Brian_D
Gold Member
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- Homework Statement
- Solve ##\left(-2+2 \,\mathrm{I}\right)^{\frac{1}{3}}##
- Relevant Equations
- $$a+\mathit{bi}=r\! \left(\cos\! \left(\theta \right)+\mathrm{I} \sin\! \left(\theta \right)\right)$$
$$r=\sqrt{a^{2}+b^{2}}$$
$$\theta =\arctan\! \left(\frac{b}{a}\right)$$
$$\mathit{nth} \mathit{root} \mathit{of} a \mathit{complex} \mathit{number}=r^{\frac{1}{n}} \left(\cos\! \left(\alpha \right)+\mathrm{I} \sin\! \left(\alpha \right)\right)$$
$$\alpha=\frac{\theta+k\cdot \left(2\pi\right)}{n}$$
k=0,1,..n-1
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this problem, ##\left(r\right)^{\left(\frac{1}{n}\right)}## is ##\left(-2+2 \,\mathrm{I}\right)^{\frac{1}{3}}## which is 1+i, which we can see is correct because ##\left(1+\mathrm{i}\right)^{3}## equals ##-2+2i## So if 1+i is the first root, when you apply the formula you get three more supposed roots when k=0, 1, and 2. Can anyone explain this contradiction?
Secondly, when I apply the formula, it doesn't work. For example, let's see what happens when k=1. First, from the given problem, a=-2 and b=2, so theta equals arctan(-1), which is -pi/4 radians. Then the formula says to add k*2pi and divide by n, which gives us -pi/4+2pi divided by 3, which is about 1.83. So alpha is 1.83 and ##r^{\frac{1}{n}} \left(\cos\! \left(\alpha \right)+\mathrm{I} \sin\! \left(\alpha \right)\right)## equals (1+i)*(-.259 + .966i). Multiplying this out, we get ##\left(-1.225+ 0.707 i\right)##, which according to the formula should be one of the cube roots. But ##\left( 1.225+ 0.707 i\right)^{3}## does not equal -2+2i, so there is something wrong. Your thoughts?
Secondly, when I apply the formula, it doesn't work. For example, let's see what happens when k=1. First, from the given problem, a=-2 and b=2, so theta equals arctan(-1), which is -pi/4 radians. Then the formula says to add k*2pi and divide by n, which gives us -pi/4+2pi divided by 3, which is about 1.83. So alpha is 1.83 and ##r^{\frac{1}{n}} \left(\cos\! \left(\alpha \right)+\mathrm{I} \sin\! \left(\alpha \right)\right)## equals (1+i)*(-.259 + .966i). Multiplying this out, we get ##\left(-1.225+ 0.707 i\right)##, which according to the formula should be one of the cube roots. But ##\left( 1.225+ 0.707 i\right)^{3}## does not equal -2+2i, so there is something wrong. Your thoughts?