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Topic: Non-Euclidean Arithmetic
Replies: 108   Last Post: Sep 13, 2012 3:39 PM

 Messages: [ Previous | Next ]
 Paul A. Tanner III Posts: 5,920 Registered: 12/6/04
Re: Non-Euclidean Arithmetic
Posted: Sep 3, 2012 12:38 PM

He is pointing out that there is a problem with the usual way of

For example: The phrase "5 added to itself once" can be taken to mean
5 + 5. Likewise the phrases "5 added to itself twice and "5 added to
itself three times" respectively can be taken to mean 5 + 5 + 5 and 5
+ 5 + 5 + 5. And so on.

The usual way of putting putting forth 5 times 3 as repeated addition
is to say that it is 5 added to itself three times. But by the above,
this can be taken to mean that 5 times 3 is 5 + 5 + 5 + 5, which is
wrong.

That is, to say it right - to say it so that there is a lower
probability of such confusion - we need to make it so that what we say
implies that the number of times one of the factors appears as an
addend is equal to the other factor.

He did this by giving an alternative way of saying it, which would
result in 5 times 3 being equal to 0 + 5 + 5 + 5. The number of times
that 5 appears as an addend is equal to the other factor 3.

As I said before, I've given an alternative way of saying it many
times here at math-teach, which would be along the line of "x times y"
is "x instances of y added up" or "y instances of x added up". (Or we
could say something like "x times y" is "the sum of x instances of y"
or "the sum of y instances of x".) Or we could use some other term
than "instance" for this, as long as the language is clearer than the
usual way in terms of implying that the number of times one of the
factors appears as an addend is equal to the other factor.

When applied to this example above, saying that 5 times 3 is 5
instances of 3 added up or 3 instances of 5 added up would be saying
that 5 times 3 is equal to 3 + 3 + 3 + 3 + 3 or 5 + 5 + 5. The number
of times one of the factors appears as an addend is equal to the other
factor.

Like I said before, I prefer something along the line of this latter
approach because I prefer that the sum written out as a model for
multiplication have all the same addends. (With the former approach,
not all the addends are the same - one of the addends is 0 while all
the other addends are the same nonzero number.)

On Sun, Sep 2, 2012 at 12:17 PM, Robert Hansen <bob@rsccore.com> wrote:
> So, are you saying that 3 x 1 = 4?
>
> When a teacher says "added x number of times" they write the multiplicand x
> number of times, not the addition symbol. When I ask you what is the sum of
> 12, 34, 16 and 7 (4 addends) I am asking what is SUM(12,34,16,7) and the
> 7.
>
> There has to be a word for when people pivot on semantics like this and
> create problems that aren't even there. I have been calling them
> "semanticists" but that isn't accurate.
>
> The photocopy example is actually a solution to the problem "1 + x = 4", not
> "x * 1 = 4".
>
> Bob Hansen
>
>
> On Sep 1, 2012, at 1:58 AM, Jonathan Crabtree <sendtojonathan@yahoo.com.au>
> wrote:
>
> P.S. Think of a photocopying machine. You have one letter and you need four
> altogether. So what button do you press on the multiplication machine?
> Three! ie k-1 Press the four button and you end up with five.

Date Subject Author
9/1/12 Jonathan J. Crabtree
9/1/12 Paul A. Tanner III
9/2/12 kirby urner
9/3/12 Paul A. Tanner III
9/3/12 kirby urner
9/3/12 Paul A. Tanner III
9/4/12 kirby urner
9/4/12 Paul A. Tanner III
9/4/12 kirby urner
9/5/12 Paul A. Tanner III
9/5/12 Robert Hansen
9/6/12 kirby urner
9/1/12 kirby urner
9/1/12 Joe Niederberger
9/1/12 Wayne Bishop
9/1/12 Joe Niederberger
9/2/12 Robert Hansen
9/3/12 Paul A. Tanner III
9/3/12 Robert Hansen
9/5/12 Paul A. Tanner III
9/3/12 Joe Niederberger
9/3/12 Robert Hansen
9/5/12 Paul A. Tanner III
9/3/12 Joe Niederberger
9/3/12 Paul A. Tanner III
9/4/12 Joe Niederberger
9/5/12 Paul A. Tanner III
9/5/12 Joe Niederberger
9/5/12 Robert Hansen
9/5/12 Paul A. Tanner III
9/5/12 Joe Niederberger
9/5/12 kirby urner
9/5/12 Joe Niederberger
9/5/12 Robert Hansen
9/5/12 Joe Niederberger
9/6/12 Joe Niederberger
9/8/12 Robert Hansen
9/7/12 Jonathan J. Crabtree
9/8/12 kirby urner
9/8/12 Paul A. Tanner III
9/10/12 kirby urner
9/10/12 Paul A. Tanner III
9/10/12 kirby urner
9/10/12 Paul A. Tanner III
9/10/12 kirby urner
9/8/12 Robert Hansen
9/8/12 kirby urner
9/8/12 Robert Hansen
9/8/12 kirby urner
9/8/12 Joe Niederberger
9/8/12 Jonathan J. Crabtree
9/9/12 kirby urner
9/8/12 Clyde Greeno @ MALEI
9/8/12 Jonathan J. Crabtree
9/8/12 Jonathan J. Crabtree
9/8/12 Joe Niederberger
9/8/12 Joe Niederberger
9/9/12 Paul A. Tanner III
9/9/12 Robert Hansen
9/9/12 Paul A. Tanner III
9/9/12 Robert Hansen
9/9/12 Paul A. Tanner III
9/9/12 Robert Hansen
9/10/12 Paul A. Tanner III
9/10/12 Wayne Bishop
9/10/12 Paul A. Tanner III
9/9/12 Joe Niederberger
9/10/12 Clyde Greeno @ MALEI
9/9/12 Joe Niederberger
9/9/12 Paul A. Tanner III
9/9/12 Wayne Bishop
9/9/12 Paul A. Tanner III
9/10/12 Wayne Bishop
9/10/12 Paul A. Tanner III
9/9/12 Paul A. Tanner III
9/9/12 Joe Niederberger
9/10/12 Clyde Greeno @ MALEI
9/10/12 Joe Niederberger
9/10/12 Paul A. Tanner III
9/10/12 Joe Niederberger
9/10/12 Clyde Greeno @ MALEI
9/10/12 Joe Niederberger
9/11/12 Joe Niederberger
9/11/12 Paul A. Tanner III
9/11/12 kirby urner
9/11/12 Paul A. Tanner III
9/11/12 kirby urner
9/11/12 Paul A. Tanner III
9/11/12 kirby urner
9/12/12 Paul A. Tanner III
9/12/12 kirby urner
9/12/12 Paul A. Tanner III
9/12/12 kirby urner
9/13/12 Paul A. Tanner III
9/13/12 kirby urner
9/13/12 Paul A. Tanner III
9/11/12 Joe Niederberger
9/11/12 Joe Niederberger
9/11/12 kirby urner
9/11/12 Joe Niederberger
9/11/12 Joe Niederberger
9/11/12 Paul A. Tanner III
9/11/12 israeliteknight
9/11/12 Joe Niederberger
9/12/12 Paul A. Tanner III
9/12/12 kirby urner
9/12/12 Paul A. Tanner III
9/12/12 kirby urner
9/11/12 Jonathan J. Crabtree