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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

Perform Operations with Complex Numbers

Section 8.7

MATH 116 - 460

Mr. Keltner

complex numbers
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
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
Example 2
  • Solve the equations:
complex numbers1
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
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
Operations with Complex Numbers
  • Example 3: Find each sum or difference.

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

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

operations with complex numbers1
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
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
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.
it s an i cycle
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

Assessment

Pgs. 604-606:

#’s 7-84, multiples of 7