Tour:Inquiry problems one (beginners)

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This page is a Inquiry problems page, part of the Groupprops guided tour for beginners (Jump to beginning of tour)
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NEXT SECTION Inquiry problems: Inquiry problems two
General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part

This page lists some problems for thought/inquiry. Many of these problems are aha problems, and they should be obvious at the end of part two.

Adding a point at infinity

The nonzero reals form a group under multiplication. Zero, however, is not invertible.

Here's one way to try to remedy this. Consider the set \R_\infty = \R \cup \{ \infty \}. Define the following multiplication on \R_\infty: the product of two finite real numbers is their usual product, the product of a nonzero real number with \infty is \infty, the product of \infty and \infty is \infty, and the product of 0 and \infty is 1.

Is \R_\infty a group under multiplication?

Explore the methods you used to prove this result, and what they tell you about the nature of groups.