Once
upon a time there was a group that did math only with integers, and
another group that did it with rational numbers (that is, integers that
can be expressed as fractions.) The Integer Only group thought the
Rational Number group had gone down a slippery slope. They refused to
trust the answers the RN’s gave, because they held that the RN’s were
using illegitimate concepts. However, the RN's still used integers;
they just used them in new and different ways. They were able to
discover even more elegant and complex numbers, and do even more elegant
and complex equations.

Now
both groups were doing Math: it's just that the Integers Only folks
held that any numbers that weren’t integers were counterfeits and lies,
while the Rational Numbers crowd saw them as windows to even wider and
deeper realities. The IO's were afraid to allow anything besides whole
numbers and their corresponding negative numbers; the RN's were excited
to be able to express those same integers in many ways--even ultimately
as complex numbers. The RN’s tried to show the IO's that they didn't
need to be so worried: they could still have the number "4," even if it
could also be expressed as a complex number (4+ 0i). But the IO’s
refused to believe it, because if they accepted the reality of numbers
besides integers, they would lose their identity, and no longer have a
reason to exist.

So
you might say that the RN's were able to speak many Math languages:
Counting, Whole, Integer, Irrational, Real, Imaginary and Complex, while
the IO's spoke only three: Counting, Whole and Integer. When an RN
tried to speak Complex to the IO, it sounded like nonsense or worse, but
when the IO spoke to the RN, it often sounded simplistic and imprecise.
Now there were some equations that couldn’t be fully translated into
Counting, Whole or Integer. The IO’s said those equations were corrupt
and refused to let their children see them. Every once in a while, an IO
child would discover one and begin to wonder what it said, and the IO’s
would be greatly disturbed and forbid the child to ever encounter it
again. But some children, when they came of age, continued to wonder,
and even learned the forbidden math languages. It was those children who
eventually became the best mathematicians, because they were no longer
afraid of numbers.

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