Rational numbers

foundations

Published

October 11, 2026

To do

Read §5 Rational numbers
Before Theorem 3 and Theorem 4 (√2): cover the proof, try it yourself
§5 exercises: 1, 2, 3 (a c e g), ★4, 5 (a c e g i), 6 (a c e), 7, ★8, ★9, ★10, 11, 13, 16, 19
Paper: all listed exercises. Mark ✓ / ✗ / S. Answers: back of the book.
Notebook: write each div below in your own words. Copy the ★ exercises in clean.

Definition 1 (Rational number (racionální číslo)) Your words.

Theorem 1 (Cross-multiplying (násobení do kříže)) Your words.

Proof. Cover the book’s proof. Try 10–15 min first.

Theorem 2 (Cancellation rule for fractions (krácení zlomků)) Your words.

Proof. Cover the book’s proof. Try 10–15 min first.

Theorem 3 (Theorem 3: lowest form (zlomek v základním tvaru)) Your words.

Proof. Cover the book’s proof. Try 10–15 min first.

Theorem 4 (Theorem 4: \sqrt{2} is irrational (iracionální)) Your words.

Proof. Cover the book’s proof. Try 10–15 min first.

Intuition

Your words.

Exercises

★ 4. Let a = m/n with m \neq 0, n \neq 0. Show that there is a rational b with ab = ba = 1.

Solution

Your words.

★ 7c. Show that \binom{m}{n} = \binom{m}{m - n}.

Solution

Your words.

★ 7d. Show that \binom{m}{n - 1} + \binom{m}{n} = \binom{m + 1}{n} for 1 \le n \le m.

Solution

Your words.

★ 8. Prove: no positive rational a has a^3 = 2.

Solution

Your words.

★ 9. Prove: no positive rational a has a^4 = 2.

Solution

Your words.

★ 10. Prove: no positive rational a has a^2 = 3.

Solution

Your words.