# Tour:Inquiry problems one (beginners)

From Groupprops

This page is a Inquiry problems page, part of the Groupprops guided tour for beginners (Jump to beginning of tour)PREVIOUS: Examples peek one (beginners)|UP: Introduction one |NEXT: Introduction two (beginners)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 . Define the following commutative multiplication on : the product of two finite real numbers is their usual product, the product of a nonzero real number with is (whichever order we multiply them in), the product of and is , and the product of and is (whichever order we multiply them in).

Explicitly:

- For , the product is defined via the usual multiplication of real numbers.
- ( could be a nonzero real number or )

Is a group under multiplication?

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