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Jimmy Asked a Dangerous Question: What Is 0⁰? Sometimes the simplest questions cause the biggest mathematical headaches.
“Sir, what is zero to the power zero?”
The question came from Jimmy, who had a talent for asking awkward questions just after I had confidently finished explaining something.
On the board I had written two laws of indices.
and
“These are important rules,” I told the class. “You’ll use them often.”
Jimmy stared at the board for a moment.
Then his hand went up.
“If both rules are true,” he said carefully, “what is 0⁰?
The class fell silent.
Two Rules — One Contradiction
Jimmy’s difficulty is easy to see.
From the first rule, if we replace a with 0, we obtain
But the second rule says
If we replace n with 0, we obtain 0⁰ = 0.
So which answer is correct?
Jimmy frowned.
“Sir,” he said, “both answers can’t be right.”
Looking Near Zero
When mathematicians encounter situations like this, they often examine what happens…
