07 - Identity Element is Idempotent

Formal statement:

\[ e \cdot e = e \]

Prompt:

“Why is doing nothing twice still doing nothing? Provide a natural-language explanation using ideas like pressing a reset button twice or staying still.”

“Why Doing Nothing Twice is Still Doing Nothing

Imagine you’re holding a reset button in your hand. When you press this reset button, everything returns exactly to how it was initially—it doesn’t change anything, it just reaffirms the original state.

Now, consider pressing this reset button a second time immediately after the first. Because the first press already returned everything to the original condition, pressing it again doesn’t produce any further change. The state of things stays exactly as it was after the first press—which was already identical to the original state.

In mathematics, the identity element is exactly like this reset button. When you combine anything with this identity element, it leaves that thing exactly as it was—it does nothing to it. Specifically, when you apply the identity element to itself, nothing changes. Doing nothing, and then doing nothing again, is still just doing nothing once.

Hence, the identity element is “idempotent,” meaning applying it multiple times is equivalent to applying it just once. Formally, this is expressed as:

[ e e = e ]

Simply put: if doing nothing doesn’t change anything, then doing nothing twice still changes nothing.