Math 150 – Fall 2015
Chapter 1E
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Chapter 1E - Complex Numbers
Question. What are the solutions to x2 + 1 = 0?
Definition. We define i such that
i=
√
−1
A complex number is a number that can be written in the form a + bi, where a and
b are real numbers.
• a is called the real part
• b is called the imaginary part
• a + bi is called the standard form of the complex number
• i2 = −1
• Two complex numbers are equal if and only if their real parts and their imaginary
parts are equal.
√
√
√
√
Example 1. Write the number 24 − −64 − −81 + 54 in its standard form a + bi.
Find the real part and the imaginary part of the number.
Example 2. Find real numbers a and b such that (5a − 11) − 9i = 5 −
b
4
i
Definition. The absolute value of a complex numer z = a + bi in standard form is
defined as
p
|z| = |a + bi| = a2 + b2
• If z1 and z2 are two complex numbers, then the distance between these numbers
is defined as
|z1 − z2 |
• If z is a real number, then the absolute value of z is the same as its real number
absolute value.
Math 150 – Fall 2015
Chapter 1E
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Example 3. Find the absolute value of the following complex numbers.
(a) |5 − 3i|
(b) | − 11|
(c) |5 + 12i|
Properties of the absolute value of a complex number:
1. For z = a + bi a complex number, the absolute value of z represents the distance
from the origin (0, 0) to the point with coordinates (a, b).
2. For two complex numbers z1 and z2 , then
|z1 z2 | = |z1 | |z2 |
3. For two complex numbers z1 and z2 ,
|z1 |
z1
=
z2
|z2 |
4. For two complex numbers z1 and z2 ,
|z1 + z2 | ≤ |z1 | + |z2 |
This is called the triangle inequality.
Example 4. (a) Show that the absolute value of the product of 2 − i and 3 + 2i equals
the product of their absolute values.
(b) Show that the absolute values of 1 − i and 3 + 2i is less than or equal to the sum
of their absolute values.
Math 150 – Fall 2015
Chapter 1E
Definition. The conjugate of the complex number z = a + bi is defined as
z = a − bi.
We use z to denote the conjugate.
Example 5. Compute the conjugate of the following numbers:
(a) 5 − 9i
(b) 7i
(c) 8
(d) −5 + 13i
Properties of the Conjugate of a Complex Number:
1. Since (a + bi)(a − bi) = a2 − (bi)2 = a2 + b2 = |a + bi|2 , then
z · z = |z|2
2. If x is a real number, then x = x.
3. The conjugate of the conjugate of z equals z
z=z
4. The conjugate of a sum is the sum of the conjugates.
(z1 + z2 ) = z1 + z2
5. The conjugate of a product is the product of the conjugats
z1 · z2 = z1 · z2
6. The conjugate of a quotient if the quotient of the conjugates
z1
z2
=
(z1 )
(z2 )
Example 6. Calculate z1 + z2 and z1 + z2 for z1 = 5 + 3i and −2 − 9i.
Example 7. Put
2+3i
1−4i
in standard form.
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