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 . 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).
- For , the product is defined via the usual multiplication of real numbers.
- ( could be a nonzero real number or )
Is a group under multiplication?
Explore the methods you used to prove this result, and what they tell you about the nature of groups.