Fields

We will give the technical definition for a field, but it might help to mention examples.

The set of Rational Numbers is a field. So is the set of Integers.

The Definition

A field is a set with two operations that match the general definitions for addition and multiplication and all of the following are true:

• There are Identity elements for Addition and Multiplication
• Inverse elements exist with respect to both addition and multiplication.
• There is one exception, there is no multiplicative inverse for the zero element.
• Both addition and multiplication are Associative (Associativity)
• both addition and multiplication are commutative (Commutativity)
• multiplication distributes over addition (Distributivity)

Appendix A

Vector Space is built around a type of object in a set (with the set closed to Addition) and it requires the existence of another set containing a something else used to scale the objects in the original set. That something else comes from a field.

We might talk about a Vector Field. Lets start with giving to examples of ehat we can find:

Appendix B

Suppose we create a 2D space with coordinates (x,y) and at every point in that space there is a map in between that point to a something. This Builds on the idea of a function.

• At the point (2,5) we find the vector (4,10).
• At the poiny (3,7) we find the vector (6,14).