Survey

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Survey

Document related concepts

Transcript

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 that your grandparents can understand. 2