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ThudnBlunder
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Simply Complex
« on: Jul 27th, 2010, 12:22pm » |
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Let a,b,c,d represent complex numbers. Are the following statements True or False?
1) If a + b = 0 and |a| = |b|, then a2 = b2
2) If a + b + c = 0 and |a| = |b| = |c|, then a3 = b3 = c3
3) If a + b + c + d = 0 and |a| = |b| = |c| = |d|, then a4 = b4 = c4 = d4
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Grimbal
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Re: Simply Complex
« Reply #1 on: Jul 28th, 2010, 1:47am » |
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It looks like 1 and 2 are true, 3 not necessarily
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rmsgrey
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Re: Simply Complex
« Reply #2 on: Jul 28th, 2010, 8:07am » |
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1) a+b = 0 uniquely determines b=-a for any given a (additive inverses are unique) so the condition on the moduli is redundant. Since a2 = (-a)2, the statement is true.
2) a+b+c = 0 means c = -(a+b). The condition on the moduli then gives |a+b| = |a| = |b|, which holds true iff a=b=0, or a/b is a complex cube root of unity, in which case a/c is the other complex cube root of unity (I find it easier to see visualising it geometrically - if a+b is a point on the unit circle, then a and b are the two points on the circle a unit distance away from it, and c is the point opposite, putting a, b and c 120 degrees apart around the circle) so cubing them will give a3, 1.a3 and 12.a3, so the statement is true.
3) The conditions can be fulfilled by choosing any a, b with |a|=|b| and letting c=-a and d=-b. In general, the condition |a|=|b| is insufficient for a4[/sup=b[sup]4, so the statement is false (though there are special cases in which a,b,c,d meeting the conditions do have the same fourth power)
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