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Transcript
Complex Numbers Lesson 3.3 The Imaginary Number i 1 i i 2 1 By definition Consider powers if i i 1 2 i i i i 3 2 i i i 1 1 1 4 2 2 i i i 1 i i 5 ... 4 It's any number you can imagine Using i Now we can handle quantities that occasionally show up in mathematical solutions a 1 a i a What about 49 18 Complex Numbers Combine real numbers with imaginary numbers a + bi Imaginary part Real part Examples 3 4i 3 6 i 2 4.5 i 2 6 Try It Out Write these complex numbers in standard form a + bi 9 75 5 144 16 7 100 Operations on Complex Numbers Complex numbers can be combined with addition subtraction multiplication division Consider 9 12i 7 15i 3 i 8 2i 2 4i 4 3i Operations on Complex Numbers Division technique Multiply numerator and denominator by the conjugate of the denominator 3i 3i 5 2i 5 2i 5 2i 5 2i 15i 6i 2 25 4i 2 6 15i 6 15 i 29 29 29 Complex Numbers on the Calculator Possible result Reset mode Complex format to Rectangular Now calculator does desired result Complex Numbers on the Calculator Operations with complex on calculator Make sure to use the correct character for i. Use 2nd-i Warning 16 49 Consider It is tempting to combine them 16 49 16 49 4 7 28 The multiplicative property of radicals only works for positive values under the radical sign Instead use imaginary numbers 16 49 4i 7i 4 7 i 2 28 Try It Out Use the correct principles to simplify the following: 3 121 4 81 4 81 3 144 2 The Discriminant Consider the expression under the radical in the quadratic formula 2 b b 4ac 2a This is known as the discriminant What happens when it is Positive and a perfect square? Positive and not a perfect square? Zero Complex roots Negative ? Example 2 x 3x 5 0 Consider the solution to Note the graph No intersections with x-axis Using the solve and csolve functions Fundamental Theorem of Algebra A polynomial f(x) of degree n ≥ 1 has at least one complex zero Number of Zeros theorem Remember that complex includes reals A polynomial of degree n has at most n distinct zeros Explain how theorems apply to these graphs Conjugate Zeroes Theorem Given a polynomial with real coefficients P( x) an x an 1 x n n 1 ... a1 x a0 If a + bi is a zero, then a – bi will also be a zero Assignment Lesson 3.3 Page 211 Exercises 1 – 78 EOO