Lecture 10. Complex Numbers. January 28, 2015
√
We denote i = −1, so i2 = −1. Every number of the type z = a + bi, where a, b are real numbers,
is called a complex number. Examples:
√
√
z = 0, z = 2, z = 22 + 22i, z = −3i, z = −1 − 2i.
The set of complex numbers is denoted by C. Every real number a is also a compelx number (just
let b = 0). For z = a + bi, we let
a = Re z, b = Im z
be the real part and the imaginary part of z. We can express complex numbers as points on the
coordinate plane, where the a-axis is called the real axis, and the b-axis is called the imaginary
axis. The vector from the√origin to this point also represents z. The length of this vector is called
absolute value of z: |z| = a2 + b2 . The angle from the real axis to this vector (positive direction is
counterclockwise) is called the argument of z: arg z.
√
Example 1. z = 1 + i ⇒ |z| = 2, arg z = π/4.
q √
√
√
Example 2. z = − 3 − i ⇒ |z| = (− 3)2 + (−1)2 = 4 = 2, arg z = 7π/6.
This example shows that sometimes
arg z 6= arctg(b/a).
√
√
Here, arctg(b/a) = arctg(−1/ −3) = arctg(1/ 3) = π/6, and arg z = 7π/6.
Also, note that arg z is defined up to adding 2π. For example, we can also write in the last
example that arg z = −5π/6 or arg z = 2π + 7π/6.
For z = a + bi, the number z = a − bi is called the complex conjugate of z. It is just a reflection
across the real axis of z. Note that the product of any complex numbers and its conjugate is a real
number:
zz = (a + bi)(a − bi) = a2 − (bi)2 = a2 + b2 + |z|2 .
Arithmetric Operations. We can add and subtract complex numbers:
(1 + i) − (3 + 5i) = −2 − 4i.
We can also multiply them, recalling that i2 = −1:
(1 + i)(3 + 5i) = 3 + 3i + 5i + 5i2 = −2 + 8i.
How to divide complex numbers? Multiply the denominator by its conjugate:
(1 + i)(1 − 3i)
1 + i − 3i − 3i2
4 − 2i
2 1
1+i
=
=
=
= − i.
2
2
1 + 3i
(1 + 3i)(1 − 3i)
1 +3
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