246A, Notes 0: the complex numbers

What's new

Kronecker famously wrote, “God created the natural numbers; all else is the work of man”. The truth of this statement (literal or otherwise) is debatable; but one can certainly view the other standard number systems $latex {{bf Z}, {bf Q}, {bf R}, {bf C}}&fg=000000$ as (iterated) completions of the natural numbers $latex {{bf N}}&fg=000000$ in various senses. For instance:

  • The integers $latex {{bf Z}}&fg=000000$ are the additive completion of the natural numbers $latex {{bf N}}&fg=000000$ (the minimal additive group that contains a copy of $latex {{bf N}}&fg=000000$).
  • The rationals $latex {{bf Q}}&fg=000000$ are the multiplicative completion of the integers $latex {{bf Z}}&fg=000000$ (the minimal field that contains a copy of $latex {{bf Z}}&fg=000000$).
  • The reals $latex {{bf R}}&fg=000000$ are the metric completion of the rationals $latex {{bf Q}}&fg=000000$ (the minimal complete metric space that contains a copy of $latex {{bf Q}}&fg=000000$).
  • The complex numbers $latex {{bf C}}&fg=000000$ are the algebraic…

View original post 9,620 more words

Advertisements

About scottedwards2000

data guy
This entry was posted in Uncategorized. Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s