Warmup

3-3 Complex Numbers Perform operations with pure imaginary numbers. Perform operations with complex numbers.

What would the solution to 𝑥 2 +1=0 be? It has no real solutions, but what if we looked at its reflection across its vertex?

The simplest imaginary number is −1 or 𝑖 Pure imaginary numbers like 6𝑖, −2𝑖, or 3 𝑖 are square roots of negative numbers. For any positive real number 𝑏, − 𝑏 2 = 𝑏 2 −1 𝑜𝑟 𝑏𝑖 Steps to find imaginary parts: Break down inside amount into factors, pulling out −1 as 𝑖 If there are matching factors, simplify out to front of radical. Rewrite result.

19. Simplify −169

The complex number a + bi can be treated as if it is a binomial, and operations on complex numbers follow properties for adding, subtracting, multiplying, and dividing binomials, with one exception. That exception is to replace  𝑖 2  with –1 whenever  𝑖 2  appears in an expression.

Adding or Subtracting: To add or subtract, combine like terms. The commutative, associative and distributive properties for Multiplication and Addition hold true for complex numbers Adding or Subtracting: To add or subtract, combine like terms. 27. −3+𝑖 +(−4 −𝑖)

Multiplying 31. Simplify (3+5𝑖)(5−3𝑖)

Dividing complex numbers Two complex numbers 𝑎+𝑏𝑖 𝑎𝑛𝑑 𝑎−𝑏𝑖 are called complex conjugates. The product of complex conjugates is always a real number. You can use this to simplify quotients of complex numbers. 33. Simplify 2𝑖 1+𝑖

Simplify to a+bi form 5 −2𝑖 3𝑖

39. Solve 2 𝑥 2 +10=0