Perform Operations with Complex Numbers

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Perform Operations with Complex Numbers. Section 8.7 MATH 116 - 460 Mr. Keltner. Complex Numbers. Taking the square root of a negative number was a problem for many years. Leonhard Euler (pronounced OILER), defined the imaginary unit , i , such that: It follows, then, that:.

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### Perform Operations with Complex Numbers

Section 8.7

MATH 116 - 460

Mr. Keltner

Complex Numbers
• Taking the square root of a negative number was a problem for many years.
• Leonhard Euler (pronounced OILER), defined the imaginary unit, i, such that:
• It follows, then, that:
More Complex-ity
• It follows that any imaginary number in the form can be written in terms of i.
• Use the Product Rule of Radicals to separate a radical’s individual factors.
• Example 1: Write each imaginary number as a product of a real number and i.
Example 2
• Solve the equations:
Complex Numbers
• A complex number in standard form is any number that can be written as a + bi, where a and b are real numbers and i is the imaginary unit.
• a is called the real part and
• b is called the imaginary part.
• The form a + biis called thestandard formof a complex number.
• Real numbers, like 7 or -13/5,are just complex numbers where b = 0.
• Apure imaginary number, like 5i, is one where its real part is zero, or where a = 0.
Sums and Differences of Complex Numbers
• To add (or subtract) two complex numbers, simply add (or subtract) their real parts and their imaginary parts separately.
• In plain English: Treat the i as if it is a variable and combine like terms.
• Heads up! Be careful whenever you are subtracting a quantity, like 4 - (2 - 3i).
Operations with Complex Numbers
• Example 3: Find each sum or difference.

(-10 - 6i) + (8 + i)

(-9 + 2i) - (3 - 4i)

Operations with Complex Numbers
• To multiply two complex numbers, use the distributive property or the FOIL method just the same as we would with other algebraic expressions.
• Example 4: Multiply (2 - i ) • (-3 - 4i ).
Division with Complex Numbers
• In order to simplify a fraction containing complex numbers, we often need to use the conjugate of a complex number.
• The conjugate of a complex number in the form a + bi will be a – bi.
• Note that when we multiply a complex number and its conjugate, any imaginary terms are eliminated.
Rationalizing the Denominator
• To simplify a quotient where there is an imaginary term in the denominator, multiply by a fraction that is equal to 1, using the conjugate of the denominator.
• This process is called rationalizing the denominator.
• Example 5: Simplify the complex fraction below. Write your answer in standard form.
By evaluating the powers of I and simplifying, we note that it has a pattern of repeating every 4th time around.

i =  i2=

i3=  i4=

i5=  i6=

Use the previous observations to help evaluate each power of i.

i25

i43

i-18

It’s an I - cycle!

### Assessment

Pgs. 604-606:

#’s 7-84, multiples of 7