1 / 14

Understanding Complex Numbers in Mathematics

Explore the properties of complex numbers, their operations, and applications such as finding solutions to quadratic equations and roots of unity. Practice problems included to enhance understanding.

sheilaward
Download Presentation

Understanding Complex Numbers in Mathematics

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. Chapter 4: Polynomial and Rational Functions4.5: Complex Numbers Essential Question: What are the two complex numbers that have a square of -1?

  2. 4.5: Complex Numbers • Properties of the Complex Number System • The complex number system contains all real numbers • Addition, subtraction, multiplication, and division of complex numbers obey the same rules of arithmetic that hold in the real number system with one exception: • The exponent laws hold for integer exponents, but not necessarily for fractional ones • We don’t need to worry about this for now, I just needed to list the exception • The complex number system contains a number, denoted i, such that i2 = -1 • Every complex number can be written in the standard form: • a + bi • a + bi = c + di if and only if a = c and b = d • Numbers of the form bi, where b is a real number, are called imaginary numbers. Sums of real and imaginary numbers, numbers of the form a + bi, are called complex numbers

  3. 4.5: Complex Numbers • Example #1: Equaling Two Complex Numbers • Find x and y if 2x – 3i = -6 + 4yi • The real number parts are going to be equal • 2x = -6 • x = -3 • The imaginary number parts are going to be equal • -3i = 4yi • -3/4 = y

  4. 4.5: Complex Numbers • Example #2: Adding, Subtracting, and Multiplying Complex Numbers • (1 + i) + (3 – 7i) • Combine like terms • 4 – 6i • (4 + 3i) – (8 – 6i) • Distribute, then combine terms • 4 + 3i – 8 + 6i = -4 + 9i • 4i(2 + ½ i) • Distribute and simplify • 8i + 2i2 = 8i + 2(-1) = -2 + 8i • (2 + i)(3 – 4i) • FOIL and simplify • 6 – 8i + 3i – 4i2 = 6 – 8i + 3i – 4(-1) = 6 – 8i + 3i + 4 = 10 – 5i

  5. 4.5: Complex Numbers • Example #3: Products and Powers of Complex Numbers • (3 + 2i)(3 – 2i) • FOIL • 9 – 6i + 6i – 4i2 = 9 – 4(-1) = 9 + 4 = 13 • (4 + i)2 • 16 + 4i + 4i + i2 = 16 + 4i + 4i + (-1) = 15 + 8i

  6. 4.5: Complex Numbers • Powers of i • i1 = i • i2 = -1 • i3 = i2 • i = -1 • i = -i • i4 = i2 • i2 = -1 • -1 = 1 • i5 = i4 • i = 1 • i = i • And we keep repeating from there… • Example #4: Powers of i • Find i54 • The remainder when 54 / 4 is 2, so i54 = i2 = -1

  7. 4.5: Complex Numbers • Complex Conjugates • The conjugate of the complex number a + bi is the number a – bi, and the conjugate of a – bi is a + bi • Conjugates multiplied together yield a2 + b2 • (a – bi)(a + bi) = a2 + abi – abi – b2i2 = a2 – b2(-1) = a2 + b2 • The conjugate is used to eliminate the i from the complex number, and is used to remove the use of i in the denominator of fractions

  8. 4.5: Complex Numbers • Example #5: Quotients of Two Complex Numbers • Simplify • multiply top & bottom by the conjugate of the denominator

  9. 4.5: Complex Numbers • Assignment • Page 300 • Problems 1-35 & 55-57, odd problems • Show work where necessary (e.g. FOILing, converting to i) • Due tomorrow

  10. Chapter 4: Polynomial and Rational Functions4.5: Complex Numbers (Part 2) Essential Question: What are the two complex numbers that have a square of -1?

  11. 4.5: Complex Numbers • Square Roots of Negative Numbers • Because i2 = -1, • In general, • Take the i out of the square root, then simplify from there • Example #6: Square Roots of Negative Numbers

  12. 4.5: Complex Numbers • Complex Solutions to a Quadratic Equation • Find all solutions to 2x2 + x + 3 = 0

  13. 4.5: Complex Numbers • Zeros of Unity • Find all solutions of x3 = 1 • Rewrite equation as x3 - 1 = 0 • Use graphing calculator to find the real roots (1) • Factor that out • (x – 1)(x2 + x + 1) = 0 • x = 1 or x2 + x + 1 = 0

  14. 4.5: Complex Numbers • Assignment • Page 300 • Problems 37-71 (odd) (skip 55/57, you did that last night) • Due tomorrow • You must show work

More Related