Ch 6 Complex Numbers

Imaginary Numbers Previously, when we encountered square roots of negative numbers in solving equations, we would say “no real solution” or “not a real number”. That’s still true. However, we will now introduce a new set of numbers. Imaginary numbers which includes the imaginary unit i.

Imaginary Numbers The imaginary unit, written i, is the number whose square is ‒1. That is,

Examples Write using i notation. a. b. c.

Example Multiply or divide as indicated. a. b.

Standard Form of Complex Numbers A complex number is a number that can be written in the form a + bi where a and b are real numbers. a is a real number and bi would be an imaginary number. If b = 0, a + bi is a real number. If a = 0, a + bi is an imaginary number.

Adding and Subtracting Complex Numbers

Example (4 + 6i) + (3 – 2i) (8 + 2i) – (4i) Add or subtract as indicated. a. b. (4 + 6i) + (3 – 2i) (8 + 2i) – (4i)

Multiplying Complex Numbers To multiply two complex numbers of the form a + bi, we multiply as though they were binomials. Then we use the relationship i2 = – 1 to simplify.

Example Multiply: 8i · 7i

Example Multiply. 5i(4 – 7i)

Example Multiply. (6 – 3i)(7 + 4i)

Complex Conjugate In the previous chapter, when trying to rationalize the denominator of a rational expression containing radicals, we used the conjugate of the denominator. Similarly, to divide complex numbers, we need to use the complex conjugate of the number we are dividing by.

Complex Conjugate The complex numbers a + bi and a – bi are called complex conjugates of each other. (a + bi)(a – bi) = a2 + b2

Example Divide.

Example Divide.

Patterns of i The powers recycle through each multiple of 4. Divide by 4 R = 0  1 R = 2  -1 R = 1  i R = 3  -i

Example Find each power of i. a. b.