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Algebra 2 CH 1.6 Perform Operations with Complex Numbers.notebook
1
August 28, 2013
Perform Operations with Complex Numbers
1.6 Perform Operaons with Complex Numbers
• Imaginary unit
• Imaginary number
• Complex number
• Complex conjugates
• Complex plane
• Absolute value of a complex number
Perform Operations with Complex Numbers
How have we classified numbers in the past?
Whole Numbers
Rational Numbers
Real Numbers
Perform Operations with Complex Numbers
Natural Numbers1, 2, 3, 4...
Whole Numbers0, 1, 2, 3, 4...
Integers ... 3, 2, 1, 0, 1, 2, 3, 4...
Rational NumbersIrrational
Numbers
Imaginary
Numbers
Real Numbers
Complex Numbers
Perform Operations with Complex Numbers
What is an imaginary Number?
Have we ever been able to take the square root of a negative number?
Perform Operations with Complex Numbers
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Perform Operations with Complex Numbers
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Algebra 2 CH 1.6 Perform Operations with Complex Numbers.notebook
2
August 28, 2013
Perform Operations with Complex Numbers
Solve the following problem using imaginary numbers.
Perform Operations with Complex Numbers
Perform Operations with Complex Numbers Perform Operations with Complex Numbers
A complex number is a combination of a real number and an imaginary number.
Example:
General form:
Where is real and is imaginary
Perform Operations with Complex Numbers
Now we will add, subtract, multiply and divide complex numbers.
Perform Operations with Complex Numbers
Addition:
Algebra 2 CH 1.6 Perform Operations with Complex Numbers.notebook
3
August 28, 2013
Perform Operations with Complex Numbers
Subtraction:
Perform Operations with Complex Numbers
Multiplication: (Use the FOIL method)
Perform Operations with Complex Numbers
Division:
We need to use a special technique when we encounter division. The complex conjugate of the denominator must be used.
The complex conjugate of 1+6i is 16i
Perform Operations with Complex Numbers
Division:We will multiply both the numerator and denominator by the complex conjugate of the denominator.
Perform Operations with Complex Numbers
Division:Perform Operations with Complex Numbers
Division:
This is a + bi form.
Algebra 2 CH 1.6 Perform Operations with Complex Numbers.notebook
4
August 28, 2013
Perform Operations with Complex NumbersGraphing Complex Numbers
real axis
imaginary axis
Complex Plane
Perform Operations with Complex NumbersGraphing Complex Numbers
real axis
imaginary axisGraph:
4+2i
3+6i
74i
83i
5i
i
Perform Operations with Complex Numbers
The Absolute Value of a Complex Number
A complex number can be written as:
Perform Operations with Complex Numbers
The absolute value of the complex number is:
• This absolute value of a complex number is also called the modulus.
• It is also the distance from the origin to the point z
Perform Operations with Complex NumbersGraphing Complex Numbers
real axis
imaginary axis
Find
if:
Perform Operations with Complex Numbers
Why do we need to know about complex numbers?
Complex numbers are used in the calculation of impedance, which is an electrical circuits total opposition to the flow of current.
Algebra 2 CH 1.6 Perform Operations with Complex Numbers.notebook
5
August 28, 2013
Perform Operations with Complex NumbersIf a circuit has the following components, we can calculate the total impedance, which is measured in ohms ( Ω ).
A 5 ohm resistor has an impedance of 5Ω
A 10 ohm capacitor has an impedance of 10iΩ
A 7 ohm inductor has an impedance of 7i Ω
The total impedance is: 5 10i + 7i
= 5 3i Ω
Perform Operations with Complex Numbers
1.6 Perform Operaons with Complex Numbers
• Imaginary unit
• Imaginary number
• Complex number
• Complex conjugates
• Complex plane
• Absolute value of a complex number
