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In the twentieth century, it was discovered that the basic ideas of classical algebraic geometry can be applied to any commutative ring with a unit, such as the integers.
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.
The only real difference I can find is the reference to quantities (but what is a quantity? There are also often remarks about geometry referring to 'real world' objects such as shapes and solids and relations between them.
I'm hesitant to accept these because they fail to recognise the distinction between observations and mathematics.
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"...As a consequence, algebraic geometry became very useful in other areas of mathematics, most notably in algebraic number theory.For instance, Deligne used it to prove a variant of the Riemann hypothesis.To apply algebra in this context, you don't need any new algebra skills, but you do need to have some understanding of geometry and an ability to translate the somewhat abstract ideas of algebra to a more concrete use in geometry.Let's start with a couple of practice problems to illustrate.Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.Visit Stack Exchange "the branch of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations." "the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogues" which don't answer the question much further than 'geometry is what geometers do' and an even more cringeworthy 'algebra is black scribbles on a page for people doing algebra'.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.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.