Download 5.4 Clock Arithmetic

5.4
Clock Arithmetic
In chapter 4 we talked about mathematical systems. A mathematical system has 3 major
properies.
1. It is a set of elements
2. It has one or more operations to combine these elements (ie. Multiplication, addition)
3. It has one or more relations for comparing those elements.
We will be looking a the 12-hour clock system which is based on an ordinary clock face, except
that 12 is considered to be the 0 hour and we don’t consider minutes, only whole number divisions.
The set we use is
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}.
We add numbers the same way we would add times on a clock. For example if we start at 5 and
add 2 hours to it we end up at 7. If we start with 5 and add 7 hours to it we end up at 12 but
in clock arithmetic that is considered to be 0. If we start at 5 and add 8 hours to it we get one
hour past zero so we end up at 1. So
5 + 2 = 7, 5 + 7 = 0, and 5 + 8 = 1
in clock arithmetic.
Example 1: Perform the following operations using 12 hour clock arithmetic.
8+8=
2+8=
0+3=
Does clock arithmetic satisfy the following properties?
• closure property?
• commutative property?
• associative property?
• identity property?
1
4+9=
Additive Inverse
When we talk about the additive inverse we want to know what number, x, to we have to
add to a so that a + x = 0? In regular addition this is easy but in clock addition it is a little
more difficult. Since 0 is the same as 12 in 12-hour clock arithmetic then the additive inverse is
whatever number you need to add to a to get 12. For example 5 + 7 = 0 so 7 is the additive
inverse of 5 in 12-hour clock arithmetic.
Fill in the following table:
Clock value a
Additive inverse of a
0
1
2
3
4
5
7
6
7
8
9
10
11
Since we can do addition on a clock we can also do subtraction. Find the following.
8−3=
2−8=
0−3=
4−9=
We can also do multiplication:
5 × 5 = 5 + 5 + 5 + 5 + 5 = 10 + 10 + 5 = 8 + 5 = 1
Example 2: Find each product using 12 hour clock arithmetic.
8×3=
2×8=
0×3=
4×9=
2