Ok, here is the issue: if you imagine that there is any way to compare complex numbers, e.g.z < w, then you would want the following properties to be true:

∗ For any two complex numbers z and w, either z < w,w < z, OR w = z.

Now, imagine that we have such a way to compare imaginary numbers. Since i 6= 0, the abovecondition tells us that either 0 < i or i < 0. If it is the first one, then we find that

0 < i ⇒ 0 · i < i2 ⇒ 0 < −1.

However, you also have that

0 < i ⇒ 0 · i < i4 ⇒ 0 < 1,

which means that0 < −1 ⇒ 0 + 1 < −1 + 1 ⇒ 1 < 0;

thus we find that 0 < 1 AND 1 < 0, contradicting the property above. A similar argumentworks for the case i < 0, and so no such ordering is possible.

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