Difference between revisions of "Tour:Inquiry problems one (beginners)"

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(New page: {{tour-special| target = beginners| pagetype = Inquiry problems| secnum = one| next secnum = two| next = Introduction two (beginners)| previous = Examples peek one (beginners)}} This page...)
 
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The ''nonzero'' reals form a group under multiplication. Zero, however, is not invertible.
 
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 <math>\R_\infty = \R \cup \{ \infty \}</math>. Define the following multiplication on <math>\R_\infty</math>: the product of two finite real numbers is their usual product, the product of a nonzero real number with <math>\infty</math> is <math>\infty</math>, and the product of <math>0</math> and <math>\infty</math> is <math>1</math>.
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Here's one way to try to remedy this. Consider the set <math>\R_\infty = \R \cup \{ \infty \}</math>. Define the following multiplication on <math>\R_\infty</math>: the product of two finite real numbers is their usual product, the product of a nonzero real number with <math>\infty</math> is <math>\infty</math>, the product of <math>\infty</math> and <math>\infty</math> is <math>\infty</math>, and the product of <math>0</math> and <math>\infty</math> is <math>1</math>.
  
 
{{quotation|Is <math>\R_\infty</math> a group under multiplication?}}
 
{{quotation|Is <math>\R_\infty</math> a group under multiplication?}}
  
 
Explore the methods you used to prove this result, and what they tell you about the nature of groups.
 
Explore the methods you used to prove this result, and what they tell you about the nature of groups.

Revision as of 17:26, 23 June 2012

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 \R_\infty = \R \cup \{ \infty \}. Define the following 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, the product of \infty and \infty is \infty, and the product of 0 and \infty is 1.

Is \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.