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complex numbers Unit 3
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N-CN.A.1: I know there is a complex number i such that , and every complex number has the form with a and b being real numbers. N-CN.A.2: I can add, subtract and multiply complex numbers.
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What is a complex number?
COMPLEX means something consists of many different and connected parts. A NUMBER is an arithmetic value used to represent a quantity.
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A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, that satisfies the equation i2 = −1. In this expression, a is the real part and b is the imaginary part of the complex number.
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What is the answer to this
expression? Is this possible? Identify how many solutions you see each quadratic function.
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Not all quadratic equations have real number solutions.
For instance, has no real number solution because the square of any real number x is never negative. To overcome this problem, mathematicians created an expanded system of numbers using the imaginary unit, i, defined as . Note that The imaginary unit i can be used to write the square root of any negative number.
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I can solve quadratic equations with complex solutions.
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guided practice Example 1
Solve the equation.
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guided practice Example 1
Solve the equation.
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guided practice Example 2
Solve the equation.
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guided practice Example 2
Solve the equation.
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independent practice Example 3
Solve the equation.
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independent practice Example 3
Solve the equation.
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independent practice Example 4
Solve the equation.
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independent practice Example 4
Solve the equation.
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independent practice Example 5
Solve the equation.
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independent practice Example 5
Solve the equation.
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independent practice Example 6
Solve the equation.
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independent practice Example 6
Solve the equation.
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Checking for understanding…
Why is this true…shouldn’t both answers produce 5?
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I can calculate i to any power.
optional instructional video illustrating the pattern of imaginary values
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PATTERNS OF IMAGINARY (i) NUMBERS
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guided practice Example 7
Evaluate.
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guided practice Example 7
Evaluate.
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guided practice Example 8
Evaluate.
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guided practice Example 8
Evaluate.
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Independent practice Example 9
Evaluate.
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Independent practice Example 9
Evaluate.
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Independent practice Example 10
Evaluate.
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Independent practice Example 10
Evaluate.
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guided practice Example 11
Evaluate.
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guided practice Example 11
Evaluate.
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guided practice Example 12
Evaluate.
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guided practice Example 12
Evaluate.
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Independent practice Example 13
Evaluate.
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Independent practice Example 13
Evaluate. =
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Independent practice Example 14
Evaluate.
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Independent practice Example 14
Evaluate. =
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I can add, subtract and multiply expressions of a complex number and write the final answer in standard form.
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guided practice Example 15
Simplify the expressions containing complex numbers and write your final answer in standard form.
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guided practice Example 15
Simplify the expressions containing complex numbers and write your final answer in standard form.
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guided practice Example 16
Simplify the expressions containing complex numbers and write your final answer in standard form.
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guided practice Example 16
Simplify the expressions containing complex numbers and write your final answer in standard form.
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guided practice Example 17
Simplify the expressions containing complex numbers and write your final answer in standard form.
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guided practice Example 17
Simplify the expressions containing complex numbers and write your final answer in standard form. FOIL METHOD
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independent practice Example 18
Simplify the expressions containing complex numbers and write your final answer in standard form.
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independent practice Example 18
Simplify the expressions containing complex numbers and write your final answer in standard form.
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Watch the instructional video
I can divide expressions of a complex number and write the final answer in standard form. Watch the instructional video
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independent practice Example 19
Simplify the expression and write your final answer in standard form.
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independent practice Example 19
Simplify the expression and write your final answer in standard form.
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independent practice Example 19
Simplify the expression and write your final answer in standard form.
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independent practice Example 19
Simplify the expression and write your final answer in standard form.
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independent practice Example 19
Simplify the expression and write your final answer in standard form.
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independent practice Example 19
Simplify the expression and write your final answer in standard form.
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Checking for understanding…
True or False Every complex number is an imaginary number.
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Checking for understanding…
True or False Every complex number is an imaginary number. False; 1 is complex (1 + 0i), but it is not imaginary.
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Checking for understanding…
True or False Every irrational number is a complex number.
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Checking for understanding…
True or False Every irrational number is a complex number. True
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Checking for understanding…
True or False The sum of two imaginary numbers is always an imaginary number.
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Checking for understanding…
True or False The sum of two imaginary numbers is always an imaginary number. False; example below (3 – 4i) + (5 + 4i) = 8 + 0i
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Checking for understanding…
True or False In the complex number 3 – 7i, the real part is 3 and the imaginary part is 7i.
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Checking for understanding…
True or False In the complex number 3 – 7i, the real part is 3 and the imaginary part is 7i. False; - 7i
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