Lesson 5-6 Complex Numbers

Recall Remember when we simplified square roots like: √128 = √64 ● √2 = 8√2 ? Remember that you couldn’t take the square root of a negative number, because it produced imaginary numbers like: √-64 = imaginary We can get around this restriction with a simple definition.

Definition √-1 = i, which stands for imaginary. So, √-64 = √64 ● √-1 = 8i Ex: √-104 = √2 ● √2 ● √2 ● √13 ● √-1 = 2i √26 You try: √-54 = ?

Definition A Complex Number has the form; a + bi. a and b are real numbers. a is called the real part of the complex number, and bi is called the imaginary part of the complex number. Ex: 5 - √-4 = 5 - 2i You try: 6 + √-49 = ?

Assignment Page 278 Numbers 1-18.

Additive Inverse of a Complex Number A Complex Number has the form; a + bi. a and b are real numbers. To obtain the additive inverse of a complex number, simply change both numbers’ signs. Ex. 5 + 2i becomes – 5 – 2i.

Adding and Subtracting Complex Numbers To add, simply add the real parts and the imaginary parts together. Ex. (2 + 4i) + (5 – i) = 7 + 3i Ex. ( i ) + (– 2 – 3i) = – 2 – 2i To subtract, change the second complex number into its additive inverse, then add. Ex. (2 + i) – (5 + 2i) becomes (2 + i) + (– 5 – 2i) = – 3 – i

Assignment Page 278 Numbers 24-34.

Multiplying Complex Numbers To multiply complex numbers: Distribute or FOIL the real and imaginary parts. Apply the fact that i2 = -1 Combine the two real terms and the two imaginary terms. Example: (2 + i)(5 + 2i) = 10 + 4i + 5i + 2i2 = 10 + 4i + 5i – 2 = 8 + 9i

Your Turn Assignment (6 – 3i)(2 + 2i) = ? Worksheet on Multiplying Complex Numbers

Absolute Value of Complex Numbers To calculate the absolute value of complex numbers use the Pythagorean theorem for the numbers of the complex number. Remember to simplify radicals. Examples: │2 + 3i│= √(2)² + (3)² = √4 + 9 = √13 │4 – 6i│= √(4)² + (–6)² = √16 + 36 = √52 = 2√13

Your Turn Assignment │5i│= ? │4 + 8i│= ? Worksheet on the Absolute Value of Complex Numbers