Lecture 11

  Modular arithmetic is arithmetic in which numbers do not continue forever.  Modulo 7 has numbers 0, 1, 2, 3, 4, 5, and 6.  Modulo 5 has numbers 0, 1, 2, 3, and 4.  In general, modulo n has numbers 0, 1, 2, …, n-1. Modular Basics

  Clocks start over at 12 and/or 24 hours.  Calendars start over after  7 days for one week  52 weeks for one year  365 days for one year  You must pay attention to going forward in time and going backward in time in order to add or subtract from where you are. Clocks and Calendars

  Class starts at 2:30 PM on Thursday.  If I assign a paper due 145 hours from the start of class, when will it be due?  If you had a paper due that was assigned 80 hours prior, when was it assigned? Clock Examples

 If we divide 145 hours by 24 (hours in a day), we find 6 whole days and a decimal. Subtracting 6 from our quotient we find remaining. That is, of a 24 hour period. Multiply this decimal portion by 24 to see how many hours it represents – 24( ) = 1. That is, this assignment is due 145 hours, or 6 days and 1 hour from the start of class. It would be Wednesday at 3:30 PM. 145 hours after 2:30 PM

 Once again we divide 80 by 24 to get This means three days prior and of 24 hours…24( ) = 8 hours. Three days prior to Thursday is Monday and 8 hours prior to 2:30 is 6:30 AM. This assignment was given at 6:30 AM on Monday and due 80 hours later on Thursday at 2:30 PM. 80 hours prior to 2:30 PM

  April 20 th, 2014 falls on a Sunday.  What day of the week was 4/20 in 1980?  What is the next year that 4/20 falls on a Sunday? Calendar Examples

  The first thing we should do is determine which years were leap years (366 days) and which were not (365 days).  Leap day occurs in any year that is evenly divisible by 4 (with an exception that doesn’t arise in our example).  Leap years between 1980 and now were in 1980, 1984, 1988, 1992, 1996, 2000, 2004, 2008, and /20/1980

  We know that from April 20 th to April 20 th is a year. We can count the whole years and the leap years, multiplying appropriately to find the number of days.  1980 is a leap year, but April 20 th is after leap day so it won’t count. There are 8 leap years and 26 non- leap years.  This makes 366x8=2928 and 26x365=9490 days, respectively, for a total of 12, 418 days.

  In order to determine the day of the week, we now divide this number of days by 7 to get 1774 with no decimal.  No decimal tells us it was exactly 1774 weeks prior…4/20 fell on a Sunday in 1980.

  In 2015, it will be a Monday.  2016 is a leap year so will be Wednesday.  2017 will be Thursday.  2018 will be Friday.  2019 will be Saturday.  2020 is a leap year so will be a Monday… D’oh!  2021 Tuesday, 2022 Wednesday, 2023 Thursday, 2024 Saturday, 2025 will be the next time 4/20 falls on a Sunday. 4/20 next time on Sunday

  Check digits are a system for us to make sure the input numbers are correct in a variety of sources.  Check digits allow us to detect errors through a variety of systems. Check Digits