Math 2.0

definition

Ordered field

Definition

An ordered field is a field FF with a total order ≤\le on FF. The order must have these two properties for all aa, bb and cc in FF:

  1. if a≤ba \le b, then a+c≤b+ca + c \le b + c;
  2. if 0≤a0 \le a and 0≤b0 \le b, then 0≤ab0 \le a b.

Remarks

In an ordered field, each square is nonnegative: a2≥0a^2 \ge 0 for each aa. In particular 1=12>01 = 1^2 > 0 and −1<0-1 < 0. An equivalent definition uses a positive cone PP. This is a subset of FF that is closed under addition and multiplication. Also, FF must be the disjoint union of PP, {0}\{0\} and −P-P. The field C\mathbb{C} of complex numbers has no such order.

Explanations

L1 human

An ordered field is a number system where you can add, multiply, divide and compare numbers. The rules of comparison are the usual ones. If you add the same number to both sides of an inequality, the inequality stays true. The product of two positive numbers is positive.

What this node needs for W2
  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What depends on this

193 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Ordered_field, Total order

What links here

Used by