Question:
A binary operation \( * \) is defined on \( \mathbb{R} \) by:
\[ a * b = \sqrt{a^2 + b^2} \]
Find the identity element.
Concept:
An identity element \( e \) satisfies:
\[ a * e = a \quad \text{for all } a \in \mathbb{R} \]
Solution:
Step 1: Apply definition
\[ a * e = \sqrt{a^2 + e^2} = a \]
Step 2: Solve equation
\[ \sqrt{a^2 + e^2} = a \]
Square both sides:
\[ a^2 + e^2 = a^2 \]
\[ e^2 = 0 \Rightarrow e = 0 \]
Step 3: Check carefully
\[ a * 0 = \sqrt{a^2} = |a| \]
This equals \( a \) only when \( a \geq 0 \), not for all real numbers.
—Final Conclusion:
There is no identity element for this operation on \( \mathbb{R} \), because \( a * 0 = |a| \neq a \) for negative values of \( a \).