Although the real numbers satisfy the field axioms given in the previous section, these axioms don't fully characterize R. For one thing, we're missing the notion of order. (We're going to need one more thing to fully characterize R, but we'll get there later.)

3. Axiom (The Order Axioms). There is a relation called < that's defined for all real numbers - that is, given any numbers x, y R, then x < y is either true or false. This relation satisfies the following properties: (a) Trichotomy ("tri-" = three; "-chotomy" = cutting): For any x, y R, exactly one of the following three possibilities is true: x < y, y < x, or x = y.

Taking y = 0, we can define two new words: 0 < x, in which case we say that x is positive; x < 0, in which case we say that x is negative; or x = 0.

(b) Transitivity: If x < y and y < z, then x < z.

(c) Addition preserves order: If x < y, then x + z < y + z for any real number z.

(d) Multiplication preserves positivity: If 0 < x and 0 < y, then 0 < x y. (So we say that the positive numbers are closed under multiplication.)

Use all of the things above using only the < sign, but it will also be convenient for us to use the > sign in the usual way: x > y means the same thing as y < x. Same with , which means "either x < y or x = y", and , which means something similar.

4. Pick a few of the following to prove using the order axioms. (How many is "a few"? More than one and less than all.)

(a) Let a be a real number. Show that if a > 0, then a < 0. Similarly, show that if a < 0, then a > 0 (and thus that a < a!).

(b) Prove that 1 > 0.

(c) Prove that there exists a negative number.

(d) Prove that if a > b and c < 0, then ac < bc. ("Multiplying by a negative number flips the sign.")

(e) (Corollary) Prove that a positive number times a negative number is negative.

(f) Prove that if a < b, then b < a. (g) Suppose that a, b > 0 and a > b. Then 1 a < 1 b .