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        riddles >> easy >> Simply Complex
        
(Message started by: ThudanBlunder on Jul 27^th, 2010, 12:22pm)

Title: Simply Complex
Post by ThudanBlunder on Jul 27^th, 2010, 12:22pm

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

Title: Re: Simply Complex
Post by Grimbal on Jul 28^th, 2010, 1:47am

It looks like [hide]1 and 2 are true, 3 not necessarily[/hide]

Title: Re: Simply Complex
Post by rmsgrey on Jul 28^th, 2010, 8:07am

1) [hide] 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.[/hide]

2) [hide] 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.[/hide]

3) [hide]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 a^{4[/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)[/hide]