|Murals at North Hollywood High|
|Mathematics is the study of patterns. One of the ways in
which we may use number patterns is in the creation of unique and artistically
pleasing designs. In this web page, you will learn how to make designs
based on modular arithmetic tables.
Modular or "clock" arithmetic is arithmetic on a circle instead of a
number line. For a detailed explanation, visit Susan Addington's web page
Arithmetic. (Like this page, it will require a JAVA-capable browser.)
A simple explanation of modular arithmetic follows.
|Let us begin by examining the clock at the left. Observe that there have been two alternations. Since the number preceding 1 in ordinary arithmetic is 0 (the identity element for addition), the 12 has been replaced by a 0. Since our aim is to do clock arithmetic rather than tell time, both the hour and minute hands have been omitted from the dial.|
If it is 3 o'clock and we add 5 hours, the time will be 8 o'clock. We write: 3 + 5 = 8 . But if it is 9 o'clock and we add 5 hours, the time will be 2 o'clock. So we write: 9 + 5 = 2 . Every time we go past 0 on the dial, we start counting the hours at 1 again. All multiples of 12 are equivalent to 0. To convert whole numbers to their mod 12 equivalent, we simply divide by 12 and record the remainder. It is the remainder alone that interests us.
In modulo 12 (or simply mod 12), we use only the twelve
whole numbers from 0 through 11. It is a finite system. Any integer
can be expressed as one of the numbers from 0 through 11. Classic sums
and products can be presented visually in operations tables. The tables
for addition and multiplication mod 12 are presented below.
The addition table is rather boring. The only pattern worth mentioning is the presence of identical numbers on diagonal rows. There are many patterns in the multiplication table.
|The familiar 12 hour clock is very old. However there isn't anything special about the number 12. Consider the 4-hour clock at the right. The modulus 4 has been been replaced by the additive identity 0, as on the 12-hour clock. The clock has four numbers from 0 through 3. Every time we go past 0 on the dial, we start counting the hours at 1 again.|
All multiples of 4 are equivalent to 0. To convert whole numbers into their mod 4 equivalent, we divide by the modulus 4 and record the remainder as before. Using the arrow buttons, set the modulus in the applet below to "4" so you can examine the mod 4 addition table. Verify the results for yourself.
Change the operation to multiplication by clicking on the "x" button, then study the resulting table.
A light switch with four positions (OFF, LOW, MEDIUM, HIGH) operates like a mod 4 system. Likewise the 7-day cycle of common weekdays operates like a mod 7 system.
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Copyright Jill Britton
|Applet copyright Alexander Bogomolny of Interactive Mathematics Miscellany and Puzzles|