1 / 12

Perform Operations with Complex Numbers

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:.

jessamine-castro
Download Presentation

Perform Operations with Complex Numbers

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Perform Operations with Complex Numbers Section 8.7 MATH 116 - 460 Mr. Keltner

  2. 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:

  3. 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.

  4. Example 2 • Solve the equations:

  5. 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.

  6. 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).

  7. Operations with Complex Numbers • Example 3: Find each sum or difference. (-10 - 6i) + (8 + i) (-9 + 2i) - (3 - 4i)

  8. 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 ).

  9. 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.

  10. 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.

  11. 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!

  12. Assessment Pgs. 604-606: #’s 7-84, multiples of 7

More Related