算术与公设 ℝ

Introduction: Is Q Enough

Definition 1.8 (Terminating Decimals)

What is $0.375$ as a fraction in lowest terms?

Definition 1.9 (Repeating Decimals)

Which of the following is not a repeating decimal?

Definition 1.10 (Non-Repeating Decimals)

Why do non-repeating decimals expose the gaps in Q?

Three Properties of the Real Number Postulates

What does the completeness (property III) of the real number postulates say?

The Seven Field Axioms

Which field axiom is $a(b+c)=ab+ac$?

Example: Proof of the Zero Product Theorem

Convert $0.\overline{9}$ to a fraction in lowest terms. (Enter a fraction such as p/q)

In the proof of $a\cdot 0=0$, after setting $e=a\cdot 0$ we obtain $e=e+e$. Which axiom is used in this step?

How are subtraction $a-b$ and division $a\div b$ ($b\neq 0$) defined in $\mathbb{R}$ using inverses? Separate the two answers with a semicolon.

Is $a\cdot 0=0$ an axiom or a corollary?

In your own words, explain: why is $\mathbb{Q}$ not enough, so that we need to postulate $\mathbb{R}$? What problem does each of the three properties of the postulated $\mathbb{R}$ solve?

Comprehensive Check