Doug Miller wrote :
FromTheRafters wrote in news:nikq90$udm$1
@news.albasani.net:

The natural number one with an exponent of two can equal two under the
right circumstances, but I wouldn't expect a seventh grader to know
this or that adding *all* natural numbers together gives -1/12.

The set of natural numbers is the infinite set {1, 2, 3, 4, ...} and its sum
is likewise infinite.

Sure, doing it *that* way. Of course you would never get there to *all*
natural numbers since it is an infinite process. It doesn't make sense
in arithmetic to us seventh gradeers now just like the previous
division by zero being undefined wouldn't make sense to a seventh
grader. That was my point. however with some redefining of things into
a different system of which seventh graders are not aware you can get
different results. Results that work in that system.

And under no circumstances at all is one to the second power equal to two.

Perhaps you should wait to post again until after you've sobered up. ;-)

In an additive group in the integers (which the naturals are a subset
of) exponentiation is the repeated application of the group operator
just as it is in multiplicative groups. It just so happens that
repeated addition is equivalent to multiplication and we aren't used to
thinking of it as exponentiation. I said exponent and right
circumstances. I didn't say second power, because that would have been
misleading.