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1.4 – Complex Numbers

1.4 – Complex Numbers. Real numbers have a small issue; no symmetry of their roots To remedy this, we introduce an “imaginary” unit, so it does work The number i is defined such that . Simplifying Negative Roots. For a positive number a,

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1.4 – Complex Numbers

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  1. 1.4 – Complex Numbers

  2. Real numbers have a small issue; no symmetryof their roots • To remedy this, we introduce an “imaginary” unit, so it does work • The number iis defined such that

  3. Simplifying Negative Roots • For a positive number a, • Follow all other rules to simplify the remaining radical • Example. Simplify:

  4. Complex Numbers • A complex number, a+bi, has the following: • Real part, a • Imaginary part, bi • Only equal if both parts are equal (real/imaginary) • 5 + 10i • With imaginary numbers, only combine the like terms (real with real, imaginary with imaginary) • Multiplication, follow same rules as polynomials (FOIL, like terms, etc.)

  5. Example. Simplify: • Example. Simplify:

  6. Quotients • Similar to radical expressions, denominators of fractions cannot contain imaginary numbers or a complex number • Use the complex conjugate = for given complex number a+bi, the complex conjugate is a-bi

  7. Example. Simplify the quotient: • Note the denominator contains the complex number, 4-3i • What is the complex conjugate? 4+3i

  8. Example. Simplify the following:

  9. Roots and Complex Numbers • When dealing with negative roots, we can simplify using the rules introduced • Now, we can simplify radicals in a second way • Example. Simplify: • How can we write ?

  10. Powers of i • The imaginary number, i, has a particular pattern • i2= -1 • i3 = i2 x i = -1 x i= -i • i4 = i2 x i2 = -1 x -1 = 1 • i = i • Pull out powers that are multiples of 4; those will become 1

  11. Example. Simplify: • i15 = i12x i3 = • 4i25

  12. Assignment • Page 61 • #1-41 odd

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