Tour:Inquiry problems one (beginners)

This page is a Inquiry problems page, part of the Groupprops guided tour for beginners (Jump to beginning of tour)
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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 ${\mathbb{R}}_{\mathrm{\infty }}=\mathbb{R}\cup \{\mathrm{\infty }\}$. Define the following commutative multiplication on ${\mathbb{R}}_{\mathrm{\infty }}$: the product of two finite real numbers is their usual product, the product of a nonzero real number with $\mathrm{\infty }$ is $\mathrm{\infty }$ (whichever order we multiply them in), the product of $\mathrm{\infty }$ and $\mathrm{\infty }$ is $\mathrm{\infty }$, and the product of $0$ and $\mathrm{\infty }$ is $1$ (whichever order we multiply them in).

Explicitly:

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

Is

${\mathbb{R}}_{\mathrm{\infty }}$

a group under multiplication?

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