# 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:

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
\(x + a = y + a \longrightarrow x = y\)

Right zero element for multiplication
\(a0 = 0\)

\((-a)b = -ab\)

\(ab = 0 \longrightarrow a = 0 \vee b = 0\)

\(x \gt 0 \Longrightarrow -x \lt 0\)

\(x \lt 0 \Longleftrightarrow -x \gt 0\)

\(a \le b \Longrightarrow -b \le -a\)