complex numbers Unit 3

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.

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.

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.

What is the answer to this expression? Is this possible? Identify how many solutions you see each quadratic function.

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.

I can solve quadratic equations with complex solutions.

guided practice Example 1 Solve the equation.

guided practice Example 1 Solve the equation.

guided practice Example 2 Solve the equation.

guided practice Example 2 Solve the equation.

independent practice Example 3 Solve the equation.

independent practice Example 3 Solve the equation.

independent practice Example 4 Solve the equation.

independent practice Example 4 Solve the equation.

independent practice Example 5 Solve the equation.

independent practice Example 5 Solve the equation.

independent practice Example 6 Solve the equation.

independent practice Example 6 Solve the equation.

Checking for understanding… Why is this true…shouldn’t both answers produce 5?

I can calculate i to any power. optional instructional video illustrating the pattern of imaginary values https://www.youtube.com/watch?v=rDLDGQMKT3M

PATTERNS OF IMAGINARY (i) NUMBERS

guided practice Example 7 Evaluate.

guided practice Example 7 Evaluate.

guided practice Example 8 Evaluate.

guided practice Example 8 Evaluate.

Independent practice Example 9 Evaluate.

Independent practice Example 9 Evaluate.

Independent practice Example 10 Evaluate.

Independent practice Example 10 Evaluate.

guided practice Example 11 Evaluate.

guided practice Example 11 Evaluate.

guided practice Example 12 Evaluate.

guided practice Example 12 Evaluate.

Independent practice Example 13 Evaluate.

Independent practice Example 13 Evaluate. =

Independent practice Example 14 Evaluate.

Independent practice Example 14 Evaluate. =

I can add, subtract and multiply expressions of a complex number and write the final answer in standard form.

guided practice Example 15 Simplify the expressions containing complex numbers and write your final answer in standard form.

guided practice Example 15 Simplify the expressions containing complex numbers and write your final answer in standard form.

guided practice Example 16 Simplify the expressions containing complex numbers and write your final answer in standard form.

guided practice Example 16 Simplify the expressions containing complex numbers and write your final answer in standard form.

guided practice Example 17 Simplify the expressions containing complex numbers and write your final answer in standard form.

guided practice Example 17 Simplify the expressions containing complex numbers and write your final answer in standard form. FOIL METHOD

independent practice Example 18 Simplify the expressions containing complex numbers and write your final answer in standard form.

independent practice Example 18 Simplify the expressions containing complex numbers and write your final answer in standard form.

Watch the instructional video I can divide expressions of a complex number and write the final answer in standard form. Watch the instructional video https://www.youtube.com/watch?v=Z8j5RDOibV4

independent practice Example 19 Simplify the expression and write your final answer in standard form.

independent practice Example 19 Simplify the expression and write your final answer in standard form.

independent practice Example 19 Simplify the expression and write your final answer in standard form.

independent practice Example 19 Simplify the expression and write your final answer in standard form.

independent practice Example 19 Simplify the expression and write your final answer in standard form.

independent practice Example 19 Simplify the expression and write your final answer in standard form.

Checking for understanding… True or False Every complex number is an imaginary number.

Checking for understanding… True or False Every complex number is an imaginary number. False; 1 is complex (1 + 0i), but it is not imaginary.

Checking for understanding… True or False Every irrational number is a complex number.

Checking for understanding… True or False Every irrational number is a complex number. True

Checking for understanding… True or False The sum of two imaginary numbers is always an imaginary number.

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

Checking for understanding… True or False In the complex number 3 – 7i, the real part is 3 and the imaginary part is 7i.

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