I was getting my seven year old to do sums in his head. He counted and did several small sums in base 10, then base 5, 4, 3. He reached base 2. Zero, one, ten, eleven, a hundred. He had the hang of it.
And then he pushed down to base 1 despite there being no such thing.
Counting: Zero, zero zero, zero zero zero. Then he remembered taking off leading zeros so he tried zero, zero, zero, zero, zero.
He did all the adds, take aways and timeses the base could muster.
Zero plus zero is zero.
Zero less zero is zero.
Zero zeroes are zero.
I wasn't convinced by his zero divided by zero is zero but couldn't think of an alternative. And at that point he decided there were no other sums anywhere in the base 1 system so he's not heard of exponentiation.
Base 1 according to Wikipedia is unary but that's not the same. I failed to find it discussed by searching. Has he fully covered the topic?