Cube Roots of NumbersDate: 11/05/97 at 12:29:41 From: Jared Martin Subject: Cube roots of numbers If x^3 = N, where N is some expression (which could be a constant), then you have a degree three equation so there must be three roots. If you take i (sqrt(-1)), then the cube root is -i. But since x^3 = i is degree three, there should be three different values of x. What are they? How do you determine these three values for other numbers? Is there a formula? Please help. Date: 11/05/97 at 16:55:49 From: Doctor Anthony Subject: Re: Cube roots of numbers You are quite right that there will be 3 cube roots of a number. You do need to work with complex numbers, however, to understand how to find the three roots, so if what I show you is not clear, it will become so when you have studied complex numbers. Suppose z^3 = 8 Now taking the cube root of each side you would say that z = 2, however, there are two other cube roots which we shall now find. Since cos(2k.pi) = 1 and sin(2k.pi) = 0 where k is any integer, we could write the equation z^3 = 8(cos(2k.pi) + i.sin(2k.pi)) Take cube root of both sides, and use deMoivre's theorem which shows that: [cos(x) + i.sin(x)]^(1/3) = cos(x/3) + i.sin(x/3) to get z = 2[cos(2k.pi/3) + i.sin(2k.pi/3)] k = 0, 1, 2 k=0 gives z1 = 2(cos(0) + i.sin(0)] = 2 (the one real root) k=1 gives z2 = 2(cos(2.pi/3) + i.sin(2.pi/3)) = 2(-1/2 + i.sqrt(3)/2) k=2 gives z3 = 2(cos(4.pi/3) + i.sin(4.pi/3)) = 2(-1/2 - i.sqrt(3)/2) If we give k more values, 3, 4, 5, ..... we simply repeat the three roots already found. If you represent the three roots on an Argand diagram that has real values along the x axis and imaginary values on the y axis, the three roots will appear as the three spokes of a wheel, with the z values lying on a circle of radius 2 units. One root will lie along the positive x axis, and the other two at +120 degrees and -120 degrees to the x axis. So the roots are symmetrically spaced round the circle. In fact this is always the way that cube roots of a real number will look. If you take the cube root of an imaginary number, say i, then you still get three spokes but they will be rotated round to lie along the 30 degree, 150 degree and 270 degree lines on the unit circle. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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