3.1.8 Study Project: Inverses of sums and products 1 Study Project: Inverses of sums and products For reference, the nine algebraic axioms of (, +, i) are listed in the supplement at the beginning of in this section. 1. Translating words into mathematical notation a) In the real number system (, +, i) , when we refer to a product, we mean ab, where and are real numbers, and when we refer to a multiplicative inverse, we mean where is a nonzero real number. • Translate the multiplicative inverse of a product into mathematical notation. • Translate the product of multiplicative inverses into mathematical notation. b) Suppose we wish to prove the following: "The multiplicative inverse of a product is the product of the multiplicative inverses." • Translate the statement into mathematical notation. • If we wish to prove this statement, what are the hypothesis and the conclusion? c) In (, +, i) , reference to a sum, means where and are real numbers, and reference to an additive inverse means where is a real number. • Translate the additive inverse of a sum into mathematical notation. • Translate the sum of additive inverses into mathematical notation. d) Suppose we wish to prove the following: "The additive inverse of a sum is the sum of the additive inverses." • Translate the statement into mathematical notation. • If we wish to prove this statement, what are the hypothesis and the conclusion? Barbara A. Shipman, Active Learning Materials for a First Course in Real Analysis www.uta.edu/faculty/shipman/analysis. Supported in part by NSF grant DUE-­‐0837810 3.1.8 Study Project: Inverses of sums and products 2 2. The multiplicative inverse of a product – first proof Complete the proof, using only the algebraic axioms of (, +, i) . The multiplicative inverse of a product is the product of the multiplicative inverses: Hypothesis: Let a, b ∈  , a, b ≠ 0 . . Conclusion: Show that Proof: Multiplicative identity, used twice Multiplicative inverses, used twice ___________ , used multiple times Commutativity of , several times Multiplicative inverses _____________________________ 3. The additive inverse of a sum – first proof Prove that the additive inverse of a sum is the sum of the additive inverses in six steps, by changing each line of the proof in Part (2) from the multiplicative context to the additive context. Notice how the argument is exactly the same; only notation has changed from multiplicative to additive. 4. The multiplicative inverse of a product – second proof a) We have shown that the multiplicative inverse of a nonzero real number is unique. • Using this fact, if we wish to show that is the multiplicative inverse of , then it is enough to show that ___________________ . • Using the uniqueness of multiplicative inverses, if we wish to show that is the multiplicative inverse of _____________ . , then it is enough to show that Barbara A. Shipman, Active Learning Materials for a First Course in Real Analysis www.uta.edu/faculty/shipman/analysis. Supported in part by NSF grant DUE-­‐0837810 3.1.8 Study Project: Inverses of sums and products 3 b) Complete the proof, using only algebraic axioms and the fact that multiplicative inverses are unique. The multiplicative inverse of a product is the product of the multiplicative inverses: Hypothesis: Let a, b ∈  , a, b ≠ 0 . Conclusion: Show that Proof: We will show that multiplicative inverse of . conclude that , that is, we will show that acts as the . Since multiplicative inverses are unique, this is sufficient to ___________ , used multiple times ___________ , used multiple times ___________ , used multiple times c) Compare the proof above with the proof in Part (2). Do you prefer one over the other? Explain. 5. The additive inverse of a sum – second proof Give a second proof that the additive inverse of a sum is the sum of the additive inverses, using the fact that additive inverses are unique. Explain how this proof is the same as the proof in Part (4), where only the notation has changed to reflect the additive rather than the multiplicative context. ■ Barbara A. Shipman, Active Learning Materials for a First Course in Real Analysis www.uta.edu/faculty/shipman/analysis. Supported in part by NSF grant DUE-­‐0837810