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

Is $\mathbb {R} _{\infty }$ a group under multiplication?

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