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5-4 Complex Numbers (Day 1) Objective: CA 5.0 Students demonstrate knowledge of how real number and complex numbers are related both arithmetically and graphically.

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Not all quadratic equations have real number solutions. has no real number solutions because the square of any real number x is never negative.

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To overcome this problem, mathematicians created an expanded system of numbers using the imaginary unit. The imaginary unit i can be used to write the square root of any negative number.

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The square root property of a negative number property 1. If r is a positive real number then:

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2. By property (1): it follows that…

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Example 1: Solve

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A complex number written in standard form is a number a + bi where a and b are real numbers. The number a is the real part of the complex number, the number bi is the imaginary part. If b 0 then a + bi is an imaginary number If a= 0 and b 0 then a + bi is a pure imaginary number.

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Every complex number corresponds to a point in the complex plane. Keep in mind: a is the real part (x –coordinate) bi is the imag. part (y-coordinate)

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Example 2: 2-3i = (2, -3) -3+2i = (-3, 2) 4i = (0, 4)

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Two complex numbers a + bi and c + di are equal if and only if a=c and b=d Sum of complex numbers Difference of complex numbers

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Simplify: -18 + -32 i18 + i32 3i2 + 4i2 7i2

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Example 3: Write the expression as a complex number in standard form. 4 – i + 3 + 2i 7 + i

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Example 4: 7 – 5i - 1 + 5i 6 + 0i 6

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Example 5: 6 + 2 - 9i - 8 + 4i -9i + 4i -5i

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To multiply complex numbers use the distributive property or the FOIL method. Multiplying Complex Numbers

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Example 5: Write each expression as a complex number in standard form. 1.

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Example 6:

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Example 7:

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Homework = Accelerated Math Objective: Add & Subtract/Multiply Complex Numbers

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