Visualizing Complex NumbersDate: 03/19/2001 at 22:30:39 From: Anonymous Subject: Complex Numbers What exactly are imaginary numbers and how are they used? I do not quite understand how the square root of -1 is possible. Thank you. Date: 03/20/2001 at 09:23:20 From: Doctor Floor Subject: Re: Complex Numbers Hi, Thanks for writing. This is a difficult question! Imaginary numbers are all numbers that are the product of a real number and the square root of -1. The set of complex numbers is the set of numbers that can be found as the sum of a real and an imaginary number. First of all, to understand complex numbers we have to get rid of the number line alone. Instead we consider a number plane, with a grid. The usual real numbers are found along the x-axis, so that's our usual number line. So the point (x,0) in the plane we take for the real number x. To find the square root of -1, we first try to find out what multiplying by the square root of -1 should do. We do that by geometric interpretation. By such interpretation, multiplying by -1 is the same as rotating about the origin through 180 (or -180) degrees. Multiplying by the square root of -1 should do this halfway, so that after doing it twice it would be the same as multiplying by -1. And halfway rotating through 180 (or -180) degrees, is not difficult: that's the same as rotating through 90 (or -90) degrees. So we can take rotating through 90 degrees as "multiplying by the square root of -1" (through -90 degrees is then muliplying by minus the square root of -1). Now, the square root of -1 as a number in the plane is found as 1 multiplied by the square root of -1, or (1,0) after rotation through 90 degrees about (0,0), i.e. the point (0,1). And we see that imaginary numbers will all be found on the y-axis. Therefore, when we talk about the plane of complex numbers, the x-axis is called the real axis, and the y-axis the imaginary axis. I hope this helps you. Best regards, - Doctor Floor, The Math Forum http://mathforum.org/dr.math/ |
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