Ordered field

In this section, you'll learn the formal definition for ordered fields, and prove several of their basic properties.

Watch this to learn what a field is:

What is a field ?
Dr Peyam

Then watch the first approximately 2 minutes of this video, to learn the formal definition for ordered fields:

Then go through these proofs:

Do all the problems at the end of the video:
Left cancellation law for addition $$a + x = a + y \longrightarrow x = y$$
Right cancellation law for addition $$x + a = y + a \longrightarrow x = y$$
What is a field ?
Dr Peyam
Right zero element for multiplication $$a0 = 0$$
$$(-a)b = -ab$$
What is a field ?
Dr Peyam
$$ab = 0 \longrightarrow a = 0 \vee b = 0$$
What is a field ?
Dr Peyam
$$x \gt 0 \Longrightarrow -x \lt 0$$
$$x \lt 0 \Longleftrightarrow -x \gt 0$$
$$a \le b \Longrightarrow -b \le -a$$
Field and Order Axioms
NSC - MATH 457 Videos (Introduction to Real Analysis)