# Algebra And Geometry Construction Case Studies Project Management This problem is a bit more complicated than the previous one.

We want to find the area of the shaded region; let's first figure out what we know.

is either 3 or –3; the negative value makes no sense in this context, however, so we reject it as a spurious solution (a solution that does not make any sense in the context of the problem).

This leaves us with = 2(3) 2(3)(7) = 6 42 = 48 The perimeter of the rectangle is thus 48 units.  For instance, the unit circle is the set of zeros of and is an algebraic variety, as are all of the conic sections.So to ask the question more precisely, how do you take a set of axioms and know they describe a 'geometry' or an 'algebra' or any other subject for that matter?It seems unlikely that mathematicians would label things differently without having a clear distinction between them.Grothendieck defined schemes as the basic geometric objects, which have the same relationship to the geometry of a ring as a manifold to a coordinate chart.The language of category theory evolved at around the same time, largely in response to the needs of the increasing abstraction in algebraic geometry.Traditions die hard, even in the face of supposedly rational considerations.200 years ago, anyone who could "really prove" things, as opposed to giving a physical/physics-y quasi-heuristic, was a "geometer"...We know how to calculate the area of a square of side . The Pythagorean theorem tells us that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse.In this case, the legs of the right triangle are both of length Once again, we can reject the negative number as a spurious solution.It is the interplay between symbols and geometry that causes a great deal of very cool mathematics to come to light.Algebraic geometry is the study of geometries that come from algebra, in particular, from rings.

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1. The literature review is a written overview of major writings and other sources on a selected topic.