完备性与根

Introduction: Filling the Gaps on the Number Line

The Completeness Postulate

What does the completeness postulate guarantee?

Rational Numbers and Irrational Numbers

Which of the following numbers is irrational?

The Digit-by-Digit Construction of the Square Root of 2

Using the digit-by-digit method to find $\sqrt{2}$, what is the tenths digit? (Given $1.4^2=1.96<2<2.25=1.5^2$)

nth Roots and Proposition 1.7

According to Proposition 1.7, is $\sqrt[3]{8}$ rational or irrational?

Example: The Irrationality of the Square Root of 2 plus the Square Root of 3

Which of $\sqrt{4}$ and $\sqrt{2}$ is rational and which is irrational? Enter "rational" or "irrational" for each, separated by a semicolon.

Is $\sqrt[3]{2}$ rational or irrational? (Enter rational or irrational)

Determine whether $\dfrac{\sqrt{18}}{\sqrt{8}}$ is rational or irrational. (Enter rational or irrational)

What does the completeness postulate guarantee?

In your own words, explain: why is $\mathbb{Q}$ not complete while $\mathbb{R}$ is complete? What does the completeness postulate guarantee? How does Proposition 1.7 distinguish between the existence and the rationality of $\sqrt[k]{n}$?

Comprehensive Check