Clock Arithmetic
There are times when it is helpful to be able to add, subtract, multiply or divide numbers
from a finite set.
For example, when we add elapsed time to a beginning time to determine an ending time,
no sum can be greater than 12. Similarly, if we add a number of days to a particular day
of the month, the sum cannot be greater than 31, or 30.
This assignment will build on your understanding of whole-number arithmetic to
examine how arithmetic can be defined on a finite set of numbers.
We will be using a clock model. Another way to refer to clock arithmetic is to call it
arithmetic mod 12. This means that 12 is the number of digits in the set used for
arithmetic.
Part 1: Addition and subtraction
The number line model is one model used to present addition and subtraction of whole
numbers. This type of model can also be used when adding or subtracting whole numbers
on a clock.
1. To see how this applies draw a clock face and label the times. Use this clock to find the
following sums and differences on the clock.
4+7=
7 + 11 =
12 + 11 =
9+4=
7–6=
6–7=
12 – 11 =
5–9=
11 – 11 =
2. In the case of subtraction mod 12, must it be that the second number is less than the
first number, as in the case of whole numbers?
3. Recall that zero is the additive identity for whole numbers. What is the additive
identity for clock addition?
4. What are the additive inverses for the numbers in clock arithmetic? Are there any
numbers that do not have an additive inverse? If so, what are they and how are they
related to the base number of the clock?
5. Are there any numbers that are their own inverses? If so, what are they and how are
they related to the base number of the clock?
Part 2: Multiplication
One way to define multiplication of whole numbers is as repeated addition. For example,
4 μ 3 is defined as 3 + 3 + 3 + 3 = 12. Multiplication mod 12 can be defined the same
way. Based on this, 5 μ 3 = 3 because 3 + 3 = 6, 6 + 3 = 9, 9 + 3 = 12, and 12 + 3 = 3.
6. Find the basic facts of multiplication mod 12 by completing the multiplication table
below.
μ
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
7. Recall from multiplication of whole numbers that the product of zero with any number
is zero. And if the product of any two whole numbers is zero, one of the numbers must be
zero. Is this true for multiplication mod 12?
8. Recall that one is the multiplicative identity for whole numbers. What is the
multiplicative identity for clock addition?
9. What are the multiplicative inverses for the numbers in clock arithmetic? Are there any
numbers that do not have a multiplicative inverse? If so, what are they and how are they
related to the base number of the clock?
10. Are there any numbers that are their own inverses? If so, what are they and how are
they related to the base number of the clock?