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2.5 Introduction to Complex Numbers

2.5 Introduction to Complex Numbers. 11/7/2012. Quick Review. If a number doesn’t show an exponent, it is understood that the number has an exponent of 1. Ex: 8 = 8 1 , x = x 1 , -5 = -5 1 Also, any number raised to the Zero power is equal to 1

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2.5 Introduction to Complex Numbers

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  1. 2.5 Introduction to Complex Numbers 11/7/2012

  2. Quick Review If a number doesn’t show an exponent, it is understood that the number has an exponent of 1. Ex: 8 = 81 , x = x1 , -5 = -51 Also, any number raised to the Zero power is equal to 1 Ex: 30 = 1 -40 = 1 Exponent Rule: When multiplying powers with the same base, you add the exponent. x2 • x3 = x5 y • y7 = y8

  3. The square of any real number x is never negative, so the equation x2 = -1 has no real number solution. To solve this x2 = -1 , mathematicians created an expanded system of numbers using the IMAGINARY UNIT, i.

  4. Simplifying i raised to any power Do you see the pattern yet? The pattern repeats after every 4. So you can find i raised to any power by dividing the exponent by 4 and see what the remainder is. Raise i to the remainder and determine its value. • Step 1. 22÷ 4 has a remainder of 2 • Step 2. i22 = i2 • Step 1. 51 ÷ 4 has a remainder of 3 • Step 2. i51= i3

  5. Find the value of Checkpoint • 1. i 15 • 2. i 20 • 3. i 61 • 4. i 122 • i3 = -i • i0 = 1 • i1= i • i2 = -1

  6. Complex Number Is a number written in the standard form a + bi where a is the real part and bi is the imaginary part.

  7. State the real and imaginary part of each complex number. 9i Real Imaginary 0 9i 0

  8. Adding and Subtracting Complex Numbers Add/Subtract the real parts, then add/subtract the imaginary parts

  9. SOLUTION ( ( = – – Group real and imaginary terms. 3 + 2i ( + 1 i ( 3 + 1 + 1 2i i = Write in standard form. 4 + i Add Complex Numbers ( ( Write as a complex number in standard form. – 3 + 2i ( + 1 i (

  10. ( ( – – – 6 2i ( 1 2i ( SOLUTION ( ( = + 2i – – – – Group real and imaginary terms. 6 2i ( 1 2i ( 6 1 – 2i = Simplify. 5 + 0i = Write in standard form. 5 Subtract Complex Numbers Write as a complex number in standard form. -1 + 2i

  11. Checkpoint ANSWER 5 + i ( ( 2. ANSWER – 3 i ( + 2 + 4i ( 5 + 3i ( ( 3. ANSWER – 4 + 6i ( 2 + 3i ( 2 + 3i ( ( 4. ANSWER – – – – 2 + 4i ( 2 + 7i ( 4 3i Add and Subtract Complex Numbers Write the expression as a complex number in a + biform. ( ( 1. – 4 2i ( + 1 + 3i (

  12. ( 4 – 3i ( SOLUTION ( 2i – 1 + 3i ( ( ( a. 6 + 3i ( – – Multiply using distributive property. 2i 6i2 2i 1 + 3i ( + = Use i21. – = ( – – 2i + 6 1 ( = Write in standard form. – – 6 2i = Multiply Complex Numbers Write the expression as a complex number in standard form. b. a.

  13. ( – – – 4 3i ( 24 18i + 12i 9i2 = – – 24 6i 9i2 = Simplify. ( – – – 24 6i 9 1 ( = – 33 6i = Write in standard form. ( 6 + 3i ( Use i21. – = Multiply Complex Numbers b. Multiply using FOIL.

  14. Divide Complex Numbers multiply top and bottom by i = = -

  15. Complex Conjugates Two complex numbers of the form a + bi and a - bi Their product is a real number because Ex: (3 + 2i)(3 – 2i) using FOIL 9 – 6i + 6i - 4i 2 9 – 4i2i2 = -1 9 – 4(-1) = 9 + 4 = 13 Is used to write quotient of 2 complex numbers in standard form (a + bi)

  16. 3 + 2i 1 – 2i Write as a complex number in a + bi form. = 8 1 5 5 – i Write in standard form. + = SOLUTION Multiply the numerator and the denominator by 12i, the complex conjugate of 12i. 1 + 2i 3 3 + + 2i 2i • = + – – + 1 2i 1 1 2i 2i – – 1 + 8i 5 + 3 + 6i 4i2 + 2i Multiply using FOIL. = – 1 + 2i – 4i2 2i Simplify and use i21. – = ( – 3 + 8i + 4 1 ( = ( – – 1 4 1 ( Divide Complex Numbers Simplify.

  17. Checkpoint ( 1. ANSWER – 3i 2 i ( ( ( 2. ANSWER 4 + 3i – 1 + 2i ( 2 i ( 3 + 6i 2 + i 1 3 3. ANSWER i + – 1 i 2 2 Multiply and Divide Complex Numbers Write the expression as a complex number in standard form.

  18. Graphing Complex Number Imaginary axis Real axis

  19. Ex: Graph 3 – 2i To plot, start at the origin, move 3 units to the right and 2 units down 3 2 3 – 2i

  20. Ex: Name the complex number represented by the points. Answers: A is 1 + i B is 0 + 2i = 2i C is -2 – i D is -2 + 3i D B A C

  21. Homework WS 2.5 #1-12all, 13-27odd, 31-34all

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