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Imaginary Problems That Won't Go Away...
Even When Ordered

Question 1: Are ii and i-i real numbers?

Question 2: If yes, which is larger?

 


Hint: Think Euler...or De Moivre...or...

 


Solution Commentary: By Euler's equation, eix = cos(x) + i sin(x).

For x = pi/2, we get e(i*pi)/2 = cos(pi/2) + i sin(pi/2) = 0 + i = i.

Now, raising each side to the ith power, e(i*i*pi)/2 = ii.

But, i*i = -1, which implies that e(-pi)/2 = ii is a real number.

Finally, e(pi)/2 > e(-pi)/2 where e(pi)/2 = i-i. Thus, i-i > ii.

For a variation on this argument, see Churchill's Complex Variables and Applications (1960), pp. 60-62.