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**COMPLEX NUMBERS Objectives**

Use the imaginary unit i to write complex numbers. Add, subtract, and multiply complex numbers. Use complex conjugates to write the quotient of two complex numbers in standard form. Perform operations with square roots of negative numbers Solve quadratic equations with complex imaginary solutions

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**Complex Numbers C R Real Numbers R Irrational Numbers Q -bar**

Integers Z Imaginary Numbers i Whole numbers W Natural Numbers N Rational Numbers Q

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**What is an imaginary number?**

It is a tool to solve an equation and was invented to solve quadratic equations of the form π π π +ππ+π. . It has been used to solve equations for the last 200 years or so. βImaginaryβ is just a name, imaginary do indeed exist; they are numbers.

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**The Imaginary Unit i π= βπ π π =βπ**

Previously, when we encountered square roots of negative numbers in solving equations, we would say βno real solutionβ or βnot a real numberβ. π= βπ π π =βπ

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**Complex Numbers & Imaginary Numbers**

a + bi represents the set of complex numbers, where a and b are real numbers and i is the imaginary part. a + bi is the standard form of a complex number. The real number a is written first, followed by a real number b multiplied by i. The imaginary unit i always follows the real number b, unless b is a radical. Example: 2+3π If b is a radical, then write i before the radical. π 2

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**Adding and Subtracting Complex Numbers**

(5 β 11i) + (7 + 4i) Simplify and treat the i like a variable. = 5 β 11i i = (5 + 7) + (β 11i + 4i) = 12 β 7i Standard form

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**Adding and Subtracting Complex Numbers**

(β 5 + i) β (β 11 β 6i) = β 5 + i i = β i + 6i = i

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(5 β 2i) + (3 + 3i) 5+3β2π+3π π+π

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(2 + 6i) β (12 β i) 2+6πβ12+π 2β12+6π+π βππ+ππ

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**Multiplying Complex Numbers**

4i (3 β 5i) 4π 3 β4π(5π) 12πβ20 π 2 π π =βπ 12πβ20(β1) 12π+20 ππ+πππ Standard form

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**Multiplying Complex Numbers**

(7 β 3i )( β 2 β 5i) use FOIL β14β35π+6π+15 π 2 π π =βπ β14β29π+15(β1) β14β29πβ15 βππβπππ Standard form

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7i (2 β 9i) 7π 2 +7π(β9π) 14πβ63 π 2 π π =βπ 14πβ63(β1) 14π+63 ππ+πππ Standard form

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**(5 + 4i)(6 β 7i) 30β35π+24πβ28 π 2 π π =βπ 30β11πβ28(β1) 30β11π+28**

30+28β11π ππβπππ Standard form

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Complex Conjugates The complex conjugate of the number a + bi is a β bi. Example: the complex conjugate of π+ππ is πβππ The complex conjugate of the number a β bi is a + bi. Example: the complex conjugate of 3βππ is 3+5π

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Complex Conjugates When we multiply the complex conjugates together, we get a real number. (a + bi) (a β bi) = aΒ² + bΒ² Example: 2+3π 2β3π =4β9 π 2 4β9 π 2 =4β9 β1 =4+9 =13

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Complex Conjugates When we multiply the complex conjugates together, we get a real number. (a β bi) (a + bi) = aΒ² + bΒ² Example: 2β5π 2+5π =4β25 π 2 4β25 π 2 =4β25(β1) =4+25 =29

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**Using Complex Conjugates to Divide Complex Numbers**

Divide and express the result in standard form: 7 + 4i 2 β 5i The complex conjugate of the denominator is 2 + 5i. Multiply both the numerator and the denominator by the complex conjugate.

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**Using Complex Conjugates to Divide Complex Numbers**

14+35π+8π+20 π 2 4β25 π 2 = 14+43π+20 β1 4β25 β1 14+43πβ = β6+43π 29 β π ππ + ππ ππ π =

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**Divide and express the result in standard form:**

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**Roots of Negative Numbers**

β16 = β’ β1 i = β1 = 4 β’ i =ππ

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**Operations Involving Square Roots of Negative Numbers**

See examples on page 282.

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**The complex-number system is used to find zeros of functions that are not real numbers.**

When looking at a graph of a function, if the graph does not cross the x-axis, it has no real-number zeros.

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**A Quadratic Equation with Imaginary Solutions**

See example on page 283.

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