Clock Arithmetic
It’s 6 o’clock at night. You eat supper for one hour, watch TV for
2 hours, study for 4 hours. What time are you done studying? If we
add the time intervals, we get 6 + 1 + 2 + 4 = 13. But 13 o’clock is not
a valid time (except in the army). The time will be 1 in the morning,
which we get by subtracting 12 from 13. In mathematics we write
this as:
13 mod 12 = 1.
On a standard 12-clock, once you reach the number 12, the time resets
to 0. Similarly, 25 mod 12 = 1 and −2 mod 12 = 10, that is, 2 hours
before 12 is 10 o’clock.
We are familiar with different clocks besides the 12-clock. For
example, we can think of the days of the week as lying on a 7-clock.
It is convenient to assign the days the values: Sun = 0, Mon = 1,
Tues = 2, · · · , Sat = 6. Suppose you begin a trip for Florida on
Wednesday. You spend 2 days driving. You stay in a hotel in Orlando
for 6 days, and a hotel in Miami for 5 days. It takes 2 days to drive
home. On what day do you return home? Since Wednesday is day
number 3, we can add up all the days to get: 3 + 2 + 6 + 5 + 2 = 18.
On a 7-clock, 18 is the same as 4, which we write as:
18 mod 7 = 4.
So you return on day number 4, or Thursday.
The number of pennies needed in the exact change for a purchase
can be computed using a 5-clock. For example, suppose you buy a
$1.99 sandwich, a $.89 drink, and the tax is 26 cents. How many
pennies do you need to pay the bill? Adding the three numbers as
cents, we have
(199 + 89 + 26) mod 5 = 314 mod 5 = 4.
You need 4 pennies (plus a dime and 3 dollars).
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