Name: Date: Page 1 of 4 Activity 2.3.2 Imaginary Numbers In 50 A.D., Heron of Alexandria ran into a problem when he tried to take the square root of a negative number. He quit when he could not figure it out! In the 1500’s Girolamo Cardano solved an equation and came up with 15 as one of the solutions. He found no pleasure in working with these numbers and considered them useless. Many other mathematicians agreed with him. In 1637 Rene Descartes called these numbers “imaginary”, meaning it in a derogatory way. Other mathematical greats such as Isaac Newton and Albert Girard called these “solutions impossible.” In the 18th century, Euler introduced the notation = √−1 More and more mathematicians found uses for these numbers and helped the world understand them. 1. Discover the interesting pattern occurs when you raise i to a power: a. Fill in the table: 1 = 1 = 2 = ∙ = −1 2 = −1 3 = ( · ) · = −1 · = − 3 = ____ 4 = ( · ) · ( · ) = (−1) · (−1) = 1 4 = ___ 5 = ( · · · ) · = 1 · = 5 =____ 6 = ( ∙ ∙ ∙ ) ∙ ( ∙ ) = 4 ∙ 2 = 1 ∙ 2 = −1 6 =____ 7 = 8 = 9 = Activity 2.3.2 7 =____ 8 =____ 9 =____ Connecticut Core Algebra 2 Curriculum Version 1.0 Name: Date: 10 = Page 2 of 4 10 =____ 11 = 11 =____ 12 = 12 =____ b. What is the pattern that occurs when i is raised to the exponents 1, 2, 3, 4, 5 …in order? c. How many terms does it take till the pattern is repeated? 2. a. Fill in the blank: 16 =____ 20 =____ 44 =____ 100 =____ b. Notice that 16=4·4, 20=4·5, 44=4·11, and 100=4·25 . We say that given any natural number ‘n’ 4n is a multiple of 4. Also 4n is divisible by 4. Conclusion: 4 =____ 3. a. Consider 17 =____ 45 =____ 101 =____ 8889 =____ b. If we divide the exponent of i into groups of 4, and the remainder is 1 then we have 4+1 = 4 · 1 = 1 ∙ = ________ Activity 2.3.2 Connecticut Core Algebra 2 Curriculum Version 1.0 Name: Date: Page 3 of 4 4. Try these, then write a procedure for simplifying i is raised to a natural number. a. 60 =____ b. 61 =____ g. 26 =____ c. 62 =____ h. 19 =____ d. 63 =____ e. 64 =____ j. 412 =____ f. 65 =____ k. 413 =_____ 5. In full sentences, explain how to simplify i is raised to a natural number. 6. Remember the rules for a square root that give real numbers? Do these rules hold if a square root is an imaginary number? Recall that √9 ∙ √25 = √9 ∙ 25 , because the left side is 3·5 and the right side is √225 that equals 15, also. In general, if a and b are positive real numbers, then √ ∙ √ = √ ∙ Let’s look at this ‘proof’ that -1 = 1. Either -1 really does equal 1, or you have to find a flaw in the argument: 2 = −1 and 2 also equals ∙ . Since ∙ is equivalent to √−1 ∙ √−1 , which we then write as √−1 ∙ −1. Multiplying -1 times -1 gives 1, so 2 = √−1 ∙ −1 = √1 = 1 . So , from the beginning: 2 = −1, and 2 = 1. In other words −1 = 2 = 1 , and -1 =1 What is the flaw in this argument? 7. When multiplying or dividend by the square root of a negative number, you must rewrite the radical so that it is an real number multiplied by Example: √−3 ∙ √−3 = (√−1 ∙ √3)(√−1 ∙ √3) = √3 ∙ √3 = 2 √9 = −1 ∙ 3 = −3 a. √−5 ∙ √−5 = ______ Activity 2.3.2 b. 2√−6 ∙ √−6 = ______ Connecticut Core Algebra 2 Curriculum Version 1.0 Name: c. 3√−4 ∙ 2√24 = ______ Date: Page 4 of 4 d. √−30 ∙ √2 = ______ e. 2√2 ∙ √−10 = ______ g. 5√−6 ∙ 4√2 = ______ h.3√−25 ∙ 6√5 = ______ \ f. 4√−2 ∙ 3√−8 = ______ 2 i. (5√−2)2 = ______ j. −2√−12 ∙ √−3 = ______ k. (−2√−17) = ______ 2 l. 6√−7 ∙ 2√−14 = ______ Activity 2.3.2 m. √−3 3 ______ n. √−40 2 = ______ Connecticut Core Algebra 2 Curriculum Version 1.0