When one of the two inputs of a binary operation[1] that is acting on a set[2] is an identity element, the output will be the other element that was used as an input.
Very loosely speaking, the identity element doesn’t change anything when combined with things. E.g., multiplying 3 by 1 (a multiplicative identity) results in 3, or adding 0 (an additive identity) to 5 results in 5.
Additional points to note:
- Which element of the set is the identity element depends on which binary operation is acting on the set.
- The identity element is often just called “the identity” if it is clear which binary operation and set you are talking about.
- If the order that an element goes into the binary operation matters, there can be elements which behave like an identity element if put in first but not when put in second. It’s also possible to have elements which behave like an identity element if put in second but not when put in first, e.g., 5-0=5 but 0-5=-5.
- It is only possible for there to be zero or one identity elements which behave like an identity element regardless of the order it is put into the binary operation.
- For examples of identity elements, see https://en.wikipedia.org/wiki/Identity_element#Examples
.
Source: “Identity element,” Wikipedia, retrieved 10/10/2020
[1] See “What is the Gist of a “Binary Operation”?“
[2] See “What is the Gist of a “Set” (Mathematics)?“
.
Disclaimer:
I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).
