Day of the Week A formula known as Zeller's Congruence makes use of the greatest integer functio

a. For December 7, 1941, we have ( c=19 ), ( y=41 ), ( d=7 ), and ( m=10 ). Plugging the

Day of the Week A formula known as Zeller's Congruence makes...

Day of the Week A formula known as Zeller's Congruence makes use of the greatest integer function [x] to determine the day of the week on which a given day fell or will fall. To use Zeller's Congruence, we first compute the integer z given by z = [[(13m - 1) / 5]] + [[y / 4]] + [[c / 4]] + d + y - 2c The variables c, y, d, and m are defined as follows: c = century y = year of the century d = day of the month m = month, using 1 for March, 2 for April, ... , 10 for December. January and February are assigned the values 11 and 12 of the previous year. For example, for the date September 30, 2009, we use c = 20, y = 9, d = 30, and m = 7. The remainder of z divided by 7 gives the day of the week. A remainder of 0 represents a Sunday, a remainder of 1 a Monday, ... , a remainder of 6 a Saturday. a. Verify that December 7, 1941, was a Sunday. b. Verify that January 1, 2020, will fall on a Wednesday. c. Determine on what day of the week Independence Day (July 4, 1776) fell. d. Determine on what day of the week you were born.

Transcript

00:01 Okay this is a this is like a math problem you can do at a math party you know your birthday party obviously because we're talking about birthdays or dates zellar's congruence makes use of the greatest integer function basically the greatest energy function you put you put your number in and the number the integer that is less than or equal to the number the nearest energy the next one less than the next one less than the number or equal to x is the one is your is the what you would get out so we're talking about and less than or equal to x less than or equal to n plus one so so we're if we were on the number line and these were two integers and your number here this is your number here then the the if we put this into our function right we put this into our function x we would get out a because it is the integer just less than the nearest integer less than or equal to our value.

01:12 So maybe 31 .5 if we put that in there, we would get 31.

01:18 Okay, that was background for me and for you to remember.

01:25 Okay.

01:26 So now the other thing we have to do is figure out how they're defining their variables.

01:34 So c is century.

01:38 So if you have something like 1995, c would be 19.

01:44 That's the century.

01:46 And then the year of the century would be 95.

01:52 The century is 100 years, right? so 19 would be 1900 years, 19 centuries, 1900 years.

02:02 The year would be 95.

02:04 That's the 95th year and the 9th.

02:06 19th century.

02:08 They don't count it that way, but in this math problem they do.

02:12 D is the day of the month.

02:16 So if it were october 31st, 1995, the day would be 31 and then this is here.

02:26 This is really crazy the month using one for march, two for april.

02:35 This is really weird.

02:36 So march, april, june, july, august, september, october, november, december.

02:52 We go one, two, three, four, five, six, seven, eight, nine, ten.

03:00 And then it's really weird because it says, january fair, assigned the values 11 and 12th of the previous year.

03:08 And i don't know why that's so.

03:13 Obviously zeller knew why we're going to have to give that a roll so let's see what we get here so i am going to make some space for some work and notice here here's an example september 30th 2009 see it was 20 that's the century the year is 9 the day is 30th for 30 and the month is 7 because it goes march, april, may, june, july, august, september, september.

03:54 Okay.

03:57 And then we are going to take, it says the remainder of z divided by seven gives the day of the week.

04:07 So whatever number we get out, we're going to divide it by seven.

04:11 And whatever the remainder is, it's going to give us the day.

04:15 So zero corresponds to sunday and six corresponds to saturday.

04:22 So let's try this.

04:23 Really, this is just getting things substituted in, calculating them correctly, and then figuring it out.

04:32 Okay, so a, december, i see, so of course they start with a hard one.

04:38 Let's get the year first, right? let's get the year first.

04:41 So this is a, i'm looking at a, i need all the space i can get.

04:46 So let's go.

04:50 C would be 19.

04:54 Y would be 41.

04:58 Your day, d would be 7.

05:02 The month would be 12.

05:11 And, okay, so now we're going to put all this in.

05:19 So i'm just going to write it out.

05:20 So z equals, when you use these square brackets, 13m there.

05:31 Let's just put in the end.

05:33 Let's just put in the m when we get it.

05:35 So 13 times 12 over 5 plus 41 over 4 plus our century is 19 over 4 plus 7 plus 41.

06:02 Times 19.

06:04 Okay.

06:06 I'm going to pause this for just a second.

06:08 I'll pause it just a second to do my calculations.

06:14 Hold on.

06:15 I know.

06:16 It felt like forever.

06:17 I was gone.

06:18 I just paused myself.

06:19 You're like, wait, you're right back.

06:21 Yeah.

06:21 Okay.

06:21 So i did this.

06:23 Now, let's just, let's take, i put these in the decimal so that we could get these put in.

06:27 I'm going to want to, i'm going to write the numbers up here.

06:34 In the right corner in blue.

06:36 So if i'm doing the greatest integer function for this one, i'm gonna, what's the, what is the integer just below 31 .2? it's on the number line, it's 31.

06:49 So i'm gonna write 31 plus, what's the integer that's just below 10 .25? these aren't negative signs, i was just, i was just saying that's the answer to that.

07:01 10 .25, so that'll be plus 10, down here, 4 .7 .7.

07:05 75, that'll be plus 4, and then we're going to do plus 7, plus 41, and then minus 38.

07:19 What is that equal to? i'm actually going to do this live, i'm not going to pause...

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