MATC Mathematics Club
Madison Area Technical College
Professor Marion Cohen, Arcadia U, PA
"ARITHMETIC: more of the story"
Abstract: Remember arithmetic? Just plain arithmetic. In particular, addition, multiplication, and exponentiation, along with the fact that multiplication is “repeated” addition, while exponentiation is repeated multiplication. As a teen-ager, newly enraptured by math, I was fascinated by these rather simple (in comparison to other math-stuff) ideas. I asked myself, suppose humankind had defined addition differently, and then “repeated” this new addition, resulting in a new multiplication? What exotic arithmetic might result?
Getting a bit more mathematical, I define an arithmetic to be a pair of binary operations on the set of positive integers, such that the second operation is formed by “repeating”, or “iterating” the first. So the pair (addition, multiplication) is an arithmetic and so is (multiplication, exponentiation). Are there other examples of arithmetics? Answer: many! All we need do is specify our first operation, then define the second to be its iteration. It’s fun to start with various first operations and calculate the resulting iterations. You’ll have a chance to try your hands at that during my talk. (Recognizing patterns is the name of the game.)
In my freshman year at NYU, I went the other way, asking: Given the second operation, can we form an arithmetic by finding a “compatible” first? Again, under certain rather general conditions, yes. And it’s fun to start with various second operations and calculate what they’re iterates of. You’ll have a chance to try your hands at that, too.
Towards the end of my talk, I’ll tell you about more recent, and more difficult, problems I’ve been solving and still working on concerning “alternative arithmetics”. The one that’s stumping me at the moment is, can we find arithmetics, other than (addition, multiplication), both of whose operations are associative? Answer: yes. Characterizing those arithmetics is another story. (I have a structure theorem, but proving sufficiency is harder.) Finally: Yes, there are “non-ordinary” arithmetics both of whose operations are commutative. They’re easier than “associative arithmetics” to express in “closed form” and that’s how my talk will end. (If there’s time, we can check it together.)
Marion Deutsche Cohen received her B.A. in mathematics from New York University, and her M.A. and Ph.D. in math from Wesleyan University in Middletown Connecticut. She teaches at Arcadia University in Glenside, Pennsylvania; in particular, she has developed an interdisciplinary course (university seminar), Truth and Beauty: Mathematics in Literature, which has been very popular with students.
Her article about the course (and with the same title as the course) appeared in the March 2013 issue of The Mathematics Teacher, as well as a shorter version in the December 2009-January 2010 issue of FOCUS. She has done work in abstract algebra and calculus. She is also a poet, writer, and memoirist, author of 20 books including Crossing the Equal Sign (Plain View Press, TX), poetry about the experience of and passion for math.
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