The most basic question possible
Tau
most simple idea.
You know how integers get greater and greater as you go through the number
scale? Well how do you speak about negative numbers? Someone told me that it
is considered that -1 is a greater value than -2 and greater still than -3.
But because 3 is a greater value than 2 and 2 greater than 1, I have written
that the sequence -1, -2, -3 is one of an increasing negative value. Is that
wrong? If so, how should I have phrased it? If it's that it's said to be a
decreasing value, I certainly don't want to give the reader of the piece I
am writing any kind of impression that this sequence progresses towards an
increasingly small value if it might be construed that increasingly small
means increasingly negligible (i.e. negligibly different from zero) - quite
the opposite.
Thanks in advance,
Tau
David W. Cantrell
> Hi, not being a mathematician I am having real difficulty expressing the
> most simple idea.
> You know how integers get greater and greater as you go through the
> number scale? Well how do you speak about negative numbers? Someone told
> me that it is considered that -1 is a greater value than -2 and greater
> still than -3. But because 3 is a greater value than 2 and 2 greater than
> 1, I have written that the sequence -1, -2, -3 is one of an increasing
> negative value. Is that wrong?
It could, at least, lead to confusion.
> If so, how should I have phrased it?
Maybe something like:
The sequence -1, -2, -3,... is one of negative numbers which increase in
absolute value.
David
Albert Y. C. Lai
> You know how integers get greater and greater as you go through the number
> scale? Well how do you speak about negative numbers? Someone told me that it
> is considered that -1 is a greater value than -2 and greater still than -3.
> But because 3 is a greater value than 2 and 2 greater than 1, I have written
> that the sequence -1, -2, -3 is one of an increasing negative value. Is that
> wrong?
Cities X and Y are at temperatures -5 degrees and -10 degrees,
respectively. Who is at a "higher" temperature? Who is in "stronger"
freeze? Does it mean "higher" is the opposite of "stronger"?
Persons X and Y have net worths -$5K and -$10K, respectively. (Yes,
they're both broke.) Who has "more" worth? Who is in "greater" debt?
Does it mean "more" is the opposite of "greater"?
Mathematics is a neutral tool. Perspectives, beliefs, and languages
provide interpretations. Some of them are mutually conflicting
interpretations. Some of them are just mincing words.
Interpretations are necessary, but beware of premature interpretations.
Don't say "greater" before you have thought through "greater what".
Tau
news:f2sna...@news4.newsguy.com...
It's all down to conventions - they continue to ascribe meanings to words
while not ascribing meanings to words, if you know what I mean....
Stan Brown
<nocorres...@yahoo.co.uk>:
> But because 3 is a greater value than 2 and 2 greater than 1, I have written
> that the sequence -1, -2, -3 is one of an increasing negative value. Is that
> wrong? If so, how should I have phrased it? If it's that it's said to be a
> decreasing value
The values are decreasing, because -1 is less than -1 a you said.
They are also increasingly negative (not, IMHO, increasing negative:
you want an adverb here not a second adjective.
--
Stan Brown, Oak Road Systems, Tompkins County, New York, USA
http://OakRoadSystems.com/
Darrell
news:lnh4i.337$y45...@newsfe6-win.ntli.net...
<...>
> Someone told me that it is considered that -1 is a greater value than -2
> and greater still than -3.
Correct.
> But because 3 is a greater value than 2 and 2 greater than 1, I have
> written that the sequence -1, -2, -3 is one of an increasing negative
> value. Is that wrong?
My opinion is yes, that's wrong. To say "increasing negative value" to me,
at least, implies two things: The sequence is increasing, and the values
are negative. Even if "increasing" is understood to be an adverb of
"negative" it's still incorrect, because these "negative values" are
decreasing not increasing.
Another person may reasonably interpret it as increasing in the "negative"
direction, which is really just another way of saying "decreasing." Not
unlike saying subtraction is the addition of the opposite. True, but we
usually use the shorthand "subtract" when we need to "add the opposite."
Technically, the sequence is not increasing but decreasing (an important
concept in and of itself, so I would get it understood now and early.) If I
were to rephrase your description, and stay as close to your description as
possible, I would say "The sequence -1, -2, -3, ... is one of decreasing
negative values. This is true because a)the sequence is decreasing, and
b)the values are negative. It even passes the adverb test.
But that's not my real point. My point is, the integers have a property
called "order." In essence, this means we can relate any pair of integers
with the operator >= or =< even negative integers. When we speak of order
among the negatives it can be nonintuitive to a beginner, who may have a
mindset of "How can one number less than 0 be greater than another number
also less than 0? ...implying he thinks "greater" to mean at the very least
more than zero.
In the spirit of your subject line (the most basic question) its simply an
axiom, i.e. we construct the integers such that we have "order" e.g. for any
two integers a,b either a>=b or a=<b. At the early stage when this is
introduced, it's often explained on a number line. The greater number is
always rightward of the lesser number:
<---(-3)---(-2)---(-1)---(0)---(1)---(2)---(3)--->
> If so, how should I have phrased it? If it's that it's said to be a
> decreasing value, I certainly don't want to give the reader of the piece I
> am writing any kind of impression that this sequence progresses towards an
> increasingly small value if it might be construed that increasingly small
> means increasingly negligible (i.e. negligibly different from zero) -
> quite the opposite.
Just ensure the reader understands the integers are unbounded, i.e. there is
no greatest or least integer. IOW, make sure that when you say things like
"increasing" and "decreasing" it does not imply the number approaches zero,
which is something entirely different. These are terms that have specific
meaning in the context of sequences, series, functions, etc.
--
Darrell
[Mr.] Lynn Kurtz
<the_sta...@fastmail.fm> wrote:
>Mon, 21 May 2007 13:44:49 GMT from Tau
><nocorres...@yahoo.co.uk>:
>> But because 3 is a greater value than 2 and 2 greater than 1, I have written
>> that the sequence -1, -2, -3 is one of an increasing negative value. Is that
>> wrong? If so, how should I have phrased it? If it's that it's said to be a
>> decreasing value
>
>The values are decreasing, because -1 is less than -1 a you said.
>They are also increasingly negative (not, IMHO, increasing negative:
>you want an adverb here not a second adjective.
Is that like a woman being increasingly pregnant?
--Lynn
Tau
"Darrell" <dr6...@comcastsnip.net> wrote in message
news:p_SdnT0zQM02x8_b...@giganews.com...
The issue I have is that I am speaking of two effects either side of a mean
(which I represent as a zero value), and I refer to the one cancelling out
the other; so I am speaking of the negative effect getting bigger as the
negative numbers get further from zero. So I'm a bit put off using the word
'decreasing'.
Darrell
>
> "Darrell" <dr6...@comcastsnip.net> wrote in message
> news:p_SdnT0zQM02x8_b...@giganews.com...
>> "Tau" <nocorres...@yahoo.co.uk> wrote in message
>> news:lnh4i.337$y45...@newsfe6-win.ntli.net...
> The issue I have is that I am speaking of two effects either side of a
> mean (which I represent as a zero value), and I refer to the one
> cancelling out the other; so I am speaking of the negative effect getting
> bigger as the negative numbers get further from zero. So I'm a bit put off
> using the word 'decreasing'.
>
What you now describe can be modeled by ...-3,-2,-1,0,1,2,3... Additive
inverse property: For any integer n there exists -n such that n+(-n)=0.
It's a simple matter of reading the numbers left to right. If they get
larger, the sequence is increasing, if they get smaller the sequence is
decreasing. Notice each number "cancels" out the corresponding number of
the different sign. IOW, (-3)+3=0, (-2)+2=0, etc.
_This_ sequence is increasing, not the one you posted.
Don't be afraid to call a decreasing sequence by its name when you need to.
Informal adverbs like "increasingly" and "decreasingly' wrt the adjective
negative or positive are misleading. Negative is negative, not just a
little negative or a great deal negative (think pregnancy, one either is or
is not regardless of trimester.) Wrt a sequence of integers, negative or
positive, or both, your original sequence -1,-2,-3,... you seem to want to
say its increasing(ly) negative when actually its decreasing
--
Darrell.
Gary S. Simon
"Tau" <nocorres...@yahoo.co.uk> wrote:
> The issue I have is that I am speaking of two effects either side of a mean
> (which I represent as a zero value), and I refer to the one cancelling out
> the other; so I am speaking of the negative effect getting bigger as the
> negative numbers get further from zero. So I'm a bit put off using the word
> 'decreasing'.
If it can be presumed that your readers will understand the terms
"positive effect" and "negative effect" to mean what you intend them to
mean, then there should be no problem with stating that the effects
"increase".
Tau
"Gary S. Simon" <gars...@thisisatypo.pipeline.com> wrote in message
news:garscosi-F6AD8E...@news.west.earthlink.net...
Ah this leads to further problems, as I have spoken of the value, _d_, being
'negatively affected' when its positive value has any amount subtracted from
it, even such as to render the value disproportionately in the negative
numbers. In the context you describe, I think the wording might convey that
a positive effect is an increasing absolute difference from zero, which it
needn't be.
My readers will be trained mathematicians.
Darrell
> My readers will be trained mathematicians.
Then they should have no problem understanding your sequence -1,-2,-3,... is
decreasing, and the one I proposed ...-3,-2,-1,0,1,2,3,... is increasing.
--
Darrell
Tau
Yeh but as I say my problem is more the concern of a mix of messages. I am
not actually going to write about a 'sequence' in such terms; rather, I'm
going to make sporadic references to what such-and-and-such implies. The
essay centres, unwaveringly, around the issue of a deviation (sorry, I guess
I use that term inadvisedly, as I know it has a specific connotation in
mathematics) of a specific value from a mean value. So I have a bit of a
jumble of contexts in which I use 'increase' and 'decrease' which ideally I
want to iron out. Thanks to guys for all your advice - it's really helpful.
Darrell
Fine and dandy, but totally different from your original question, which
concerned the specific sequence -1, -2, -3, ... and whether it's OK to say
it's "increasing." The answer to _that_ question remains no, since it's
decreasing. The answer to what you are now suggesting (funny how what you
suggest seems to change with each post) is anyone's guess.
Hopefully you at least learned the sequence is decreasing, not increasing,
and why.
--
Darrell