The Absolute Value of a Complex NumberDate: 05/06/98 at 21:00:16 From: Russell Nadel Subject: Complex Numbers In our Alg II class, we were working with complex numbers (a + bi format). Our textbook and graphing calculators said that the absolute value of a + bi was the same thing as sqrt(a^2 + b^2). We couldn't understand where the i went. Could you help us? Date: 05/11/98 at 10:56:02 From: Doctor Nick Subject: Re: Complex Numbers Hi Russell - One way of understanding absolute value is the following: the absolute value of a number is the distance from that number to zero. That is, if we think of a number as a point in the x-y plane, the absolute value of that number is the distance between that point and the origin. Now, if the number is real, as a point it lies on the x-axis, so the distance from the origin to the number is just the usual absolute value. But, if we take a complex number a + bi, it sits at the point (a,b) in the x-y plane. How far is (a,b) from (0,0)? Well, we can make a triangle to help us out: draw a line (the hypotenuse) between the origin and (a,b); draw a line from (a,0) to (a,b); draw a line (a,0) to (0,0) That gives us a right triangle that lies on the x-axis. Now, since it's a right triangle, we know that the length of the hypotenuse is the square root of the sum of the squares of the other sides. How long are these other sides? Well, one is the absolute value of a, and the other is the absolute value of b. (If a and b are positive, you don't need the absolute value, but we need to allow a and b to be negative, so that a + bi can be any complex number.) Now, the Pythagorean theorem says that the distance from (a,b) to (0,0) is the square root of |a|^2 + |b|^2. But, |a|^2 = a^2 for all real numbers a. (Why? If a >= 0, then |a| = a, so it's true. If a < 0, then |a| = -a, so |a|^2 = (-a)^2 = ((-1)^2)*a^2 = (1)a^2 = a^2.) So the distance from (a,b) to (0,0) is the square root of a^2 + b^2, and that's why it's the absolute value of a + bi. Notice that this formula works for real numbers as well as complex ones. If x is real, then x = x + 0i, and the absolute value of x is the square root of x^2 + 0^2, which is just the square root of x^2. What's the square root of x^2? The usual absolute value of x. Have fun, -Doctor Nick, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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