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)