231

Complex Numbers

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Complex NumbersThe square root of a negative number has no real solution,

but it does have an imaginary one:

An expression is complex (also called imaginary) if it has an i in it.

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Rational Numbers

Irrational Numbers

Imaginary

Numbers

Real Numbers

Whole Numbers

Natural Numbers

Complex Numbers: All numbers are technically considered complex numbers. Real Numbers can be written as a + 0i - no imaginary component.

Integers

Complex Numbers

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Complex Numbers

Why does this work?

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Higher order i's can be simplified into i, -1, -i, or 1.

If the power of i is even:...and the exponent is a multiple of 4 , then it simplifies to 1.

...and the exponent is a multiple of 2,but not 4 , then it simplifies to -1.

If the power of i is odd:

...factor out one i to create an even exponent. Use the rules for even exponents and leave the factored i.

Complex Numbers

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Simplify the following:

RememberComplex Numbers

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More examples:

RememberComplex Numbers

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142 Simplify:

A i

B -1

C -i

D 1

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143 Simplify:

A i

B -1

C -i

D 1

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144 Simplify:

A i

B -1

C -i

D 1

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145 Simplify:

A i

B -1

C -i

D 1

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Simplify radical expressions that have a negative by taking out i first. Then, perform the indicated operation(s). Simplify any expression

that has a power of i greater than one.

Complex Numbers

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Examples:Complex Numbers

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146 Simplify:

A

B

C

D

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147 Simplify:

A

B

C

D

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148 Simplify:

A

B

C

D

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149 Simplify:

A

B

C

D

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150 Simplify:

A

B

C

D

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Working with Complex Numbers

Operations, such as addition, subtraction, multiplication and division, can be done with i.

Treat i like any other variable, except at the end make sure i is at most to the first power.

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Answers of complex numbers are left in standard form. The standard form of a complex number is a + bi.

Examples of standard form of a complex number:

3 - 2i 0 + 3i 8 + 0i

Working with Complex Numbers

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When adding or subtracting complex numbers, collect like terms. Leave answers in standard form.

Adding or Subtracting Complex Numbers

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When multiplying, multiply numbers, multiply i's and simplify any i with a power greater than one.

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Multiplying Complex Numbers

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When multiplying, multiply numbers, multiply i's and simplify any i with a power greater than one.

Multiplying Complex Numbers

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Multiply and leave answers in standard form.Multiplying Complex Numbers

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Multiply and leave answers in standard form.Multiplying Complex Numbers

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151 Simplify:

A

B

C

D

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152 Simplify:

A

B

C

D

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153 Simplify:

A

B

C

D

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154 Simplify:

A

B

C

D

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155 Simplify:

A

B

C

D

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Dividing with iSince i represents a square root, a fraction is not in simplified form if

there is an i in the denominator. And, similar to roots, if the denominator is a monomial just multiply top and bottom of the fraction

by i to rationalize.

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Simplify:

Dividing with i

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Simplify:

Dividing with i

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156 Simplify:

A B

C D

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157 Simplify:

A B

C D

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158 Simplify:

A B

C D

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Example:

If the denominator is a binomial including i, rationalize it by multiplying top and bottom by its conjugate. Remember using conjugates earlier in this unit: the conjugate of 4 - 3i is 4 + 3i.

Rationalizing Complex Numbers

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Simplify:

Rationalizing Complex Numbers

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159 Simplify:

A B

C D

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160 Simplify:

A

B

C

D

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161 Simplify:

A

B

C

D

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162 Simplify:

A

B

C

D

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