Is Infinity the Identity Element in Some Operations?

[Ontophobe]

If zero is the additive identity and one is the multiplicative identity, is there an operation for which infinity is the identity?

 

[Hornbein]

If zero is the additive identity and one is the multiplicative identity, is there an operation for which infinity is the identity?

Sure. If the only number you have is infinity, then it is the additive, subtractive, and multiplicative identity.

 

[HallsofIvy]

In what number system? 0 is the additive identity and 1 the multiplicative identity in the integers, rational numbers, real numbers, complex numbers but none of those include "infinity". To have "infinity" you have to go to the "extended real number system" but that does NOT have the usual operations so has no "identities".

 

[aikismos]

I think, dear OP, you might want to draw a fine point on the fact that 0 and 1 are numbers (whether natural, integral, real, or complex) and infinity is not a number, so much as a symbol to represent a process. As such, when talking about identities you have to ask yourself, even in an abstract algebra, does infinity ever play the role of an identity. How could you possibly interpret it as identity? For example:

Adding something to nothing gives you that same something.
Multiplying something by one gives you that same something.
What operation could you do with infinity so that no matter what the number, you get that same something?

Infinity is essentially a symbol we invoke when we want to represent iteration potentially without end, more than it is a number which we can count to or perform standard operations such as addition, multiplication, and exponentiation (all of which are just shortcuts for counting).

 

[pwsnafu]

As such, when talking about identities you have to ask yourself, even in an abstract algebra, does infinity ever play the role of an identity.

Answer is yes. Infinity is the identity element of elliptic groups.