Presentation on theme: "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."— Presentation transcript:
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 COMPLEX NUMBERS Objectives
R Real Numbers R Rational Numbers Q Integers Z Whole numbers W Natural Numbers N Irrational Numbers Q -bar Complex Numbers C Imaginary Numbers i
What is an imaginary number?
Previously, when we encountered square roots of negative numbers in solving equations, we would say no real solution or not a real number. The Imaginary Unit i
Complex Numbers & Imaginary Numbers
Adding and Subtracting Complex Numbers (5 11i) + (7 + 4i) Simplify and treat the i like a variable. = 5 11i i = (5 + 7) + ( 11i + 4i) = 12 7i Standard form
Adding and Subtracting Complex Numbers ( 5 + i) ( 11 6i) = 5 + i i = i + 6i = 6 + 7i
(5 – 2i) + (3 + 3i)
(2 + 6i) (12 i)
Multiplying Complex Numbers 4i (3 5i) Standard form
Multiplying Complex Numbers (7 3i )( 2 5i) use FOIL Standard form
7i (2 9i) Standard form
(5 + 4i)(6 7i) Standard form
Using Complex Conjugates to Divide Complex Numbers Divide and express the result in standard form: 7 + 4i 2 5i The complex conjugate of the denominator is 2 + 5i. Multiply both the numerator and the denominator by the complex conjugate.
Using Complex Conjugates to Divide Complex Numbers
Divide and express the result in standard form: 5 + 4i 4 i
Roots of Negative Numbers
Operations Involving Square Roots of Negative Numbers See examples on page 282.
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.
A Quadratic Equation with Imaginary Solutions See example on page 283.