Q.If $*$ is defined on the set $R$ of all real numbers by $a * b = \sqrt{a^2 + b^2}$, find the identity element, if it exists, in $R$ with respect to $*$.

Concept: Identity element — an element $e \in R$ such that $a * e = a$ and $e * a = a$ for all $a \in R$.

Step 1: For an identity $e$, we require $a * e = a$ for all $a \in R$.

Using the definition:

$a * e = \sqrt{a^2 + e^2} = a$.

Step 2: Square both sides (valid since both sides are non-negative):

$a^2 + e^2 = a^2 \implies e^2 = 0 \implies e = 0$. …

Concept and Intuition

An identity element $e$ for a binary operation $*$ on a set is a special number that leaves every other number unchanged when combined with it. Formally, for all $a$ in the set, we need:

For our operation $a * b = \sqrt{a^2 + b^2}$, the operation is commutative (since $a^2 + b^2 = b^2 + a^2$), so we only need to check one condition — say $a * e = a$ for all real $a$.

The key question: can we find a single real number $e$ such that $\sqrt{a^2 + e^2} = a$ for every real $a$? That means the square root of $a^2 + e^2$ must equal $a$ itself. But a square root is always non-negative, so this can only work if $a$ itself is non-negative. That’s the first red flag.

Let’s work through it carefully.

Step-by-step reasoning

1. Set up the identity condition.

We need an $e \in \mathbb{R}$ such that for all $a \in \mathbb{R}$:

2. Square both sides (but carefully).

Squaring gives $a^2 + e^2 = a^2$, which simplifies to $e^2 = 0$, so $e = 0$.

So if an identity exists, it must be $e = 0$.

3. Test $e = 0$ for all $a$.

Compute $a * 0 = \sqrt{a^2 + 0^2} = \sqrt{a^2} = |a|$.

For $a \ge 0$, $|a| = a$, so $a * 0 = a$ works.

But for $a < 0$, $|a| = -a \neq a$. For example, $(-3) * 0 = \sqrt{9} = 3 \neq -3$. …

Method: Finding the Identity Element for a Given Operation

Use this when a specific operation is defined and you must find its identity element or show none exists.

Steps

Step 1: Write the defining equation for $e$

Impose $a*e=a$ using the rule. Here $\sqrt{a^2+e^2}=a$.

Step 2: Solve for the candidate

Manipulate to isolate $e$ (square both sides when both are non-negative): $a^2+e^2=a^2 \Rightarrow e=0$.

Step 3: Verify for all $a$, on both sides …

Common Mistakes

Mistake 1: Writing $\sqrt{a^2}=a$ instead of $|a|$

Why it's wrong: the square root returns the non-negative root, so $\sqrt{a^2}=|a|$, which equals $a$ only for $a\ge 0$. Correct approach: always simplify $\sqrt{a^2}$ to $|a|$ and check the sign.

Mistake 2: Stopping after finding the candidate $e=0$

Why it's wrong: solving $a*e=a$ only gives a candidate; you must verify it for every $a$, and here it fails for negatives. Correct approach: substitute the candidate back over the whole set before concluding. …