Real Numbers: Positivity

Numbers do not just sit in a set — they sit on a line. Order is the property of $\mathbb{R}$ that makes a number bigger or smaller than another. The whole theory of inequalities, calculus limits, and convergence rests on this single idea, and remarkably, the entire order relation can be built from one tiny piece of data: a designated set of "positive" numbers.

The positivity axioms

The real numbers contain a special subset $P$ — the positive reals — with three properties:

Trichotomy. For every $a \in \mathbb{R}$, exactly one of the following holds: $a \in P$ (positive), $-a \in P$ (negative), or $a = 0$.

Closure under addition. If $a, b \in P$, then $a + b \in P$.

Closure under multiplication. If $a, b \in P$, then $ab \in P$.

Everything else — $a > b$, $a < b$, $a^2 \geq 0$, the triangle inequality — is derived from these three.

Defining order from positivity

We write $a > b$ to mean $a - b \in P$. In words: $a$ is greater than $b$ exactly when their difference is positive. Similarly $a < b$ means $b - a \in P$.

This definition has a useful consequence: order behaves predictably under arithmetic. If $a < b$ and $c \in P$, then $ac < bc$ (multiplying both sides by a positive preserves the inequality). If $a < b$ and $c$ is negative, then $ac > bc$ (multiplying by a negative flips the inequality — this is the sign-flip rule we will use constantly in §3.4).

(The number line is the geometric picture of order: "right of" means "greater than." Drag the markers and watch the relation $a > b$ flip when one crosses the other.)

Squares are non-negative — the most-used fact

For every real $a$, $a^2 \geq 0$, with equality only when $a = 0$. The proof is a one-line trichotomy argument:

If $a > 0$, then $a \in P$, so $a \cdot a = a^2 \in P$ by closure under multiplication.

If $a < 0$, then $-a \in P$, so $(-a)(-a) = a^2 \in P$.

If $a = 0$, then $a^2 = 0$.

In every case $a^2 \geq 0$. This humble fact is the seed of the arithmetic-geometric mean inequality, the Cauchy-Schwarz inequality, and most of the optimisation arguments you will meet in calculus.

Try it

• Use the positivity axioms to prove: if $0 < a < b$, then $a^2 < b^2$. (Hint: factor $b^2 - a^2$.)
• Show that if $a > b > 0$, then $\dfrac{1}{a} < \dfrac{1}{b}$. Where exactly do you use closure under multiplication?
• Prove the AM-GM inequality: for $a, b > 0$, $\dfrac{a + b}{2} \geq \sqrt{ab}$. (Hint: start from $(\sqrt{a} - \sqrt{b})^2 \geq 0$.)
• If $x^2 = 9$, does it follow that $x = 3$? Why or why not?

Pause: which of the three positivity axioms is being used when you multiply both sides of an inequality by a positive number? Which is used when you cancel a positive factor from both sides?

A trap to watch for

The classic sign mistake: students square both sides of an inequality to remove a square root, forgetting that squaring is not monotonic. Try $-3 < 2$: square both sides and you get $9 < 4$, which is false. The rule "$a < b \Rightarrow a^2 < b^2$" only works when BOTH sides are non-negative. The fix: before squaring an inequality, check that both sides have known sign. If either could be negative, split into cases.

What you now know

You can derive the entire order relation from three positivity axioms, and you understand exactly when multiplying or squaring preserves or flips an inequality. The next two sections push these ideas: powers and roots use the order structure to define $\sqrt[n]{a}$ uniquely, and inequalities become a discipline of their own.

Quick check

References

• Lang, S. (1971). Basic Mathematics. Springer. Chapter 3, §2 — the positivity axioms and their consequences.
• Spivak, M. (2008). Calculus, 4th ed. Publish or Perish. Chapter 1 — the order axioms presented alongside the field axioms.
• Hardy, G. H., Littlewood, J. E., and Pólya, G. (1934). Inequalities. Cambridge. The canonical reference for the AM-GM and related inequalities derived from positivity.