both program and data

I like to say “both verb and noun” — keep it closer to elementary school level when possible ;)

]]>I calculated 2 l square + l basis elements for symplectic algebra.

]]>This is a purely algebraic function, and every time we start with a rational input we’ll get a rational output. That is, . You can even show that it’s Cauchy. But what’s the limit of the sequence? If you only have the rational numbers to work with, this is a sequence that cannot converge even though we have a really simple, practical reason to want it to.

]]>I was actually thinking more along the lines that you say that lack of Dedekind-completeness is a “flaw” in ℚ, which is a word common in mathematical rhetoric that I’ve never understood. (Ditto for “nice”.) I get why ℚ doesn’t have this property, but not why that’s bad.

]]>But this similarity does point the way towards showing that the two completions of the real numbers are equivalent. The explicit isomorphism is in the next post.

]]>