3.1.9 Study Project: The inverse of an inverse 1 Study Project: The inverse of an inverse 1
1
In everyday algebra with real numbers, we use familiar calculations such as − =
, −(−9) = 9 , 7 −7
and 1 / (1 / 5) = 5 , for example. In this study we show how these properties of additive and multiplicative inverses follow from the nine algebraic axioms of (, +, i) , which are listed in Supplement 3.1.1. 1. The additive inverse of an additive inverse a) Consider the statement "The additive inverse of the additive inverse of is ." • Translate this statement into mathematical notation. • If we wish to prove the statement, what are the hypothesis and the conclusion? b) Complete the proof, using only the algebraic axioms of (, +, i) . The additive inverse of the additive inverse of is : Hypothesis: Let a ∈  . Conclusion: Show that . Proof: _________________________ Additive inverses Associativity of _____________________________ _____________________________ 2. The multiplicative inverse of a multiplicative inverse a) Consider the statement "The multiplicative inverse of the multiplicative inverse of is " • Translate this statement into mathematical notation. • If we wish to prove the statement, what are the hypothesis and the conclusion? b) Changing each line of the proof in Part (1b) from the additive to the multiplicative context, prove that the multiplicative inverse of the multiplicative inverse of is . Notice how the argument is exactly the same; only the notation has changed from additive to multiplicative. 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 3.1.9 Study Project: The inverse of an inverse 2 3. Combining additive and multiplicative inverses a) Consider the following statement: "The additive inverse of a multiplicative inverse is the multiplicative inverse of the additive inverse." • Do a few experiments with numbers to see if you believe the statement. • Translate the statement into mathematical notation. • If we wish to prove the statement, what are the hypothesis and the conclusion? b) Suppose we wish to show that for a nonzero real number , . Using the fact that the additive inverse of a real number is unique, it suffices to show that . Complete the proof below, using one axiom to justify each step, if possible. Two steps require lemmas. Which steps are they, and what are the lemmas? The additive inverse of a multiplicative inverse is the multiplicative inverse of the additive inverse: Hypothesis: Let , . . Conclusion: Show that Proof: Using the fact that the additive inverse of a real number is unique, it suffices to . show that __________________________ Multiplicative inverses Associativity of multiplication __________________________ Associativity of multiplication Multiplicative identity Distributive property __________________________ __________________________ c) State and prove the two lemmas that you need in order to complete the proof above. Prove each lemma using only one algebraic axiom of (, +, i) in each step. ■ 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