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

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(Adding a point at infinity)
 
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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>, 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>.
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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 commutative 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> (whichever order we multiply them in), 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> (whichever order we multiply them in).
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Explicitly:
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* For <math>a,b \in \R</math>, the product <math>ab</math> is defined via the usual multiplication of real numbers.
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* <math>0 \infty = \infty 0  = 1</math>
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* <math>a \infty = \infty a = \infty \ \forall \ a \in \R_\infty \setminus \{ 0 \}</math> (<math>a</math> could be a nonzero real number or <math>\infty</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.

Latest revision as of 16:51, 25 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 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)
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.