Lesson 2.4 Complex Numbers Essential Question: How do you perform operations with complex numbers?

Before we start… Solve 𝑥 2 +9=0

What is an imaginary number? An imaginary number is defined as

A number in the form a is a real number A number in the form bi is a pure imaginary number

What is a complex number? A number composed of a real number and an imaginary number. a + bi, where a is the real part and bi is the imaginary part. 5 – 6i

How do you add and subtract complex numbers? To add or subtract complex numbers, combine like terms. Combine the real parts Combine the imaginary parts Write the answer in standard form a + bi

3−𝑖 + 2+3𝑖

1+2𝑖 − 4+2𝑖

3− −2+3𝑖 + −5+𝑖

3+2𝑖 + 4−𝑖 − 7+𝑖

How do you multiply complex numbers? You will use the distributive property. Remember that 𝑖 2 =−1

5 −2+3𝑖

2−𝑖 4+3𝑖

3+2𝑖 3−2𝑖

4𝑖 −1+5𝑖

3+2𝑖 2

Complex Conjugates Sometimes the product of two complex numbers can be a real number. This occurs with pairs of complex numbers of the forms a + bi and a – bi called complex conjugates. 𝑎+𝑏𝑖 𝑎−𝑏𝑖 =𝑎2 –𝑎𝑏𝑖+𝑎𝑏𝑖 –𝑏2𝑖2 =𝑎2 –𝑏2(–1) = 𝑎2 + 𝑏2

Multiply each complex number by its complex conjugate. 1+𝑖

Multiply each complex number by its complex conjugate. 3−5𝑖

How do you divide complex numbers? Multiply the numerator and denominator by the conjugate of the denominator.

Write the quotient in standard form. 2+3𝑖 4−2𝑖

Write the quotient in standard form. 2+𝑖 2−𝑖

Write the quotient in standard form. 3−4𝑖 6−𝑖

Solutions to Quadratic Equations We know that often solutions to our quadratic equations result in answers such as −5 , which isn’t a real number. We can write it as an imaginary number.

Principal Square Root If a is a positive number, then the principal square root of the negative number –a is defined as −𝑎 = 𝑎 𝑖

Write the complex number in standard form. −3 −12

Write the complex number in standard form. −48 − −27

Write the complex number in standard form. −1+ −3 2

Solve 𝑥 2 +4=0

Solve 𝑥 2 +8=0

Solve 3𝑥 2 −2𝑥+5=0

Solve 4𝑥 2 +16𝑥+17=0

How do you perform operations with complex numbers?

Ticket Out the Door Simplify 3−2𝑖 −4+5𝑖 − 2−6𝑖