Imaginary Numbers Multiples of i are imaginary numbers. Square roots of negative numbers are imaginary.
Complex Numbers Any number that can be written in the form “ a+b i ” is a complex number. This is called the standard form. In “a+bi” both a and b are real numbers a is called the real part b is called the imaginary part A complex number is simplified iff it is written in standard form, “ a+b i ”
Operations with Complex Numbers Addition: Combine like terms Add the real parts, add the imaginary parts Subtraction: Combine like terms Subtract the real parts, subtract imaginary parts Don’t forget to distribute the negative Multiplication: Use distributive property or FOIL Replace i 2 with -1 Always write final answer in standard form, “ a+b i ”
Dividing Complex Numbers 1. Multiply the numerator and denominator by the complex conjugate of the denominator 2. FOIL the numerator and simplify 3. Denominator becomes a 2 +b 2 4. Write the result in standard form
Principal Square Root of Negative Numbers Product and quotient properties of square roots do not apply to square roots of negative numbers. Take out the i before multiplying or dividing square roots
Example 4 (like HW #29-44) Perform the indicated operations and write the result in standard form
You-Try #4 (like HW #29-44) Perform the indicated operations and write the result in standard form
Quadratic Formula It works to find complex solutions of quadratic equations, not just real ones.
Example 5 (like HW #45-50) Solve for x. Express solutions in standard form.
You-Try #5 (like HW #45-50) Solve for x. Express solutions in standard form.