Tour:Inquiry problems one (beginners)

From Groupprops
Jump to: navigation, search
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 \R_\infty = \R \cup \{ \infty \}. Define the following commutative 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 (whichever order we multiply them in), the product of \infty and \infty is \infty, and the product of 0 and \infty is 1 (whichever order we multiply them in).

Explicitly:

  • For a,b \in \R, the product ab is defined via the usual multiplication of real numbers.
  • 0 \infty = \infty 0  = 1
  • a \infty = \infty a = \infty \ \forall \ a \in \R_\infty \setminus \{ 0 \} (a could be a nonzero real number or \infty)
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.