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## 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 7-day week and 12-hour time cycle. In these two real-world 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 x2 = 4? Are there one or two solutions? Can any equation xn = 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 on-line websites involving modulo arithmetic, congruences, etc.