The Art and Craft of Problem Solving

4.2 COMPLEX NUMBERS 131

have the same state, then we can change the state of these n/ d lights. The sum of these

is

sa + sa+d + sa+ 2 d + ... + sa+(�-l)d = Sa( 1 + Sd + S 2 d + ... + S(il-1)d).

The terms in parentheses in the right-hand side form a geometric series that sums easily

and the right-hand side simplifies to

This surprising fact tells us that if we add up all the roots of unity that are "on," the

sum will never change, since whenever we change the state of a bunch of lights, they

add up to zero! The original sum was equal to 1, and the goal is to get all the lights

turned on. That sum will be

1 + S + S^2 + ... + Sn-l =

(^1) - Sn =
� = 0 =F 1.
(^1) - S (^1) - S
Hence we can never tum on all the lights! •
Problems and Exercises
4.2.18 Use complex numbers to derive identities for
cosna and sinna, for n = 3,4, 5.
4.2.19 Test your understanding of Example 4.2.16.
Given that 17 = 42 + I and 101 = 102 + I, mentally
calculate integers u, v such that 17. 10 I = u2 + v^2.
4.2.20 Prove (without calculation, if you can!) that
.. (t -s) .(H/)
e'l +e'S = 2cos 2 e' T.
4.2.2 1 Show that if x+ � = 2cosa, then for any inte­
ger n, x
I
� + -= 2 cosna.
xn
4.2.22 Find (fairly) simple formulas for sina +
sin2a + sin3a + ... + sinna and cosa + cos2a +
cos3a+··· +cosna.
4.2.23 Factor z^5 + z + I.
4.2.24 Solve z^6 + z^4 + z 3 + z2 + I = o.
4.2.25 Let n be a positive integer. Find a closed-form
expression for
sm -sm-s. n. 2n m-. 3n ···sm. (n-I)n.
n n n n
4.2.26 Consider a regular n-gon that is inscribed in a
circle with radius I. What is the product of the lengths
of all n(n - 1)/2 diagonals of the polygon (this in­
cludes the sides of the n-gon)?
4.2.27 (USAMO 1976) If P(x),Q(x),R(x),S(x) are
all polynomials such that
P(x^5 ) +xQ(x^5 ) +x^2 R(x^5 ) = (x^4 +x^3 +x^2 +x+ I)S(x),
prove that x - I is a factor of P(x).
4.2.28 The set of points (x, y) that satisfies x3 - 3xyZ 2:
3x^2 y - y^3 and x + y = -I is a line segment. Find its
length.
4.2.29 (T. Needham) Try to derive Euler's formula
eil = cost + isint in the following way:
(a) Assume that the function f(t) = eil can be dif­
ferentiated with respect to t "in the way that you
would expect"; in other words, that !, (t) = ieil.
(Note that this is not automatic, since the range
of the function is complex; you need to define
and check lots of things, but we will avoid that
for now-this is an intuitive argument!)
(b) If you view the variable t as time, and the func­
tion f(t) as tracing a curve in the complex
plane, the equation f' (t) = ieil has a rate-of-