Multiplicative inverses

ImportantTo do

• Read §6 Multiplicative inverses
• Before each proof: cover it, try it yourself
• §6 exercises: 1 (a c e g i), ★2, ★3, ★4, ★5, 6, 8, 9, 12, 13, 14, 16
• Weak spots (diagnostic 8, 10): 12 and 13 are linear systems. In 2–5, cancel factors, never terms
• 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 (Multiplicative inverse (převrácené číslo)) Your words.

Theorem 1 (If ab = 1, then b = a^{-1}) Your words.

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

Theorem 2 (If ab = 0, then a = 0 or b = 0) Your words.

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

Theorem 3 (Cancellation law for multiplication) Your words.

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

TipIntuition

Your words.

Exercises

★ 2d. Prove: \dfrac{x^n - 1}{x - 1} = x^{n-1} + x^{n-2} + \dots + x + 1 for x \neq 1.

NoteSolution

Your words.

★ 4c. Let x = \dfrac{1 - t^2}{1 + t^2} and y = \dfrac{2t}{1 + t^2}. Show that x^2 + y^2 = 1.

NoteSolution

Your words.

★ 5c. Let n be odd. Prove: \dfrac{x^n + 1}{x + 1} = x^{n-1} - x^{n-2} + \dots - x + 1 for x \neq -1.

NoteSolution

Your words.