[Math] difference between “unity” and 1 in applied mathematics

terminology

Is there a difference between "unity" and 1 in applied mathematics?
I know mathematicians have "roots of unity" and "partitions of unity",
but at least those have become standardized. In applications,
"unity" seems to be used randomly and maybe
interchangeably with the number 1. Is there some subtle meaning
there that I'm missing?

Best Answer

In my experience, "unity" is just a fancy word for the number "1". In the two examples you gave, that's what it means, certainly.

"Partition of 1" looks odd, to me. "Roots of one" risks confusion with other meanings of the word "one". So "unity" is useful, rather than merely ostentatious.

Any algebraic structure might have an element that has properties analogous to the number "1", but this thing is typically referred to as a "unit", not as "unity".

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[Math] What’s the difference between “duality” and “symmetry” in mathematics

No, duality and symmetry are not the same thing. Although in many contexts "the dual of" is a symmetric relation, this is not invariably the case (e.g. the dual of the dual of a topological vector space need not be the original).

Moreover symmetry is not just about symmetric relations; it has to do mainly with automorphisms of algebraic, geometric or combinatorial structures. Those structure preserving automorphisms (including trivial the identity mapping) form a group, and we'd refer to it as the symmetry group of the structure.

As you note, there are many kinds of symmetry. Some symmetries have order two but many do not. Indeed the group of symmetries may combine elements that have finite order with those having infinite order, elements that have discrete action with some that are continuous mappings. The symmetries of a right circular cylinder, for example, would include discrete actions like reflection in a midplane as well as continuous actions of rotation about the axis.

If you are looking for a fundamental difference, perhaps it should be noted that duality often involves different categories, i.e. the dual may belong to a different category than the original, while symmetry involves not only the same category but actually a mapping of the same object to itself.

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Euler's number, $e \approx 2.71828$, is very important in many fields of mathematics. An important function that uses this constant is $\exp x = e^x$.

However, $\operatorname{pow}(a, b) = a^b$, a more general case.

You will notice that, if $a=e$,

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