1. Introduction to Modern Algebra

2. In arithmetic, you learned four operations n n Addition n n Subtraction n n Multiplication n n Division

3. For example, you learned that n n 2+2=4 n n 0*7 = 0 n n (-1)*(-1) = 1 n n You cannot divide by zero!

4. But did you ever ask... n n Why? n n Does it have to be this way? n n Could there be a different answer? n n Can we divide by zero? n n Do two negatives always make a positive?

5. n n In Modern Algebra, we will answer these questions and many more by looking at the structure of mathematical systems. n n By looking at many different examples of mathematical structures, we will begin to see things that are true about every structure. n n This is why we sometimes call this course "abstract algebra.”

6. Algebraic structure: a set with one or more operations. n n Groups:Sets like integers with +&– n n Rings: Sets like integers with +,– & * (not ÷) n n Fields: Sets like rational numbers with +,–, * & ÷ (except 0)

7. Some of the interesting specific questions we will address are n n How do credit card machines know if you entered a legitimate card number? n n How many ways are there rotate a soccer ball so that no one can tell that it has been moved?

8. Other interesting specific questions we will address are n n Why is the product of two negative numbers always positive? n n Why can't you trisect any angle with a straight edge and compass?

9. In the process, you will learn to n n Calculate using a variety of mathematical objects and operations (not just numbers or polynomials). n n Generalize and prove conjectures. n n Make a giant leap toward being a mature mathematician.

10. n n The End