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5.7 Complex Numbers 12/4/2013 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 • x 3 = x 5 y • y7 = y8 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 i using the IMAGINARY UNIT, . i 1 2 i 1 Simplifying i given any powers io 1 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. Based on that remainder, you can determine it’s value. i1 i i 1 2 i 3 i 2 i1 1 i i i i i 1 1 1 4 2 2 i 5 i 4 i1 1 i i i 6 i 4 i 2 1 1 1 i 7 i 4 i 3 1 i i i8 i 4 i 4 11 1 Do you see the pattern yet? Ex : i 22 • Step 1. 22÷ 4 has a remainder of 2 • Step 2. i22 = i2 i 22 1 Ex : i 50 • Step 1. 51 ÷ 4 has a remainder of 3 • Step 2. i51 = i3 i 51 i Checkpoint 1. i 15 2. i 20 3. i 61 4. i 122 Find the value of Complex Number Is a number written in the standard form a + bi where a is the real part and bi is the imaginary part. Ex: 3 + 2i Adding and Add/Subtract the real parts, Subtracting then add/subtract the Complex Numbers imaginary parts Example 2 Add Complex Numbers Write ( 3 + 2i + ( 1 – i as a complex number in standard form. ( ( SOLUTION ( 3 + 2i + ( 1 – i = 3 + 1 + 2i – 1i ( =4+i Group real and imaginary terms. Write in standard form. ( Example 3 Subtract Complex Numbers Write ( 6 – 2i – ( 1 – 2i as a complex number in standard form. ( ( SOLUTION ( 6 – 2i – ( 1 – 2i = 6 – 1 – 2i + 2i ( ( -1 + 2i = 5 + 0i =5 Write in standard form. Checkpoint Add and Subtract Complex Numbers Write the expression as a complex number in standard form. 6. ( 4 – 2i + ( 1 + 3i ANSWER 5 +i ( ANSWER 5 + 3i 8. ( 4 + 6i – ( 2 + 3i ( ANSWER 2 + 3i 9. ( – 2 + 4i – ( 2 + 7i ANSWER – 4 – 3i ( ( ( 7. ( 3 – i + ( 2 + 4i ( ( ( Checkpoint Add and Subtract Complex Numbers Write the expression as a complex number in standard form. ( ( 12. ( 2 – i – ( – 1 – 4i ( 11. ( 1 – 2i + ( 4 + 5i ANSWER 5 + 3i ANSWER 3 + 3i ( Example 4 Multiply Complex Numbers Write the expression as a complex number in standard form. remember : b. (6 + 3i ( 4 – 3i ( ( ( a. 2i ( – 1 + 3i i 2 1 SOLUTION a. 2i ( – 1 + 3i = – 2i + 6i 2 ( = – 6 – 2i ( = – 2i + 6 ( – 1 Multiply using distributive property. Use i 2 = –1. Write in standard form. Example 4 Multiply Complex Numbers b. (6 + 3i ( 4 – 3i = 24 – 18i + 12i – 9i 2 Multiply using FOIL. ( = 24 – 6i – 9i 2 Simplify. = 24 – 6i – 9 ( – 1 Use i 2 = –1. ( = 33 – 6i Write in standard form. ( Homework WS 5.7 Do problems 13-38 only Complex Conjugates Two complex numbers of the form a + bi and a - bi Their product is a real number because (3 + 2i)(3 – 2i) using FOIL 9 – 6i + 6i -4i2 9 – 4i2 i2 = -1 9 – 4(-1) = 9 + 4 = 13 Is used to write quotient of 2 complex numbers in standard form (a + bi) Example 5 Divide Complex Numbers 3 + 2i as a complex number in standard form. 1 – 2i a + bi SOLUTION Write Multiply the numerator and the denominator by 1 + 2i, the complex conjugate of 1 – 2i. 3 + 2i 3 + 2i 1 + 2i • = 1 – 2i 1 – 2i 1 + 2i 3 + 6i + 2i + 4i 2 = 1 + 2i – 2i – 4i 2 3 + 8i + 4 ( – 1 = 1 – 4( – 1 ( ( = – 1 + 8i = – Multiply using FOIL. Simplify and use i 2 = – 1. Simplify. 5 1 8 + i 5 5 Write in standard form. Checkpoint Multiply and Divide Complex Numbers Write the expression as a complex number in standard form. 2+i 1– i ANSWER 1 3 + i 2 2 Properties of Square Root of Negative Number i 1 r 1 r r i r Ex: −5 = 𝑖 5 i 2 1 Example 1 Solve a Quadratic Equation Solve the equation. a. 7x 2 = – 49 b. 3x 2 – 5 = – 29 SOLUTION a. 7x 2 = – 49 x 2 = –7 Divide each side by 7. x = + – –7 Take the square root of each side. x = + –i 7 Write in terms of i. Example 1 Solve a Quadratic Equation b. 3x 2 – 5 = – 29 3x 2 = – 24 x 2 = –8 Write original equation. Add 5 to each side. Divide each side by 3. x = + – –8 Take the square root of each side. x = + –i 8 Write in terms of i. 8 4 22 2 x = + – 2i 2 Simplify the radical. Checkpoint Solve a Quadratic Equation Solve the equation. 1. x 2 = – 3 ANSWER i 3, – i 3 2. x 2 = – 20 ANSWER 2 i 5, – 2 i 5 3. x 2 + 3 = – 2 ANSWER i 5, – i 5 Graphing Complex Number Imaginary axis Real axis Ex: Graph 3 – 2i To plot, start at the origin, move 3 units to the right and 2 units down 3 2 3 – 2i Ex: Name the complex number represented by the points. Answers: D A is 1 + i B B is 0 + 2i = 2i C is -2 – i A D is -2 + 3i C i r i r i r i i 2 Since i i i 1 2 i r 1r 2 and or r r r r r r