COMPLEX NUMBERS Objectives Use the imaginary unit i to write complex numbers. Add, subtract, and multiply complex numbers. Use complex conjugates to write the quotient of two complex numbers in standard form. Perform operations with square roots of negative numbers Solve quadratic equations with complex imaginary solutions

Complex Numbers C R Real Numbers R Irrational Numbers Q -bar Integers Z Imaginary Numbers i Whole numbers W Natural Numbers N Rational Numbers Q

What is an imaginary number? It is a tool to solve an equation and was invented to solve quadratic equations of the form 𝒂 𝒙 𝟐 +𝒃𝒙+𝒄. . It has been used to solve equations for the last 200 years or so. “Imaginary” is just a name, imaginary do indeed exist; they are numbers.

The Imaginary Unit i 𝒊= −𝟏 𝒊 𝟐 =−𝟏 Previously, when we encountered square roots of negative numbers in solving equations, we would say “no real solution” or “not a real number”. 𝒊= −𝟏 𝒊 𝟐 =−𝟏

Complex Numbers & Imaginary Numbers a + bi represents the set of complex numbers, where a and b are real numbers and i is the imaginary part. a + bi is the standard form of a complex number. The real number a is written first, followed by a real number b multiplied by i. The imaginary unit i always follows the real number b, unless b is a radical. Example: 2+3𝑖 If b is a radical, then write i before the radical. 𝑖 2

Adding and Subtracting Complex Numbers (5 − 11i) + (7 + 4i) Simplify and treat the i like a variable. = 5 − 11i + 7 + 4i = (5 + 7) + (− 11i + 4i) = 12 − 7i Standard form

Adding and Subtracting Complex Numbers (− 5 + i) − (− 11 − 6i) = − 5 + i + 11 + 6i = − 5 + 11 + i + 6i = 6 + 7i

(5 – 2i) + (3 + 3i) 5+3−2𝑖+3𝑖 𝟖+𝒊

(2 + 6i) − (12 − i) 2+6𝑖−12+𝑖 2−12+6𝑖+𝑖 −𝟏𝟎+𝟕𝒊

Multiplying Complex Numbers 4i (3 − 5i) 4𝑖 3 −4𝑖(5𝑖) 12𝑖−20 𝑖 2 𝒊 𝟐 =−𝟏 12𝑖−20(−1) 12𝑖+20 𝟐𝟎+𝟏𝟐𝒊 Standard form

Multiplying Complex Numbers (7 − 3i )( − 2 − 5i) use FOIL −14−35𝑖+6𝑖+15 𝑖 2 𝒊 𝟐 =−𝟏 −14−29𝑖+15(−1) −14−29𝑖−15 −𝟐𝟗−𝟐𝟗𝒊 Standard form

7i (2 − 9i) 7𝑖 2 +7𝑖(−9𝑖) 14𝑖−63 𝑖 2 𝒊 𝟐 =−𝟏 14𝑖−63(−1) 14𝑖+63 𝟔𝟑+𝟏𝟒𝒊 Standard form

(5 + 4i)(6 − 7i) 30−35𝑖+24𝑖−28 𝑖 2 𝒊 𝟐 =−𝟏 30−11𝑖−28(−1) 30−11𝑖+28 30+28−11𝑖 𝟓𝟖−𝟏𝟏𝒊 Standard form

Complex Conjugates The complex conjugate of the number a + bi is a − bi. Example: the complex conjugate of 𝟐+𝟒𝒊 is 𝟐−𝟒𝒊 The complex conjugate of the number a − bi is a + bi. Example: the complex conjugate of 3−𝟓𝒊 is 3+5𝒊

Complex Conjugates When we multiply the complex conjugates together, we get a real number. (a + bi) (a − bi) = a² + b² Example: 2+3𝑖 2−3𝑖 =4−9 𝑖 2 4−9 𝑖 2 =4−9 −1 =4+9 =13

Complex Conjugates When we multiply the complex conjugates together, we get a real number. (a − bi) (a + bi) = a² + b² Example: 2−5𝑖 2+5𝑖 =4−25 𝑖 2 4−25 𝑖 2 =4−25(−1) =4+25 =29

Using Complex Conjugates to Divide Complex Numbers Divide and express the result in standard form: 7 + 4i 2 − 5i The complex conjugate of the denominator is 2 + 5i. Multiply both the numerator and the denominator by the complex conjugate.

Using Complex Conjugates to Divide Complex Numbers 14+35𝑖+8𝑖+20 𝑖 2 4−25 𝑖 2 = 14+43𝑖+20 −1 4−25 −1 14+43𝑖−20 4+25 = −6+43𝑖 29 − 𝟔 𝟐𝟗 + 𝟒𝟑 𝟐𝟗 𝒊 =

Divide and express the result in standard form:

Roots of Negative Numbers −16 = 16 • −1 i = −1 = 4 • i =𝟒𝒊

Operations Involving Square Roots of Negative Numbers See examples on page 282.

The complex-number system is used to find zeros of functions that are not real numbers. When looking at a graph of a function, if the graph does not cross the x-axis, it has no real-number zeros.

A Quadratic Equation with Imaginary Solutions See example on page 283.