To find the identity element for the binary operation $*$ defined on the set $\mathbb{R}$ of real numbers by $m * n = m + n + 2$, we need to find an element $e \in \mathbb{R}$ such that for any $m \in \mathbb{R}$:

$m * e = m \quad \text{and} \quad e * m = m$

Given the operation:

$m * e = m + e + 2$

We set this equal to $m$ to find $e$:

$m + e + 2 = m$

Subtract $m$ from both sides:

$e + 2 = 0$

Solving for $e$:

$e = -2$

Thus, the identity element $e$ is $-2$.

Would you like more details or have any questions?

Here are 8 related questions that expand on this topic:

1. What is the inverse element of a given number under this operation?
2. Can you prove that the operation is associative?
3. What would happen if the operation was defined as $m * n = m + n + c$ with a different constant $c$?
4. How does this binary operation differ from the standard addition operation on real numbers?
5. Can you find the identity element if the operation was $m * n = m + n - 2$?
6. Is the operation commutative, and how would you prove it?
7. What is the result of $3 * 5$ under this operation?
8. How would you define a similar operation on complex numbers?

Tip: When working with binary operations, always check for properties like associativity, commutativity, and the existence of identity and inverse elements.