Lecture Notes in Differential Equations - Bruce E. Shapiro

154 LESSON 18. COMPLEX ROOTS
Example 18.2. Find the roots of
r 2 + r + 1 = 0 (18.8)
We have a = b = c = 1 hence according to (18.2) the roots are
r = −1 ± √ 1 2 − (4)(1)(1)
2(1)
= −1 ± √ −3
2
(18.9)
Since −3 < 0 it does not have a real square root;
√
−3 =
√
(3)(−1) =
√
−1
√
3 = i
√
3 (18.10)
hence
r = −1 ± i√ 3
2
Properties of Complex Numbers
= − 1 2 ± i √
3
2
(18.11)
1. If z = a + ib, where a, b ∈ R, then we say that a is the real part of
z and b is the imaginary part, and we write
}
Re(z) = Re(a + ib) = a
(18.12)
Im(z) = Im(a + ib) = b
2. The absolute value of z = x + iy is the distance in the xy plane
from the origin to the point (x, y). Hence
|x + iy| = √ x 2 + y 2 (18.13)
3. The complex conjugate of z = x + iy is a complex number with all
of the i’s replaced by −i, and is denoted by z,
z = x + iy =⇒ z = x + iy = x − iy (18.14)
4. If z = x + iy is any complex number then
|z| 2 = zz (18.15)
because
x 2 + y 2 = (x + iy)(x − iy) (18.16)
5. The phase of a complex number z = x + iy, denoted by Phase(z) is
the angle between the x − axis and the line from the origin to the
point (x, y)
Phase(z) = arctan y x
(18.17)