1/31/2017 (67) Powers of the imaginary unit What are the imaginary numbers? Introduction to complex numbers Algebra II Khan Academy

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1 Powers of the imaginary unit Learn how to simplify any power of the imaginary unit i. For example, simplify i²⁷ as -i. Share to Google Classroom Share Tweet We know that i = 1 and that i = 1. But what about i? i? Other integer powers of i? How can we evaluate these? Finding i and i The properties of exponents can help us here! In fact, when calculating powers of i, we can apply the properties of exponents that we know to be true in the real number system, so long as the exponents are integers. With this in mind, let's find i and i. We know that i = i i. But since i = 1, we see that: i 3 2 = i i Similarly i = i i. Again, using the fact that i = 1, we have the following: = ( 1) i = i /6

2 i = i i = ( 1) ( 1) More powers of i Let's keep this going! Let's find the next 4 powers of iusing a similar method. i 5 = i 4 i i Since i 4 = i i 6 = i 4 i 2 ( 1) Since i 4 and i 2 = 1 = 1 i 7 = i 4 i 3 ( i) Since i 4 and i 3 = i = i 2/6

3 i 8 = i 4 i 4 1 Since i 4 The results are summarized in the table i i i i i i i i i 1 i 1 i 1 i 1 An emerging pattern From the table, it appears that the powers of i cycle through the sequence of i, 1, i and 1. Using this pattern, can we find i? Let's try it! The following list shows the first numbers in the repeating sequence. i, 1, i, 1, i, 1, i, 1, i, 1, i, 1, i, 1, i, 1, i, 1, i, 1 According to this logic, i should be equal to 1. Let's see if we can support this by using exponents. Remember, we can use the properties of exponents here just like we do with real numbers! i 4 5 = (i ) = (1) 5 4 i Simplify 3/6

4 Either way, we see that i. Larger powers of i Suppose we now wanted to find i. We could list the sequence i, 1, i, 1,... out to the term, but this would take too much time! th Notice, however, that i, i, i that i raised to a multiple of 4 is 1., etc., or, in other words, We can use this fact along with the properties of exponents to help us simplify i. Example Solution While is not a multiple of 4, the number 136 is! Let's use this to help us simplify i. 4/6

5 i = i 136 i 2 = ( i 4 34 ) i 2 = ( i 4 ) 34 i = 4 34 So i = 1. = (1) 34 i 2 1 = 1 i 4 i 2 = Now you might ask why we chose to write i as i i. Well, if the original exponent is not a multiple of 4, then finding the closest multiple of 4 less than it allows us to simplify the power down to i, i, or i just by using the fact that i. 2 This number is easy to find if you divide the original exponent by 4. It's just the quotient (without the remainder) times 4. [Can you show me an example?] [I'm still confused. Can I see some more examples?] Let's practice some problems Problem /6

6 Problem 2 16 Problem Challenge Problem 1 Which of the following is equivalent to i? 1 1 i i 6/6

N-CN Complex Cube and Fourth Roots of 1

N-CN Complex Cube and Fourth Roots of 1 Task For each odd positive integer, the only real number solution to is while for even positive integers n, x = 1 and x = 1 are solutions to x n = 1. In this problem