I. Field Axioms - Exercises pp. 33-34 (Algebraic Structures)

This first section of the course focuses on the Real Number System and the Complex Number System. In our first several classes, we examine the general laws of behaviors or axioms for real numbers. The axioms are given in three sets; 1) the Field Axioms; 2) The Order Axioms; 3) The Completeness Axioms.

The first section focuses on the field axioms (6 in number) from which we can deduce all familiar properties of real numbers that depend ONLY upon addition and multiplication.


I. Field Axioms


Exercises

1. Prove that if a, b is an element of R and b not equal 0, then -a/b = a/-b = -(a/b).
2. Let a, b be nonzero real numbers. Show that (ab)-1 = a-1/b-1.
3. Let a, b elements of R and b not equal to zero. Prove: if x is a nonzero real number, then ax/bx = a/b.
4. Let a,b,c,d be an element of R where b is not equal to zero and d is not equal to zero.
Prove that a/b times c/d = ac/bd.
5. Verify that a/b = c/d if and only if ad = bc.
6. Assuming b is not equal to 0, verify the following:
a) a/b + c/b = (a+ c)/b b) a/b - c/b = (a-c)/b