84)
4. Identities. For all complex numbers z,
z + 0 = 0 + z = z
z1 = 1z = z
}
(18.85)
5. Additive Inverse. For every complex number z, there exists some
unique complex number w such that z + w = 0. We call w = −z and
z + (−z) = (−z) + z = 0 (18.86)