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Excel graph of the difference between two evaluations of the smallest root of a quadratic: direct evaluation using the quadratic formula (accurate at smaller b) and an approximation for widely spaced roots (accurate for larger b). The difference reaches a minimum at the large dots, and round-off causes squiggles in the curves beyond this minimum.
The two families of lines on a smooth (split) quadric surface. In mathematics, a quadric or quadric hypersurface is the subspace of N-dimensional space defined by a polynomial equation of degree 2 over a field. Quadrics are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine ...
A quadric is said to be non-degenerate if the matrix is invertible. A non-degenerate quadric is non-singular in the sense that its projective completion has no singular point (a cylinder is non-singular in the affine space, but it is a degenerate quadric that has a singular point at infinity).
In general a quadratic field of field discriminant can be obtained as a subfield of a cyclotomic field of -th roots of unity. This expresses the fact that the conductor of a quadratic field is the absolute value of its discriminant, a special case of the conductor-discriminant formula.
Kingsoft Office Spreadsheets 2012 – For MS Windows. Both free and paid versions are available. It can handle Microsoft Excel .xls and .xlsx files, and also produce other file formats such as .et, .txt, .csv, .pdf, and .dbf. It supports multiple tabs, VBA macro and PDF converting. [10] Lotus SmartSuite Lotus 123 – for MS Windows. In its MS ...
Excel for the web is a free lightweight version of Microsoft Excel available as part of Office on the web, which also includes web versions of Microsoft Word and Microsoft PowerPoint. Excel for the web can display most of the features available in the desktop versions of Excel, although it may not be able to insert or edit them.
The idea of the proof of the class number formula is most easily seen when K = Q(i).In this case, the ring of integers in K is the Gaussian integers.. An elementary manipulation shows that the residue of the Dedekind zeta function at s = 1 is the average of the coefficients of the Dirichlet series representation of the Dedekind zeta function.
There is no loss of generality in taking a and b to be non-zero and square-free integers. According to Galois theory, there must be three quadratic fields contained in K, since the Galois group has three subgroups of index 2. The third subfield, to add to the evident ℚ(√ a) and ℚ(√ b), is ℚ(√ ab).