University:
Glendale Community College
Course:
MAT140 | College Mathematics
Academic year:

2019
Views:

487

Pages:

2
Author:

izbra0p

Section 4.8: Imaginary Numbers Chapter 4: Polynomial and Radical Operations Imaginary number “i” • • • “i” is called the imaginary number in mathematics. “i” is helpful when finding square roots of negative numbers “i” has the properties: Multiplication with “i” The rules for multiplying with “i” are the same as the rules for multiplying with any variable. There is one exception. When multiplying with “i”, 𝑖 2 can be replaced with −1. For Example: (5𝑖)(6𝑖) = 5 × 6 × 𝑖 × 𝑖 = 30𝑖 2 = 30 × (−1) = −30 For Example: (−3𝑖)(4𝑖) = −3 × 4 × 𝑖 × 𝑖 = −12𝑖 2 = −12 × (−1) = 12 For Example: (2𝑖)2 = (2𝑖)(2𝑖) = 2 × 2 × 𝑖 × 𝑖 = 4𝑖 2 = 4 = −1 = −4 How to find the square root of a negative number Step 1: Make the radicand positive by writing an “i” after the square root symbol. (Make sure the “i” is not under the square root symbol. Step 2: Reduce what is left under the radical and leave the “i” after the radical. (This is the answer.) Step 3: Check, to check square your answer and make sure it equals the original expression For Example: Reduce √−20 Step 1: √−20 = √20𝑖 Step 2: 20 = 2 × 2 × 5 √20𝑖 = 2√5𝑖 Solution: √−20 = 2√5𝑖 Step 3: 2 2 (2√5𝑖) = 22 √5 𝑖 2 = 4 × 5 × (−1) = −20 ✓ Section 4.8: Imaginary Numbers Chapter 4: Polynomial and Radical Operations Simplify using order of operations: −2+√(2)2 −4(1)(5) 2(1) First: Simplify the radicand to a single number. = = = = −2+√4−4(1)(5) 2(1) −2+√4−20 2(1) −2+√−16 2(1) −2+4𝑖 2(1) Second: Simplify the denominator = −2+4𝑖 2 Third create two fractions and simplify each. = −1 1 + 4𝑖 2 (treat the Answer: −1 + 2𝑖 4𝑖 𝑗𝑢𝑠𝑡 2 𝑎𝑠 𝑦𝑜𝑢 𝑤𝑜𝑢𝑙𝑑 4𝑥 2 𝑎𝑛𝑑 𝑐𝑎𝑛𝑐𝑒𝑙 𝑡ℎ𝑒 2 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 4.)