ModularNumbers

ModularNumbers - CSE 1400 Applied Discrete Mathematics...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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. 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 lets also rename r , the first fraction of a foot, by calling it r from now on. To compute the length r 1 , lay it along the fractional foot r = 1 3 and measure that two of these smaller remainder r 1 = almost cover r = , 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. Lets stop at this step and note what weve discovered. 1 meter = ( 3 + r ) feet 1 foot = ( 3 r + r 1 ) feet r 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 2 feet 1 foot ( 3 r + r 2 ) = 7 r 2 feet r 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....
View Full Document

This note was uploaded on 02/11/2012 for the course MTH 2051 taught by Professor Shoaff during the Fall '11 term at FIT.

Page1 / 18

ModularNumbers - CSE 1400 Applied Discrete Mathematics...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online