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
)
Isa group under multiplication?
Explore the methods you used to prove this result, and what they tell you about the nature of groups.