Clock Arithmetic

We all know how to add numbers, but did you ever think about how strangely we add the hours on a clock?

If it is 3 o'clock and we add 5 hours to the time that will put us at 8 o'clock, so we could write 3 + 5 = 8. But if it is 11 o'clock and we add 5 hours the time will be 4 o'clock, so we should write 11 + 5 = 4 !!! Now everyone knows that 11 + 5 is really 16, but there is no 16 on the clock (unless you're in the military ... "Sargent, have your squad fall in at the mess hall at 16 hundred hours! Yes, Sir!"). Every time we go past 12 on the clock we start counting the hours at 1 again. If we add numbers the way we add hours on the clock, we say that we are doing clock arithmetic. So, in clock arithmetic 8 + 6 = 2, because 6 hours after 8 o'clock is 2 o'clock.

Use clock arithmetic to add these numbers:

  • 7 + 6 = _____
  • 11 + 7 = _____
  • 3 + 6 = _____
  • 5 + 12 = _____
  • 7 + 13 = _____
  • 12 + 9 = _____
  • 1 + 8 = _____
  • 9 + 14 = _____
    • Click here to see the answers.

    The 12 hour clock that we are so familiar with is very old. The ancient Babylonians gave us the idea of breaking up time into 12 hours for half a day. There really isn't anything very special about the number 12, the Babylonians could have picked another number, like 10, and if they had our clocks would look like this:

    We can do clock arithmetic on this kind of clock too! This time, 8 + 4 = 2 because if we start at 8 and move ahead 4 hours we would be at 2 on this clock.

    Use clock arithmetic on a 10 hour clock to add these numbers:

  • 7 + 6 = _____
  • 4 + 7 = _____
  • 3 + 6 = _____
  • 5 + 12 = _____
  • 7 + 10 = _____
  • 2 + 9 = _____
  • 1 + 8 = _____
  • 9 + 14 = _____
    • Click here to see the answers.

    We could make our clocks with any number of hours and do clock arithmetic with them. We can also do subtraction. On a clock with 5 hours, 2 - 3 = 4 because if we start at 2 o'clock and move backwards 3 hours we would be at 4 o'clock. Draw your own picture of a 5 hour clock to see that this is true.

    Use clock arithmetic on a 5 hour clock to find these numbers:

  • 3 + 4 = _____
  • 2 + 1 = _____
  • 2 - 4 = _____
  • 1 - 3 = _____
  • 4 + 5 = _____
  • 4 - 5 = _____
  • 3 + 6 = _____
  • 2 - 7 = _____
    • Click here to see the answers.

    It could be very embarassing if we were doing clock arithmetic and someone looked at our work and, thinking that we were doing regular arithmetic, said, "Oh no! Most of your answers are wrong!" In order to prevent that from happening, we write clock arithmetic expressions in a special way. If we wanted to write that 4 + 3 = 1 on a 6 hour clock, we would write (4 + 3) mod 6 = 1. The "mod 6" tells us that we are doing clock arithmetic on a 6 hour clock. "Mod" is shorthand for the word "modulus" which is a fancy word for saying how long you have to go before starting over again. So, we would write some of our earlier work this way: (11 + 5) mod 12 = 4, (8 + 4) mod 10 = 2 and (2 - 3) mod 5 = 4.

    Find these numbers

  • (3 + 8) mod 7 = _____
  • (2 - 3) mod 15 = _____
  • (4 + 5) mod 6 = _____
  • (3 + 6) mod 11 = _____
  • (2 + 8) mod 8 = _____
  • (3 - 5) mod 5 = _____
  • (7 + 14) mod 7 = _____
  • (6 + 18) mod 8 = _____
    • Click here to see the answers.

    You may have noticed in some of the problems that when you go all the way around the clock exactly once you end up just where you started. Like, (5 + 7) mod 7 = 5 and (4 - 8) mod 8 = 4. In regular arithmetic, there is only one number that you could add or subtract from another number and leave that other number unchanged ... 0 of course! In clock arithmetic, going around the clock a whole number of times has the same effect as doing nothing! So, if we had a 6 hour clock, adding or subtracting 6 is the same as adding or subtracting 0. For this reason, we usually write 0 for the number of hours in the clock and change the way the clock looks so that instead of having the number of hours in the clock at the top, we put a 0 there and think of the clock as starting there instead of ending there. So, our 10 hour clock would look like this:

    Now the answer to (3 + 7) mod 10 would be written as 0 (instead of 10 as we were doing before). Here are some more examples:

    If you are wondering why anyone would want to do clock arithmetic with anything other than a 12 hour clock, think about this. Suppose you are playing a video game and the character in the game (let's call her Judy) is walking from the left side of the screen to the right side.

    When Judy gets to the right side of the screen and keeps walking, she disappears and reappears on the left side of the screen again. If the screen is 18 inches wide and we are keeping track of how far Judy is from the left side of the screen, then as soon as she is 18 inches from the left side it's as if she were back at the beginning again ... but this is just doing clock arithmetic with an 18 hour clock! The people who make these games have to use clock arithmetic to control where the images appear on the screen, and they will have to use different kinds of clocks for different kinds of screens.

    If you want to find out more about clock arithmetic, click here.
    Here are the answers to the first set of questions

  • 7 + 6 = 1
  • 11 + 7 = 6
  • 3 + 6 = 9
  • 5 + 12 = 5
  • 7 + 13 = 8
  • 12 + 9 = 9
  • 1 + 8 = 9
  • 9 + 14 = 11
    • Click here to go back

    Here are the answers for the 10 hour clock.
  • 7 + 6 = 3
  • 4 + 7 = 1
  • 3 + 6 = 9
  • 5 + 12 = 7
  • 7 + 10 = 7
  • 2 + 9 = 1
  • 1 + 8 = 9
  • 9 + 14 = 3
    • Click here to go back

    Here are the answers for the 5 hour clock.
  • 3 + 4 = 2
  • 2 + 1 = 3
  • 2 - 4 = 3
  • 1 - 3 = 3
  • 4 + 5 = 4
  • 4 - 5 = 4
  • 3 + 6 = 4
  • 2 - 7 = 5
    • Click here to go back

    Here are the answers to the last set of questions.
  • (3 + 8) mod 7 = 4
  • (2 - 3) mod 15 = 14
  • (4 + 5) mod 6 = 3
  • (3 + 6) mod 11 = 9
  • (2 + 8) mod 8 = 2
  • (3 - 5) mod 5 = 3
  • (7 + 14) mod 7 = 7
  • (6 + 18) mod 8 = 8
    • Click here to go back