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Published byCatherine Fairchild Modified over 2 years ago

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What are imaginary and complex numbers? Solve for x: x = 0 ? What number when multiplied by itself gives us a negative one? No such real number Graph it parabola does not intersect x-axis - NO REAL ROOTS

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Definition: Imaginary Numbers If is not a real number, then is a non-real or imaginary number. i A pure imaginary number is any number that can be expressed in the form bi, where b is a real number such that b ≠ 0, and i is the imaginary unit. In general, for any real number b, where b > 0: b = 5

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Powers of i i –1 i 2 = If i 2 = – 1, then i 3 = ? = i 2 i = –1( ) = –i i 2 i 2 = (–1)(–1) = 1 = i 4 i = 1( ) = i i 4 i 2 = (1)(–1) = –1 = i 6 i = -1( ) = –i i 6 i 2 = (–1)(–1) = 1 i 0 = 1 i 1 = i i 2 = –1 i 3 = –i i 4 = 1 i 5 = i i 6 = –1 i 7 = –i i 8 = 1 i 9 = i i 10 = –1 i 11 = –i i 12 = 1 What is i 82 in simplest form? 82 ÷ 4 = 20 remainder 2 equivalent to i 2 = –1 i 82 i3i3 i 4 = i 6 = i 8 = i5i5 i7i7 –1 i 2 =

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A little saying to help you remember Once I Lost one Missing eye

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Properties of i i Addition: 4i + 3i = 7i Subtraction: 5i – 4i = i Multiplication: (6i)(2i) = 12i 2 = –12 Division:

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Complex Numbers Definition: A complex number is any number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. a + bi pure imaginary number Any number can be expressed as a complex number: 7 + 0i = i = 2i real numbers a + bi

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Complex Numbers Real Numbers Rational Numbers Integers Whole Numbers Irrational Numbers Counting Numbers The Number System i i i i 75 -i 47 i i -i i3i3 i9i9 i 2 + 3i-6 – 3i 1/2 – 12i

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Model Problems Add: Express in terms of i and simplify: = 10i = 4/5i Write each given power of i in simplest terms: i 49 i 54 i 300 i 2001 = i= -1= i= 1 Multiply: Simplify:

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