Math Newsletter number 4; Wednesday, August 18, 2010

This is Math Newsletter number 4; Wednesday, August 18, 2010.
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Division of one integer by another gives both a quotient
and a remainder. Divide 7 into 22. The quotient is 3 and
the remainder is 1. This means that 3 * 7 + 1 = 22.

Define remainder(22,7) to be the remainder when 22 is
divided by 7. Extending this definition to all integers,
we get

remainder(7,2)= 1
remainder(29,3)=2
remainder(29,5)=4
remainder(29,7)=1

Notice that if two different integers have the same
remainder with respect to the same divisor, then the
difference of those two integers are exactly divisible by
that divisor.

remainder(29,7) = remainder(22,7)
This implies that 7 divides exactly (29 - 22) = 7.
remainder(7,7) = 0

In general, if a divisor divides an integer exactly, the
remainder is 0.

remainder(28,7) = 0
remainder(20,5) = 0
remainder(20,4) = 0
remainder(11,1) = 0

The reason this is important is because it enables
arithmetic of remainders.

We state the rules of remainder arithmetic most simply by
using symbolic numbers. For all integers, A, B, and all
integers m not equal to 0,

remainder(A,m) - remainder(B,m) = remainder(A-B,m)
remainder(A,m) + remainder(B,m) = remainder(A+B,m)
remainder(A,m) * remainder(B,m) = remainder(A*B,m)

We can drop the repetitious "remainder(,)" if we state which
integer, m, we are using as a divisor through out a
discussion.

For the following discussion, all arithmetical statements
are to be interperted as remainder arithmetic with respect
to the divisor 7. Another name for this chosen divisor is
"modulus".

Thus we may formally declare, "In the following discussion,
all arithmetical statements are to be interpreted as
arithemetic in modulus 7."

There are 7 days in a week. Assign
Sunday = 0
Monday = 1
Tuesday = 2
Wednesday = 3
Thursday = 4
Friday = 5
Saturday = 6

January has 31 days.
In non leap years, February has 28 days.
In leap years, February has 29 days.
March has 31 days.
April has 30 days.
May has 31 days.
June has 30 days.
July has 31 days.
August has 31 days.
September has 30 days.
October has 31 days.
November has 30 days.
December has 31 days.

Since we are working in modulus 7 arithmetic, we can say:

January has 3 days.
In non leap years, February has 0 days.
In leap years, February has 1 days.
March has 3 days.
April has 2 days.
May has 3 days.
June has 2 days.
July has 3 days.
August has 3 days.
September has 2 days.
October has 3 days.
November has 2 days.
December has 3 days.

Give the name "Jan1" to the First of January.
We calculate dates for non leap years.

February 1 is Jan1 + 3
If January first is Sunday, February first is Wednesday.
If January first is Saturday, February first is Thursday.
etc.

March 1 is (Jan1 + 3) + 0 = (Jan1 + 3)
The calendar for March resembles that of February.

April first is (Jan1 + 3) + 3 = (Jan1 + 6).
If January 1 is Sunday, April 1st is Saturday.
If January 1 is Monday, April 1st is Sunday.
etc.

May first is (Jan1+6) + 2 = Jan1 + 1.
If January 1 is Sunday, May 1st is Monday.
If January 1 is Monday, May 1st is Tuesday.
etc

June first is (Jan1 + 1) + 3 = Jan1 + 4.
If January 1 is Sunday, June 1st is Thursday.
If January 1 is Monday, June 1st is Friday.
etc.

July first is (Jan1 + 4) + 2 = Jan1 + 6.
The July calendar resembles the calendar of April.

August first is (Jan1 + 6) + 3 = (Jan1 + 2).
If January first is Sunday, August first is Tuesday.
If January first is Monday, August first is Wednesday.
etc

September first is (Jan1 + 2) + 3 = (Jan1 + 5).
If January first is Sunday, September first is Friday.
If January first is Monday, September first is Thursday.
etc

October first is (Jan1 + 5) + 2 = Jan1 + 0.
The calendar for October resembles the calendar of January.

November first is (Jan1 + 0) + 3 = (Jan1 + 3).
The calendar of November resembles the calendar of March.

December first is (Jan1 + 3) + 2 = (Jan1 + 5).
The Calendar for December resembles the calendar for
September.

December 25, Christmas, is December first + 24
= (Jan1 + 5) + 24 = Jan1 + 29 = Jan1 + 1
In non leap years, Christmas falls on the day of the week
immediately following the day of the week of New Years day.

In non leap years;
If January first is Sunday, Christmas is Monday.
If January first is Monday, Christmas is Tuesday.

December 31st, new years eve, is Christmas + 6.
New years eve of the next year is (Jan1 + 1) + 6
= Jan1 + 0 = Jan1.

For non leap years, January first, new years day, and
December 31st, the new years eve of the next year, are the
same day of the week.

Using modulus 7 arithmetic, what other interesting special
day of the week relationships can you find?

Kermit Rose