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What is the use of extension of the real and complex field?

Does every field have a non-trivial Galois extension?Extension elements always exist for field extensions?Field extension generated by $alpha$ and separabilityA question on the construction of the complex numbers and field extensionsThe real numbers are a field extension of the rationals?How many quadratic extension are there on a field?Extension of real analytic function to a complex analytic functionLinear transformation over real field vs. complex fieldDegree of the field of rational numbers extended by a complex numberDo the field of complex numbers arise necessarily and uniquely as the only field of pairs of ordered real numbers retaining some desired proprieties?

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What is the use of the extension of the real and complex fields to extended real and complex numbers (including $infty$)?

If the extended real and complex sets are no longer fields, then what is the use of such extensions?

What is the application of the extended real and complex number sets over the real and complex field?

real-analysis complex-analysis complex-numbers field-theory real-numbers

share|cite|improve this question

edited Mar 19 at 15:47

J. W. Tanner

4,4691320

asked Mar 19 at 15:38

Bijayan RayBijayan Ray

1511213

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  A subset of the real numbers might not be bounded by a real number but bounded in the extended real numbers
  $endgroup$
  – J. W. Tanner
  Mar 19 at 15:43
  

  
  
  
  


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  For the extended reals, the addition of $pminfty$ makes the order $<$ on the set "complete." Every set of extended real numbers has a supremum and infimum. This makes certain arguments and definitions easier. Both the extended reals and the extended complex numbers are "compact," a topological property that has a lot of convenience in general.
  $endgroup$
  – Thomas Andrews
  Mar 19 at 15:53
  

  
  
  
  

add a comment | 

0

1

$begingroup$

What is the use of the extension of the real and complex fields to extended real and complex numbers (including $infty$)?

If the extended real and complex sets are no longer fields, then what is the use of such extensions?

What is the application of the extended real and complex number sets over the real and complex field?

real-analysis complex-analysis complex-numbers field-theory real-numbers

share|cite|improve this question

edited Mar 19 at 15:47

J. W. Tanner

4,4691320

asked Mar 19 at 15:38

Bijayan RayBijayan Ray

1511213

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  A subset of the real numbers might not be bounded by a real number but bounded in the extended real numbers
  $endgroup$
  – J. W. Tanner
  Mar 19 at 15:43
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  For the extended reals, the addition of $pminfty$ makes the order $<$ on the set "complete." Every set of extended real numbers has a supremum and infimum. This makes certain arguments and definitions easier. Both the extended reals and the extended complex numbers are "compact," a topological property that has a lot of convenience in general.
  $endgroup$
  – Thomas Andrews
  Mar 19 at 15:53
  

  
  
  
  

add a comment | 

$begingroup$

What is the use of the extension of the real and complex fields to extended real and complex numbers (including $infty$)?

If the extended real and complex sets are no longer fields, then what is the use of such extensions?

What is the application of the extended real and complex number sets over the real and complex field?

real-analysis complex-analysis complex-numbers field-theory real-numbers

share|cite|improve this question

edited Mar 19 at 15:47

J. W. Tanner

4,4691320

asked Mar 19 at 15:38

Bijayan RayBijayan Ray

1511213

$endgroup$

share|cite|improve this question

edited Mar 19 at 15:47

J. W. Tanner

4,4691320

asked Mar 19 at 15:38

Bijayan RayBijayan Ray

1511213

share|cite|improve this question

edited Mar 19 at 15:47

J. W. Tanner

4,4691320

asked Mar 19 at 15:38

Bijayan RayBijayan Ray

1511213

edited Mar 19 at 15:47

J. W. Tanner

4,4691320

edited Mar 19 at 15:47

J. W. Tanner

4,4691320

asked Mar 19 at 15:38

Bijayan RayBijayan Ray

1511213

asked Mar 19 at 15:38

Bijayan RayBijayan Ray

1511213