Department of Mathematical Sciences
Binghamton University

Math 330: Number Systems
Term Project

Section 5, Zaslavsky | Spring 2015

Term Project Rules and Description

• The project is entirely your own work. Don't collaborate on it with anyone else. You may ask me questions, but I don't promise to answer them.
• The project involves reading, understanding, and answering questions related to Chapters 1-6.
• You will answer, as best you can, a series of questions about the topic.
• There are no rewrites on the project.
• The project is 15% of the grade.

Term Project Details

1. (The project.) Consider the set of integers, Z. A binary operation ⊗ on Z is called a "pseudo-multiplication" if it satisfies all of the three properties:
   1. ⊗ is commutative: m ⊗ n = n ⊗ m,
   2. ⊗ is associative: (m ⊗ n) ⊗ p = m ⊗ (n ⊗ p), and
   3. ⊗ is distributive over + (the usual addition of integers): m ⊗ (n + p) = m ⊗ n + m ⊗ p.
   Find all possible pseudo-multiplication operations ⊗ on Z. For each operation you find, prove it has the three properties so you know it really is a pseudo-multiplication.
   If you don't find all, find at least one (besides ·) and find as many as you can.
2. (This is part of the project.) For each pseudo-multiplication you found, answer the following questions if you can:
   1. Does it have a pseudo-multiplicative identity element?
   2. Does it have the cancellation property?
   3. Find all integers that have pseudo-multiplicative inverses.
3. Suggestions for how to get started. (These are not part of the required project.) The suggestions in part D are the most important.
   1. You may use anything you know about Z, including the usual operations of addition, subtraction, and multiplication.
   2. Do not give detailed justifications of all algebraic steps; use algebra freely.
   3. Look at examples. Make up a binary operation and test it to see if it satisfies the three properties.
      • For instance, here is a random selection of possible binary operations.
        • m ⊗ n := m · n.
        • m ⊗ n := m · n + 3.
        • m ⊗ n := m²ⁿ⁻¹.
        • m ⊗ := n · √m if m is a perfect square, and n + m² if it isn't. (This shows that any rule that defines a binary operation on the integers, no matter how complicated, is fair game to think about. I don't mean I'm recommending such rules.)
      • Try a variation on a known example. (You know one example: Ordinary multiplication, ·.)
      • Look for patterns in your examples.
      If it doesn't satisfy the properties, it may suggest what not to try. If it does, it may suggest what to try.
   4. Use the properties 1.i-iii, just as you used Axioms 1.1-5 with ordinary multiplication, to answer questions like these:
      • How is 1 ⊗ 1 related to 2 ⊗ 1 in a pseudo-multiplication?
      • What can you say about the value of 0 ⊗ 1?
      • How does (2 · m) ⊗ k compare with m ⊗ k?