√10𝑎3 𝑏 3 √5𝑎9 𝑏 4
Write the product in a + bi form:
(8 + √−25 )( 2 + √−1)
Aim: How do we divide by a complex number
and how do we find it multiplicative
inverse?
The multiplicative inverse of a + bi is
1
𝑎+𝑏𝑖
,
a +bi ≠ 0.
*You must rationalize the denominator.
Practice:
Find the multiplicative inverse and express in
a + bi form:
1) 2 – i
2) 1 + 5i
Divide:
3)
4)
5)
2
5𝑖
15𝑖
𝑖+2
2+5𝑖
3−𝑖
Two complex numbers are equal if and only if
their real parts are equal and their imaginary
parts are equal.
a + bi = c + di
iff a = c and bi = di
b = d
Determine the values of a, b, x, y for which
the equation is true.
1) a + bi = 7 + 2i
2) a + bi = 12 + i – 3
3) a + √−16 = 16 + bi
4) 3x + 5yi = 15 – 20i
5) x + 6i +3 = 12 + yi
6) x – 3yi = 18i
Aim: How do we represent addition and
subtraction of complex numbers graphically?
How do we find a vector’s magnitude and its
absolute value?
Do Now:
Express in simplest form: 𝑖 6 + 𝑖 8 + −421 + 𝑖 43
Graphing Complex Numbers
We represent complex numbers as vectors.
A vector (ray) has magnitude (length) and
direction.
2) Real Numbers  x – axis
Pure Imaginary  y – axis called yi
3) x + yi = 2 + 5i is located the same way as
(x,y) = (2,5)
4) origin = (0,0) = 0 + 0i
We can express the sum of 2 complex
numbers as a vector which is the diagonal of a
parallelogram. The diagonal of a parallelogram
is called the resultant.
1. 3 + 4 𝑖
2. 2 − 3𝑖
3. −4 + 2𝑖
4. 3 (𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝑟𝑒𝑎𝑙𝑙𝑦 3 + 0𝑖 )
5. 4𝑖 (𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝑟𝑒𝑎𝑙𝑙𝑦 0 + 4𝑖)
1) Find the sum of 3 + 4𝑖 and −4 + 2𝑖.
2) Graph 3 + 4𝑖
3) Graph −4 + 2𝑖
How would you graph the sum?
Steps:
1. Graph the two
complex numbers as
vectors.
2. Create a
parallelogram using
these two vectors as
adjacent sides.
3. The answer to the
addition is the vector
forming the diagonal
of the parallelogram
(read from the
origin).
4. This new vector is
called the resultant
vector.
2.
3.
4.
5.
6.
Motivation #3:
1. Graph
3 + 4i
2. Graph
-2 + 2i
3.
Subtract 3 + 4i from -2 + 2i
Subtraction is the process of adding the additive inverse.
(-2 + 2i) - (3 + 4i)
= (-2 + 2i) + (-3 - 4i)
= (-5 - 2i)
What do you think would be the steps used to graph the
difference of 2 complex numbers?
Steps:
1. Graph the two complex numbers as vectors.
2. Graph the additive inverse of the number being subtracted.
3. Create a parallelogram using the first number and the additive inverse.
The answer is the vector forming the diagonal of the parallelogram.
Example 1:
Plot z = 8 + 6i on the complex plane, connect the graph of z
to the origin (see graph below), then find | z | by appropriate
use of the definition of the absolute value of a complex
number.
Example 2:
Find the | z | by appropriate use of the Pythagorean Theorem
when z = 2 - 3i.
You can find the distance | z | by using the Pythagorean theorem.
Consider the graph of
2 - 3i shown at the left. The horizontal side of the triangle has
length | a |, the vertical side has length | b |, and the hypotenuse has
length | z |. By applying the Pythagorean Theorem, you have, | z |2
= a2 + b2 .
Notice: you can drop the absolute value symbols for a and b since
| a |2 = a2 and
2
2
| b | = b . You must keep the absolute value symbol for z to insure
that the final answer will be positive.
Solving this equation for | z |, you have just derived the formula for
the absolute value of a complex number:
Mixed Practice:
Show Graphically:
1) the vector difference (-3 - i) – (-2 +4i)
2) the vector sum (3 - 3i) + (4 + 4i)
3) the vector difference (-4 – i) – (2 + 4i)
Example 3:
If z = - 8 - 15i, find | z |.