On a 12-hour clock, there are only 12 numbers in the whole number system. However, every number has lots of different names. For example, the number before 1 is 0, so 12 = 0 on a 12-hour clock.

Here is a 12-hour clock showing several of the names for each number. Clock arithmetic has negative numbers, but each negative number has a positive number name.

Usually people decide on one set of standard names for the numbers on the clock, and they usually start with 0, not 1. So let's use 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11 for the standard names on the 12-hour clock.

Find the standard names for these numbers on a 12-hour clock. Try to find shortcuts to save work.

- 13
- 24
- -5
- -11
- 48
- 120
- 2457
- -34
- -763

To add negative numbers, use the minus (-) sign to change direction.
To subtract on a clock, first find standard (positive) names for the two
numbers, count clockwise for the first one, and count counterclockwise
for the second.

Examples:

8 + (-10) = -2 = 10

10 - 11 = -1 = 11

Try these problems on a 12-hour clock.

- 7 + 5
- 6 + 6
- 11 + 11
- 11 + 11 + 11 + 11 + 11
- 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7
- -11 + 7
- 48 - 22
- 1 - 9
- -6 - 6

You can also use your favorite multiplication method for regular integers,
then find the standard name for the answer., then find the standard name
for the answer.

Example:

7 x 14 = 98 (in integers)

98 / 12 = 8 r. 2

So 7 x 14 = 2 (on a 12-hour clock).

Or use a combination of these methods and shortcuts.

Example:

7 x 14 = 7 x 2 (on a 12-hour clock)

7 x 2 = 14 = 2 (on a 12-hour clock)

Example:

5 / 7 = ? (on a 12-hour clock) means

? x 7 = 5 (on a 12-hour clock)

By trial and error (there are only 12 numbers to try) or by using different names for 5

? x 7 = 5

? x 7 = 17

? x 7 = 29

? x 7 = 41

? x 7 = 65

? x 7 = 77

we find that ? = 11.

Another method is to find the multiplicative inverse of 11:

7 x m = 1

7 x m = 13

7 x m = 25

7 x m = 37

7 x m = 49

So m = 7; to divide by 7 multiply by its inverse, which happens to
also be 7.

The big problem with division is that some division questions have no answers, and some division questions have more than one answer. Which numbers can you divide by and get exactly one ansyou divide by and get exactly one answer?

13 = 1 (mod 12)

means that 13 and 1 are the same number on a 12-hour clock. Actually,
you should use an equals sign with 3 bars instead of 2, but this part of
this page is still under construction.

There is no reason to stick with 12-hour clocks. The same principles
work with any positive whole number of hours. Some clocks are especially
interesting.

About the 10-hour clock.

About the 9-hour clock.

About the 11-hour clock.

About the 2-hour clock.

Original Web Site by Susan Addington