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where the first coordinate represents the direction angle in radians, and the second coordinate represents the magnitude If we draw this vector on a polar graph, we can see that this is the polar representation of the complex number 1 + j0, which is equal to the pure real number 1 (If you like, you can use the conversion formulas to prove it) In the case of (4p /3,1)3, we get the direction angle (4p /3) 3 = 4p That s outside the allowed range of angles, but if we subtract 2p twice, then we get an angle of 0, and that s allowed As before, we get a magnitude of 1 3 = 1 Now we know that (4p /3,1)3 = (0,1) where, again, the first coordinate represents the direction angle in radians, and the second coordinate represents the magnitude This is the same as the previous result It s the polar representation of 1 + j0, which is the pure real number 1 We ve found two cube roots of 1 in the realm of the complex numbers Neither of these roots show up when we work with pure real numbers exclusively There are three different complex cube roots of 1! They are
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The pure real number 1 The complex number corresponding to the polar vector (2p /3,1) The complex number corresponding to the polar vector (4p /3,1)
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This is an open-book quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find worked-out answers in App A The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! 1 Prove that j is not equal to j, even though, when squared, they both give us 1 Here s a hint: Use the tactic of reductio ad absurdum, where a statement is proved by assuming its opposite and then deriving a contradiction from that assumption 2 Show that the reciprocal of j is equal to its negative; that is, j 1 = j 3 Find the sum and difference of the complex numbers 3 + j4 and 1 + j5 4 Find the ratio of the generalized complex conjugates a + jb and a jb That is, work out a general formula for (a + jb) / (a jb) where a and b are both nonzero real numbers 5 Prove that if we take any two complex conjugates and square them individually, the results are complex conjugates In other words, show that for all real-number coefficients a and b, (a + jb)2 is the complex conjugate of (a jb)2
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6 Find the polar product of the polar complex vectors (p /4,21/2) and (3p /4,21/2) Then convert this product vector to Cartesian form and write down the real-plus-imaginary complex number that it represents 7 Convert the polar complex vectors (p /4,21/2) and (3p /4,21/2) to the complex numbers they represent in real-plus-imaginary form Multiply these numbers and compare with the solution to Problem 6 8 Look at the results of the last challenge, where we found these three cube roots of 1: The pure real number 1 The complex number corresponding to the polar vector (2p /3,1) The complex number corresponding to the polar vector (4p /3,1) Convert the polar vectors (2p /3,1) and (4p /3,1) to their real-plus-imaginary complex-number forms 9 Graph the three cube roots of 1 as polar complex vectors Label them as ordered pairs in the form (q,r), where q is the direction angle and r is the magnitude 10 Graph the three cube roots of 1 as Cartesian complex vectors Label them as complex numbers in the form a + jb, where a and b are real numbers Also graph the unit circle, and note that the vectors all terminate on that circle
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