1.3. Complex Numbers 15
(c) %+w=F+?Tj
(d) zUr=;iv
(e) I = z
(f) Re z = rcosf3 = f(z+F) < [zl
(g) Im z = rsin0 = $(z - Z?) 5 121
(h) 1z12 = zZ
(i) I4 = I4 I4
(j) arg(zw) = arg z + arg w (up to an integer multiple of 2~)
(k) 1% + WI 5 I4 + I4
(1) 2-1’2(14 + IYl> I I4 5 I4 + IYI
(m) for 2 # 0, l/z = ??/IzI”.- The Greek geometers were interested in discovering which geomet-
ric entities could be constructed from given data using only ruler
(straightedg e ) an d compasses. Given the points representing 0, z and
w in the Argand diagram, determine ruler and compasses construc-
tions for Z, z + 20, zw and l/z. - Let c be a fixed real number and w a fixed complex number. Find the
locus of points z in the Argand diagram which satisfy the equation
Re (zw) = c. - Let /Z be a fixed positive constant. Describe the locus of the equation
121 = klz + 11. - Use complex numbers and an Argand diagram to solve the following
problem: Some pirates wish to bury their treasure on an island. They
find a tree T and two rocks U and V. Starting at T, they pace off the
distance from T to U, then turn right and pace off an equal distance
from U to a point P, which they mark. Returning to T, they pace
off the distance from T to V, then turn left and pace off an equal
distance (to TV) to a point &, which they mark. The treasure is
buried at the midpoint of the line segment PQ.
Years later, they return to the island and discover to their dismay at
the tree T is missing. One of them decides to just assume any position
for the tree and then carry out the procedure. Is this strategy likely
to succeed? - Prove De Moivm’s Theorem: For any integer n,
(r(cos0 + isine))” = r”(cosne+ isinne).
- Determine all those complex numbers z for which