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Modular Arithmetic
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Clock Arithmetic

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.

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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.
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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, and11
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.

  1. 13
  2. 24
  3. -5
  4. -11rk.
    1. 13
    2. 24
    3. -5
    4. -11
    5. 48
    6. 120
    7. 2457
    8. -34
    9. -763
    Answers.

    A useful shortcut.

    Arithmetic

    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.
    Examples:
    8 + (-10) = -2 = 10
    10 - 11 = -1 = 11

    Try these problems on a 12-hour clock.

    1. 7 + 5
    2. 6 + 6
    3. 11 + 11
    4. 11 + 11 + 11 + 11 + 11
    5. 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7
    6. -11 + 7
    7. 48 - 22
    8. 1 - 9
    9. -6 - 6
    Answers.

    Multiplication

    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.
    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)

    Division

    Division is the inverse operation of multiplication. This means that every division question is answered by answering a "find the missing number" multiplication question.
    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?

    Answers.

    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 interesting.
    About the 10-hour clock.
    About the 9-hour clock.
    About the 11-hour clock.
    About the 2-hour clock.

    Source code for the java applet. r the java applet.