Linear Algebra, Theory And Applications, 2012a

20 PRELIMINARIES
14. Suppose p (x) =a n x n + a n−1 x n−1 + ···+ a 1 x + a 0 where all the a k are real numbers.
Suppose also that p (z) =0forsomez ∈ C. Show it follows that p (z) =0also.
15. I claim that 1 = −1. Here is why.
−1 =i 2 = √ −1 √ −1=
√
(−1) 2 = √ 1=1.
This is clearly a remarkable result but is there something wrong with it? If so, what
is wrong?
16. De Moivre’s theorem is really a grand thing. I plan to use it now for rational exponents,
not just integers.
1=1 (1/4) =(cos2π + i sin 2π) 1/4 = cos (π/2) + i sin (π/2) = i.
Therefore, squaring both sides it follows 1 = −1 as in the previous problem. What
does this tell you about De Moivre’s theorem? Is there a profound difference between
raising numbers to integer powers and raising numbers to non integer powers?
17. Show that C cannot be considered an ordered field. Hint: Consider i 2 = −1. Recall
that 1 > 0 by Proposition 1.4.2.
18. Say a + ib < x + iy if a