Question:
Write the identity element for the binary operation \( * \) on the set \( \mathbb{R}_0 \) (non-zero real numbers) defined by:
\[ a * b = \frac{ab}{2}, \quad \forall a,b \in \mathbb{R}_0 \]
Concept:
An identity element \( e \) satisfies:
\[ a * e = a \quad \text{and} \quad e * a = a \]
Solution:
Step 1: Use definition
\[ a * e = \frac{a \cdot e}{2} = a \]
Step 2: Solve for \( e \)
\[ \frac{ae}{2} = a \]
Multiply both sides by 2:
\[ ae = 2a \]
Divide by \( a \) (since \( a \neq 0 \)):
\[ e = 2 \]
Step 3: Verify
\[ a * 2 = \frac{a \cdot 2}{2} = a \]
Hence, verified.
Final Answer:
\[ \boxed{2} \]