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```Math 211
Assignment 5
Due: Noon 1 December 2015
Do the following problems by hand (and show your work).
1. Express the following complex numbers in the form x + iy, where x, y ∈ R:
(a) (2 + 3i)(3 − 2i);
(b)
(1 + i)(2 − 3i)
;
3+i
(c)
2i(−8 + 6i)2
√
√ .
(4 − 2 5i)(2 − 4 5i)
√
2. (a) Express α = 1 + i and β = 1 + i 3 in polar form.
(b) Find the polar form of αβ and β/α, where α and β are as in part (a).
√
(c) Find arg(1 − i) and arg(−1 + 3i). [in radians!]
3. (a) Use properties of complex conjugation to verify the identity
(x2 + y 2 )(a2 + b2 ) = (xa − yb)2 + (xb + ya)2
for real numbers x, y, a, b ∈ R. [Hint: consider z1 = x + iy, z2 = a + ib.]
(b) Show that if n and m ∈ Z are two integers which are sums of two squares (of
integers), then so is their product mn.
4. (a) Suppose we have a polynomial f (x) = an xn +...a1 x+a0 , where ai ∈ R (0 ≤ i ≤ n).
Prove that f (z) = 0 implies f (z̄) = 0.
(b)Let f (x) = x4 + 4. First verify that f (1 + i) = 0. Then find the other three roots
from 1 + i. Hint: Use part (a). Also, multiply by −1.
5. Find and sketch all the complex solutions of the equation z 4 = i.
6. Find all the complex solutions of each of the following equations:
(a)
x3 = −i; [use special triangles to simplify the sin and cos expressions!]
(b)
x3 + 3x2 + 3x + 1 + i = 0.
[Hint for part (b): use (a) and the fact that (x + 1)3 = x3 + 3x2 + 3x + 1.]
7. Find the quotient and the remainder when you divide:
(a) x9 + x7 + 2x5 + 2x3 + x + 1 by x4 + x2 + 1 in Q[x],
(b)
x5 + x3 − x2 + x
by
x2 + i
in C[x],
8. Tony the spy is on a mission to save the world from evil forces. He intercepted an
f = 7561, which Mrs. Really Evil sent to her husband Mr.
encrypted message M
Super Evil. Tony thought he intercepted an important message, but it was just a
number of marshmallows Mrs. Really Evil had that afternoon. Tony decided to
decode the message anyways knowing that Mr. Evil uses RSA scheme with public
key (n = 84517, e = 2873).
Let M be the number of marshmallows Mrs. Really Evil had, which satisfies 0 ≤
f is the number obtained by encrypting M via RSA scheme. After
M < n. Then M
decoding the message, Tony thought that perhaps Mrs. Really Evil needs to be
concerned about diabetes. What was M ?
Bonus Describe how RSA scheme works and how it is useful in the real world in a way