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ModularNumbers

# ModularNumbers - CSE 1400 Applied Discrete Mathematics...

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CSE 1400 Applied Discrete Mathematics Modular Numbers Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Conversion Between Units 1 Modular Numbers 3 Division mod n 5 Linear Congruence Equations 6 Euclidean Algorithm for the Greatest Common Divisor 7 Relatively Prime Numbers 10 Extended Euclidean Algorithm 10 Solving Linear Congruence Equations 14 Problems on Modular Numbers 17 Abstract Conversion Between Units P retend you d like to covert one foot (12 inches ) into its equivalent length in meters . You know one foot goes into The conversion between units is 1 meter 3.2808 feet 1 foot 0.3048 meters The Euclidean algorithm compute these approximations. one meter about 3 times with a remainder of about 3 inches. The 1 foot 1 meter empirical approach to convert feet to meters is to lay the foot-long ruler along a meter stick as many times as it will go; in this case 3 times. One meter is approximately three feet. 1 meter = 3 feet + r feet where r is some fractional part of one foot. Three, the number of time a foot goes into a meter is called the quotient, and r is called the remainder or residue. 1 foot 1 meter There will be a tiny piece left over, not covered, on the meter stick. Call the length of this tiny piece r and note that r is some fraction of a foot. To compute r , lay it along the foot-long ruler and measure that about three r ’s almost covers a foot, so that r 0.3333 · · · feet and 1 meter is approximately 3.3333 · · · feet.

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cse 1400 applied discrete mathematics modular numbers 2 1 foot But there is still a tiny piece left over, not covered, on the foot-long ruler. Call the length of this tiny piece r 1 , and let’s also rename r , the first fraction of a foot, by calling it r 0 from now on. To compute the length r 1 , lay it along the fractional foot r 0 = 1 3 and measure that two of these smaller remainder r 1 = almost cover r 0 = , but by a little too much. We can continue this process until: • Some multiple of a tiny remainder exactly covers the previous one. • The approximation computed is accurate enough for its intended purpose. Let’s stop at this step and note what we’ve discovered. 1 meter = ( 3 + r 0 ) feet 1 foot = ( 3 r 0 + r 1 ) feet r 0 feet = ( 2 r 1 - r 2 ) feet We make a truncation error by ignoring the value of r 2 , effectively setting its value to 0, but we gain a conversion factors to change meters into feet and vice versa. That is, we have r 1 feet r 0 2 feet 1 foot ( 3 r 0 + r 0 2 ) = 7 r 0 2 feet r 0 feet 2/7 feet 1 meter ( 3 + 2 7 ) feet Or, 1 meter is about 23/7 = 3.28571428571 feet. Let us declare that 3 + 2/7 = 23/7 3.2857 feet is “good enough for our uses” as an approximation to 1 meter. Recall from above , 1 meter is approxi- mately 3.2808 feet. In practice we would like to have a ruler 1/7 of a foot-long so we subdivide 1 foot evenly into 7 parts and 1 meter (almost) evenly into 23 parts. Here’s how we can construct a rule one-seventh of a foot-long. Lay out three meters and you’ll discover you’ve covered 3 ( 3 + 2/7 ) = 9 + 6/7 feet. Now lay out ten foot-long rulers to 10 feet.
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ModularNumbers - CSE 1400 Applied Discrete Mathematics...

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