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4.8 Day 1 Complex Numbers.notebook November 12, 2015 4.8 Complex Numbers Normally, you can't take the square root of a negative number in the real number system, but you can in the imaginary number system What is the ? Now you can simplify problems that were previously not possible to simplify further. 1 4.8 Day 1 Complex Numbers.notebook November 12, 2015 The reason for the name "imaginary" numbers is that when these numbers were first proposed several hundred years ago, people could not "imagine" such a number. It is said that the term "imaginary" was coined by René Descartes in the seventeenth century and was meant to be a derogatory reference since, obviously, such numbers did not exist. Today, we find the imaginary unit being used in mathematics and science. Electrical engineers use the imaginary unit (which they represent as j ) in the study of electricity. 2 4.8 Day 1 Complex Numbers.notebook November 12, 2015 Imaginary Numbers are Useful... AC (Alternating Current) Electricity changes between positive and negative in a sine wave. If you combine two AC currents they may not match properly, and it can be very hard to figure out the new current. But using imaginary numbers and real numbers together makes it a lot easier to do the calculations. 3 4.8 Day 1 Complex Numbers.notebook November 12, 2015 Imaginary numbers occur when a quadratic equation has no roots in the set of real numbers. y 5 4 3 Look , no real roots! 2 1 x 5 4 3 2 1 0 1 2 3 4 5 1 2 3 4 5 4 4.8 Day 1 Complex Numbers.notebook November 12, 2015 Simplify each number using the imaginary number, i. 1. 2. 3. 4. 5 4.8 Day 1 Complex Numbers.notebook November 12, 2015 6 4.8 Day 1 Complex Numbers.notebook November 12, 2015 Remember that: Find i on your calculator. Simplify each number using the imaginary number, i. 1. 2. 3. 4. To graph in complex number plane, the point (a,b) represents the complex number a + bi. The real part is graphed on the horizontal axis and the imaginary part on the vertical axis. 7 4.8 Day 1 Complex Numbers.notebook November 12, 2015 To add or subtract complex numbers, combine the real parts and imaginary parts separately-combine the like terms. You can use your calculator too! 1. (7 - 2i) + (- 3 + i) 3. (5 - 3i) - (- 2 + 4i) 2. (- 3 + 9i) + (4 + 9i) 4. (1 + 5i) - (3 - 2i) 8 4.8 Day 1 Complex Numbers.notebook Hint: Any November 12, 2015 becomes -1. What is the product? 5. (3i)(- 5 + 2i) 6. (7i)(3i) To multiply complex numbers a + bi and c + di, use FOIL or the box method. 7. (4 + 3i)(- 1 - 2i) 8. (2 - 3i)(4 + 5i) 4 + 3i -1 - 2i 2 - 3i 4 + 5i 9 4.8 Day 1 Complex Numbers.notebook November 12, 2015 Multiply (a + bi) by its complex conjugate (a - bi). The product of complex conjugates is a real number. For quotients, we don't want to leave an i in the denominator. What is each quotient? Complex conjugate 9. to make it... -just change the sign of the imaginary term! Try with the calculator! 10. 11. 10 4.8 Day 1 Complex Numbers.notebook November 12, 2015 The Unit Imaginary Number, i, has an interesting property. It "cycles" through 4 different values each time you multiply by i. For later on.... The circle of i i 11 4.8 Day 1 Complex Numbers.notebook November 12, 2015 HW p.253; 812 all, 1931 odd 12 4.8 Day 1 Complex Numbers.notebook November 12, 2015 HW p.253; 812 all, 1931 odd 13 4.8 Day 1 Complex Numbers.notebook November 12, 2015 14 4.8 Day 1 Complex Numbers.notebook November 12, 2015 15 4.8 Day 1 Complex Numbers.notebook November 12, 2015 16