Clock (or modular) arithmetic is arithmetic you do on a clock instead
of a number line.
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
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
Find the standard names for these numbers on a 12-hour clock. Try to
find shortcuts to save work.
A Useful Shortcut
In clock arithmetic, you can add, subtract, and multiply; you can divide
by some numbers.
Addition and Subtraction
Addition and subtraction work the same as on a number line. For example,
to add 9 and 7, start at 0, count 9 along the line, then count 7 more.
You are at 16. If you count on a 12-hour clock, you will be at 4.
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.
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
Multiplication of positive numbers is repeated addition, so if you use
the standard names for numbers, you can use addition to solve a multiplication
problem. (For example, questions 2, 3, and 4 above.)
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.
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.
7 x 14 = 7 x 2 (on a 12-hour clock)
7 x 2 = 14 = 2 (on a 12-hour clock)
Division is the inverse operation of multiplication. This means that every
division question is answered by answering a "find the missing number"
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
Notation and Other Clocks
To avoid continually explaining that you are working with regular integers,
or on a 12-hour clock, there is a notation for writing equations.
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
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