Hi. I would like to know what is the square root of i , and i squared? I am looking for a response appropriate for secondary level students. Thank you, Wayne Hi Wayne The answer to your second question is straightforward since i is, by definition, one of the two numbers satisfying i squared is -1, the other is -i. Thus the answer to your second question is that i squared is -1. The first question requires more work but should be appropriate for secondary level students if they know some trigonometry. Complex numbers can be written in the form a+ib where a and b are real numbers. This gives rise to a graphical way to represent complex numbers in what is called the complex plane. Draw a coordinate system as you would for the Cartesian plane but label the horizontal axis the real axis and the vertical axis the imaginary axis. The complex number a+ib is then representated in the complex plane by the point with coordinates (a,b). For points on the unit circle (points at a distance 1 from 0) you can use some trigonometry to get a particularly useful representation. For a point z on the unit circle measure the angle (counterclockwise) between the positive real axis and the line segment joining z to 0. This angle, written , is called the argument of z and then . Now what happens when you multiply two such numbers? If and then,expanding as you normally would, using the fact that i squared is -1 and using the double angle expressions for sine and cosine, gives Hence is the complex number on the unit circle with argument Now back to your questions. First check, using this method, that i squared is -1. The complex number i has argument 90 degrees and hence i.i has argument 90 + 90 = 180 degrees. Hence i squared = cos(180 deg) + i sin(180 deg) = -1. For your first question you want z.z=i. Since the modulus of i is 90 degrees taking z with modulus 45 degrees will work, Hence z = cos(45 deg) + i sin(45 deg) and thus There is, however, another possible choice for z, and that is with modulus 225 degrees, since doubling this angle also brings you around to i. Thus a second solution to i squared = -1 is then w=cos(225 deg) + i sin(225 deg) and thus You can check both solutions by calculating z times z and w times w. Cheers, Harley

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