Interesting with a side note about toying with Excel.
When I plug in:
1/(3 sqrt(7)-8)
and
-3 sqrt(7)-8
...into Excel, there must be a rounding error. They resolve to:
-15.9372539331939000
and
-15.9372539331938000
...respectively.
Time to plug these into Wolfram. I get:
-15.93725393319377177150484726091778127713077754924735054110500337760320646969085088328117865942363083184519373501549238...
and
-15.93725393319377177150484726091778127713077754924735054110500337760320646969085088328117865942363083184519373501549238...
If you want to do exact computations, SAGE is handy, Free, and built on Python:
sage: k.<a> = QuadraticField(7); k
Number Field in a with defining polynomial x^2 - 7
sage: x = 3*a - 8
sage: x.is_integral()
True
sage: 1/x
-3*a - 8
sage: x * (1/x)
1
sage: (1/x).is_integral()
True
(Note: the is_integral() is because we define k to be QQ(sqrt(7)), not ZZ(sqrt(7)), so we want to check that our elements are really integers, i.e., lie in ZZ(sqrt(7))).
It would be nice if you could prepend them with four spaces, to place them in code blocks (they have overflow:scroll), or break them into multiple lines (or both).
When I plug in:
...into Excel, there must be a rounding error. They resolve to: ...respectively.Time to plug these into Wolfram. I get:
...respectively.