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6/11/ :01 PM Complex Numbers1 WARM-UP

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6/11/ :01 PM Complex Numbers2 C OMPLEX N UMBERS

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6/11/ :01 PM Complex Numbers3 IMAGINARY NUMBERS Imaginary Numbers are numbers can be written as a real number times i.

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6/11/ :01 PM Complex Numbers4 I MAGINARY N UMBERS Imaginary Numbers are numbers that can be written as a real number times i. Steps: 1.View the radical without the negative and simplify it 2.After simplified form, attach an i onto the real number 3.Simplify further, if needed

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6/11/ :01 PM Complex Numbers5 E XAMPLE 1 Simplify After simplified form, attach an i onto the real number

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6/11/ :01 PM Complex Numbers6 E XAMPLE 1 x = 0 Where does it cross the x-axis? +2i

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Simplify 6/11/ :01 PM Complex Numbers7 E XAMPLE 2

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Simplify 6/11/ :01 PM Complex Numbers8 E XAMPLE 3

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Simplify 6/11/ :01 PM Complex Numbers9 E XAMPLE 4

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Simplify 6/11/ :01 PM Complex Numbers10 Y OUR T URN

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Rules with Imaginaries: i 1 = i i 2 = –1 Answers CAN have an i but CAN NOT have two is Two is make –1 6/11/ :01 PM Complex Numbers11 I MAGINARY N UMBERS

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6/11/ :01 PM Complex Numbers12 W HY IS I 2 EQUAL TO –1? How do we cancel the radicals?

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Multiply What is another way of writing this problem? 6/11/ :01 PM Complex Numbers13 E XAMPLE 5

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Multiply 6/11/ :01 PM Complex Numbers14 E XAMPLE 6

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Multiply 6/11/ :01 PM Complex Numbers15 Y OUR T URN

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Multiply 6/11/ :01 PM Complex Numbers16 E XAMPLE 7

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Multiply 6/11/ :01 PM Complex Numbers17 E XAMPLE 8

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Multiply 6/11/ :01 PM Complex Numbers18 E XAMPLE 9

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Multiply 6/11/ :01 PM Complex Numbers19 Y OUR T URN

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Rationalize 6/11/ :01 PM Complex Numbers20 R EVIEW

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Rationalize 6/11/ :01 PM Complex Numbers21 E XAMPLE 10 No i s AND radicals in the denominator

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Rationalize 6/11/ :01 PM Complex Numbers22 E XAMPLE 11 No i s AND radicals in the denominator

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Rationalize 6/11/ :01 PM Complex Numbers23 E XAMPLE 12

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Rationalize 6/11/ :01 PM Complex Numbers24 Y OUR T URN

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6/11/ :01 PM Complex Numbers25 C OMPLEX N UMBERS Complex Numbers a + b i Conjugate is the complex numbers opposite sign Example: i s conjugate is 2 – 3 i Remember: NO IMAGINARY NUMBERS in the denominator SIMPLIFY RADICALS FIRST then OPERATE Real Number Imaginary Number

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Add 6/11/ :01 PM Complex Numbers26 R EVIEW

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Add 6/11/ :01 PM Complex Numbers27 E XAMPLE 13

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Subtract 6/11/ :01 PM Complex Numbers28 E XAMPLE 14

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Subtract 6/11/ :01 PM Complex Numbers29 Y OUR T URN

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Multiply FOIL (First Outer Inner Last) 6/11/ :01 PM Complex Numbers30 R EVIEW

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FOIL (First Outer Inner Last) Multiply 6/11/ :01 PM Complex Numbers31 E XAMPLE 15

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Multiply 6/11/ :01 PM Complex Numbers32 E XAMPLE 16

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Multiply 6/11/ :01 PM Complex Numbers33 Y OUR T URN

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Rationalize What is the problem with the bottom? 6/11/ :01 PM Complex Numbers34 R EVIEW We try to get rid of the radicals on the bottom so the bottom will be even With the CONJUGATE

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Rationalize What is the problem with the bottom? 6/11/ :01 PM Complex Numbers35 E XAMPLE 17 We try to get rid of the radicals on the bottom so the bottom will be even With the CONJUGATE

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Rationalize 6/11/ :01 PM Complex Numbers36 E XAMPLE 18

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Rationalize 6/11/ :01 PM Complex Numbers37 E XAMPLE 19

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Rationalize 6/11/ :01 PM Complex Numbers38 Y OUR T URN

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6/11/ :01 PMCOMPLEX NUMBERS39 I MAGINARY P OWERS Steps: 1.Look at the exponent of the imaginary number and divide by 4 2.View only the remainder or decimal 3.Convert the problem with the number REMEMBER: No is in the denominator

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6/11/ :01 PMCOMPLEX NUMBERS40 IMAGINARY NUMBERS Imaginary Exponent RemainderDecimal Imaginary Number i1i1 R10.25i i2i2 R20.50–1–1 i3i3 R30.75–i i4i4 R00.001

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6/11/ :01 PMCOMPLEX NUMBERS41 IMAGINARY NUMBERS

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Simplify 6/11/ :01 PM Complex Numbers42 E XAMPLE 20 Imaginary Exponent RemainderDecimal Imaginary Number i1i1 R10.25i i2i2 R20.50–1–1 i3i3 R30.75–i i4i4 R00.001

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Simplify 6/11/ :01 PM Complex Numbers43 E XAMPLE 21

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Simplify 6/11/ :01 PM Complex Numbers44 Y OUR T URN

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Simplify i 25 = i i 36 = 1 6/11/ :01 PM Complex Numbers45 E XAMPLE 22

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Simplify 6/11/ :01 PM Complex Numbers46 Y OUR T URN

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Simplify 6/11/ :01 PM Complex Numbers47 E XAMPLE 24

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