Clock arithmetic is arithmetic done on a clock rather than a number line. We are all...

Clock arithmetic is arithmetic done on a clock rather than a number line. We are all most comfortable with a 12-hour clock, although we could create clocks with fewer hours or with more hours. First, consider a typical clock face with 12 numbers. We might ask the question, “If it is 5 o’clock now, what time will it be in 10 hours?”

Moving in a clockwise direction moves the hour hand 10 hours forward. When we add in the usual sense, 5 + 10 = 15, but on a clock, we start counting over after we reach 12 o’clock, so the answer to the question, of course, is 3 o’clock. Notice that adding hours on a clock is a modular arithmetic problem using modulus 12, since

In performing clock arithmetic, we could think of wrapping the integer number line around and around the clock with 0 corresponding to 12. Each integer is congruent modulo 12 to one of the numbers from 0 through 11, so there are infinitely many integers associated with each clock number.

The integers associated with the number 3, for example, are . . . , -33, -21, -9, 3, 15, 27, . . . . Notice that the difference of any two integers is a multiple of 12, so each integer in this list is congruent to 3 modulo 12.

If we introduce a special notation, we can distinguish clock arithmetic from the usual arithmetic. When we added 5 and 10 using the 12-hour clock, we noted that

Sometimes this is written to indicate that we are doing clock arithmetic. Note that the modulus can change if we use a clock with a different number of hours, but we use the notation with any clock.

Subtraction can also be defined for clock arithmetic. We could ask, “If it is 5 o’clock now, what time was it 10 hours ago?” We know that it would be 7 o’clock if we count backward from 5 using the 12-hour clock, since Using clock arithmetic notation and a 12-hour clock, we would write We define subtraction on the 12-hour clock as follows: subtract whole numbers as usual, as in but if the difference is less than 0, add 12. For example

a. Use clock arithmetic to calculate each of the following using a 7-hour clock.

b. Calculate each of the following using a 5-hour clock.

c. Find two different clocks for which the integer 56 is 6 o’clock.

d. Find four different clocks for which the integer -19 is 1 o’clock.