When you study the theory of geometry, it is pure mathematics.
However, when you start using the geometry you have learned in other, more practical areas, then it becomes applied.
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That's part of geometry.That's part of geometry.That's part of geometry.That's part of geometry.
I would say that, at least in Pure Mathematics, the most challenging field in the past few years it's a sort of mathematical grand unification project, that goes by the name of "Langlands Program"; this has generated a fair amount of Fields medals in recent years, and aims to unify Number Theory, Algebraic Geometry, Arithmetic Geometry, Group Representations and a few more subjects under a common framework.
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The law of closure states that a set of numbers is closed under an operation if when the operation is performed on any two elements of the set the result is an element of the set
To be closed under an operation, when that operation is applied to two member of a set then the result must also be a member of the set. Thus the sets ℂ (Complex numbers), ℝ (Real Numbers), ℚ (Rational Numbers) and ℤ (integers) are closed under subtraction. ℤ+ (the positive integers), ℤ- (the negative integers) and ℕ (the natural numbers) are not closed under subtraction as subtraction can lead to a result which is not a member of the set.