序与幂

Introduction: Installing an Order Relation on the Real Numbers

The Five Order Axioms

Which order axiom is the basis for $-2 < -1$?

Sign Rules and Their Proofs

If $a > 0$ and $b < 0$, what is the sign of $ab$?

Rules for Manipulating Inequalities

Given $a > b$ and $c < 0$, which of the following is correct?

Definition and Monotonicity of Powers

If $0 < r < 1$ and $m > n$, what is the relation between $r^m$ and $r^n$?

Example: Comparing the Square Root of 2 with 1.5

Arrange $-2.5$, $-\dfrac{1}{2}$, $0$, $\dfrac{3}{4}$ from smallest to largest. Enter the four numbers, separated by commas.

If $a > 0$ and $b < 0$, is $ab$ positive, negative, or zero? (Enter positive / negative / zero)

Compare $2^{100}$ with $4^{50}$. Which is larger? (Enter $2^{100}$ or $4^{50}$)

What is the core conclusion of the sign rules?

In your own words, explain: why does the inequality sign reverse when multiplying by a negative number? From which order axiom is this derived?

Comprehensive Check