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Unformatted text preview: CS 173: Discrete Structures, Spring 2010 Homework 1 Solutions This homework contains 5 problems worth a total of 45 points. 1. [5 points] Demographic survey Fill out the online demographic survey at https://illinois.edu/sb/sec/8918245 Your individual answers (which will be kept confidential) will help us advise you better this term. Summaries of responses from the class as a whole will improve the CS department’s ongoing curriculum development efforts. 2. [12 points] Logarithms, Exponents, etc Simplify the following expressions as much as possible, without using a calculator (either hardware or software) . Do not approximate. Express all rational numbers as fractions. For complex numbers use i to represent the value √ 1 . Show your work. [Solution inline below] (a) log ⌊ 8 . 7 ⌋ (0 . 125 ⌈− 2 . 3 ⌉ ) = log 8 ((8 − 1 ) − 2 ) = log 8 (8 2 ) = 2 (b) log 512 (1024 3 × 4096 5 ) = log 512 (2 30 · 2 60 ) = log 512 (512 10 ) = 10 (c) − 1+(log 3 (log 5 (125 k ))) log 9 k = − 1+(log 3 ( k · log 5 (125))) log 9 k = − 1+log 3 (3 k ) log 9 k = − 1+log 3 (3)+log 3 k log 9 k = log 3 k log 9 k = (log 3 9)(log 3 k ) log 3 k = 2 (d) (log 4 (49 i )) × (log 7 (16 i )) = (2 i log 4 7) × (2 i log 7 4) = (2 i ) 2 · (log 4 7) × log 4 4 log 4 7 = 4 (e) Using your favorite reference sources, track down and state Euler’s formula giv ing the value of e ix in terms of the sin and cosine functions. Hints: e is the base of the natural logarithm. Wikipedia is a suitable reference to use. e ix = cos( x ) + i sin( x ) (f) Using your answer to (e), calculate the value of e 2 πi . Show your work. Hint: the answer and the work should be very simple....
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 Spring '08
 Erickson
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