Tour:Inquiry problems one (beginners)

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 $$\infty$$)

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