[GW-C] Re:numbers [was Re: understanding probability]

[Lennart Sorensen]

On Fri, Aug 09, 2013 at 10:36:09PM -0400, D. Hugh Redelmeier wrote:
> For example, you can do arithmetic with matrices: +, -, *, / are
> defined on them.  Are these numbers?  (The arithmethic isn't exactly
> the same as in natural numbers, so it is up to the observer to decide
> if it counts as real arithmetic to him or her.)

If the matrices have contents you can do arithmetic on, then you can do it
on the matrices too.  I am not sure you can multiply two matrices that are
full of letters without first defining what it means to multiply letters.

> IEEE 754 floating point includes infinities.  I was surprised to be
> reminded recently that its infinities are not NaNs (non-a-number).

Their infinities are still invalid to do calculations with.  You can
only really use them to compare against (not that that makes that much
sense either).

> Any system with infinities loses some properties that we assumed were
> universal.  That usually happens whenever you generalize a numeric
> system.  Classic example: complex numbers cannot be ordered in a
> useful way.

Certainly seems that way.  I guess you could order them by size, but
that still isn't really a useful way to order them.

Of course people assume things that are wrong too in many cases.
It's interesting to see all the ways of proving 0.999... = 1 and how
many people refuse to accept that, while having no issue with 0.333... =
1/3 and 3 * 1/3 = 1, yet 3 * 0.333... = 0.999... doesn't make them accept
that it is the same thing.  Intuition is often wrong.

> One simple version of infinity: just one infinity; any operation on
> infinity yields infinity.  What do we lose: lots of things, for
> example:
> 	x + 1 != x
> Infinity to the power infinity would then be infinity.

Not very useful,  Seems simpler to just not allow doing operations on it.
It's a great limit, but not a great number.

> "They often consist of numbers" is a pretty strong statement.  Do you
> mean "their elements can be denoted by numerals"?  That's more
> conservative.  Essentially, your original statement seems to imply
> that those groups are numbers.

I am sure someone has defined what it should mean.  Most groups I have
seen have elements that are numbers.

-- 
Len Sorensen
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