Roots, Radicals, and Complex Numbers Chapter 7 Roots, Radicals, and Complex Numbers

Chapter Sections 7.1 – Roots and Radicals 7.2 – Rational Exponents 7.3 – Simplifying Radicals 7.4 – Adding, Subtracting, and Multiplying Radicals 7.5 – Dividing Radicals 7.6 – Solving Radical Equations 7.7 – Complex Numbers Chapter 1 Outline

7.7 Complex Numbers Recognize a Complex Number. Addition and Subtraction Multiplication Conjugates and Division Powers of i

Recognize a Complex Number An imaginary number is a number such as −5 . It is called imaginary because when imaginary numbers were first introduced, many mathematicians refused to believe they existed! Every real number and every imaginary number are also complex numbers. Real part: a Imaginary part: b Imaginary Unit: i= −1 Square Root of a Negative Number For any positive real number n, Complex Number are numbers of the form: 𝑎+𝑏𝑖 a & b are real numbers and i is the imaginary unit 𝑺𝒕𝒂𝒏𝒅𝒂𝒓𝒅 𝒇𝒐𝒓𝒎 𝒐𝒇 𝒂 𝑪𝒐𝒎𝒑𝒍𝒆𝒙 𝑵𝒖𝒎𝒃𝒆𝒓:𝒂+𝒃𝒊

Recognize a Complex Number All 4 operations can be performed on Complex numbers. To do so: use the definition: i= −1 And 𝑖 2 =−1

Example 1: Write each complex numbers in standard form :𝒂+𝒃𝒊

Add and Subtract Complex Numbers We add complex numbers just like we add polynomials—by combining like terms. c) (3 + 4i) – (4 – 12i) 3 + 4i – 4+ 12i = 7 + 16i Example 2: Add. b) (9 + 6i) + (6 – 13i) = –3 – 7i

Multiply Complex Numbers c) (9 – 4i)(3 + i) Change all imaginary numbers to bi form. Multiply the complex numbers as you would multiply polynomials. Substitute –1 for each i2. Combine the real parts and the imaginary parts. Write the answer in a + bi form. Recall: (a + b)(a – b) = a2 – b2 (a + bi)(a – bi) = a2 + b2 Example 3: Multiply. d) (7 – 2i)(7 + 2i) b)(6i)(3 – 2i)

Divide Complex Numbers The conjugate of a complex number a + bi is a – bi. For example, Complex Number Conjugate To Divide Complex Numbers Change all imaginary numbers to bi form. Multiply both the numerator and denominator by the conjugate of the denominator. Write the answer is a + bi form.

Example 4: Divide (Write in standard form). -5

Find Powers of i Example 5: Simplify. The successive powers of i rotate through the four values of i, -1, -i, and 1. Example 5: Simplify. in = i if n = 1, 5, 9, … in = 1 if n = 4, 8, 12, … in = -1 if n = 2, 6, 10, … in = -i if n = 3, 7, 11, …

Example 6: Simplify -15