Pushing You Beyond What You May Know (Remember!)
The idea of finite arithmetic or modulo systems is well known, as we all operate on the idea of a 7day week and 12hour time cycle. In these two realworld systems, the concept that 3 + 5 = 1 and 10 + 5 = 3 are acceptable, respectively. That is, 5 days after Wednesday is Monday or 5 hours from 10 am will be 3 pm.
For example, the arithmetic table for addition mod 7 is shown. Be sure you are aware of the patterns that are evident.
Now, let's push on with some additional investigations:
 In mod 7 arithmetic, what is 3 x 6? 2 x 5? Remember: Multiplication is basically multiple additions.
 Can you construct a similar table for multiplication mod 7? What patterns can you find?
 Do you need a table for subtraction mod 7? Explain.
 The case of division mod 7 needs some creative thought. For example, using previous arithmetic understandings, 4/3 = M can be found by considering 3 x M = 4. Using this idea, construct a division table mod 7? Any difficulties arise...and how did you respond to them?
 The last question actually introduced a novel idea that occurs with mod arithmetic: Fractions may no longer be necessary! That is, can the fraction 4/3 (or any ratio of positive integers) be replaced by a single number from the set {0,1,2,3,4,5,6}. But new surprises occur...such as, can you find other fractions which are "equal" to 4/3. How would you interpret the number 2 1/3...again more surprises!
 In mod 7, can you solve the equation x^{2} = 4? Are there one or two solutions? Can any equation x^{n} = m be solved? Explain.
 Thus far, we may have created a system that represents all rational numbers. So, get creative....any way to represent irrational numbers such as SQRT(2)? pi? e? phi? 1.01002000100001....?
 Finally, what happens if you had started with arithmetic mod 6 instead of arithmetic mod 7? Any idea why problems occur?
NOTE: Modular arithmetic ideas are very useful in advanced mathematics, such as Diophantine analysis, number theory, abstract algebra, and general problem solving.
Hint: Just start playing..building tables...and then playing some more, all while reflecting on obtained results. Lots of patterns will emerge!
Solution Commentary: For more information, investigate online websites involving modulo arithmetic, congruences, etc.
