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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.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 classical algebraic geometry, the algebra is the ring of polynomials, and the geometry is the set of zeros of polynomials, called an algebraic variety.
For instance, the unit circle is the set of zeros of and is an algebraic variety, as are all of the conic sections.
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.
The result is that the square has sides of length 4 units. Let's use this to find the area of the shaded region.
Area = 8π – 16 ≈ 9.13 The area of the shaded region is thus about 9.13 square units.
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.
To the best of my understanding, algebra defines You are right, that the attempted distinction makes sense only from a rather naive viewpoint, popular though it may be.
Among professional mathematicians, these inherited "legacy" labels have a surprising endurance, which I think is mostly/only due to their widespread recognition among amateurs and professionals alike, despite their inaccuracy.