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

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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>. | + | 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). |

+ | |||

+ | Explicitly: | ||

+ | |||

+ | * For <math>a,b \in \R</math>, the product <math>ab</math> is defined via the usual multiplication of real numbers. | ||

+ | * <math>0 \infty = \infty 0 = 1</math> | ||

+ | * <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 . 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 )

Is a group under multiplication?

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