Square Roots: A Usage Survey
David W. Cantrell
roots. In the spirit of an answer to a FAQ, I have written the
following usage survey. Suggestions for improvements will be
appreciated.
David Cantrell
______________________
SQUARE ROOTS: A USAGE SURVEY
I. What is a square root?
As most often defined, a _square_root_ of a number x is any
number y such that
y^2 = x. For example, according to this definition, since both
(+3)^2 and (-3)^2 equal 9, both +3 and -3 are square roots of 9.
In general, a positive number has both a unique positive square
root and a unique negative square root. Then, by convention, the
_principal_ square root of a nonnegative number is taken to be
its nonnegative square root. For example, the principal square
root of 9 is +3.
It should be noted that some mathematicians apparently do not
use the general definition of square root as given above.
Instead, for them, in the context of real analysis, the
phrase "square root" itself indicates specifically a nonnegative
value (and thus it would be superfluous to modify "square root"
by "principal"). According to this alternative definition, +3
would be the only square root of 9. This alternative definition
is relatively uncommon, and so we will hereafter use the
definition of square root as originally stated (according to
which both +3 and -3 are square roots of 9).
II. How are square roots denoted?
There are three different ways in which square roots may be
indicated using mathematical symbols:
(1) using the common radical sign, which will be approximated
here by typing "V",
(2) using an exponent of 1/2, and
(3) using some abbreviation of "square root".
Examples of such notations are V9, 9^(1/2), and sqrt(9),
respectively.
Quite unfortunately, there are differing usages, none of which
are truly uncommon, according to which each of the examples
above can either represent only +3 or else represent either +3
or -3. [The latter case will be shown henceforth as "(+ or -)
3" since the compound symbol, having "+" above and "-" below,
cannot be typed here.]
In the most common usage, Vx and x^(1/2) both denote
specifically the principal square root, so that V9 = 9^(1/2) =
+3, for example. The principal-valued square-root function may
then be denoted either as Vx or x^(1/2).
According to another usage, Vx and x^(1/2) each represent the
square roots (plural!) of x, so that V9 = 9^(1/2) = (+ or -)3.
The multivalued square-root relation may then be denoted either
as Vx or x^(1/2). [Reference: The HarperCollins Dictionary of
Mathematics (having the well-known mathematician Jonathan
Borwein as one of its two authors) states, under the entry for
square root, "a number or quantity that when multiplied by
itself is equal to a given number or quantity, usually written
Vx in arithmetic expressions, and x^(1/2) in algebraic
expressions", and under the entry for principal value, "the
principal value of V4 is 2 although -2 is also a root."]
According to yet other usages, one of the expressions, either Vx
or x^(1/2), denotes the principal-valued square-root function
while the other denotes the multivalued relation. [Reference: An
Atlas of Functions, Spanier and Oldham, states "The symbols x^
(1/2) and Vx are often interpreted as defining equivalent
functions but in this Atlas we make a distinction between the
two. If x is positive x^(1/2) has two values, one positive and
one negative. The square-root function Vx, however, is single
valued and equal to the positive of the two x^(1/2) values."]
The abbreviation "SQRT" is used most often in computer
arithmetic, in which context it should always be interpreted as
representing the principal square root. However, outside that
context, similar abbreviations could be interpreted in different
ways. There is a convention according to which, if a multivalued
relation is denoted in lower case, then capitalization is used
to distinguish the corresponding principal-valued function.
[Referring again to the entry for principal value in the
HarperCollins Dictionary of Mathematics, "The function that has
as its values the principal values of a many-valued function is
conventionally indicated by writing it with a capital letter;
thus, Cotan^(-I) is the principal value of the inverse
cotangent, and, especially in complex functions, Ln is the
principal value of the natural logarithm."] Using this
convention, if the multivalued square-root relation were
denoted "sqrt", then the principal-valued square-root function
would be denoted "Sqrt". In this case, sqrt(9) = (+ or -)3 and
Sqrt(9) = +3. Unfortunately, another convention is the precise
reverse of that described above! [For example, in An Atlas of
Functions, "arcsin" denotes the inverse sine function
while "Arcsin" denotes the multivalued relation.] And yet
another practice is to make no notational distinction whatsoever
between a multivalued relation and the corresponding principal-
valued function! [See, for example, the treatment of inverse
trigonometric functions in the CRC Standard Mathematical Tables
and Formulae, 30th ed.] To summarize, outside of computer
arithmetic, abbreviations of "square root" can be used in
virtually all conceivable ways.
III. Don't these different alternatives lead to confusion?
They certainly can lead to confusion among those without much
experience in mathematics. This regrettable situation could be
eliminated, in theory at least, if mathematicians would come to
an agreement on a single standard usage. But this seems
unlikely. Why? Perhaps in part, because some people would not
want to abandon their preferred usage for another. But more
importantly, because professional mathematicians are seldom (if
ever) confused by the different alternatives. They are sensitive
to context, and so, in a given situation, quickly determine the
appropriate interpretation of a usage, even if it is different
from their own preference. Thus, from the professional
mathematicians' standpoint, establishing a uniform usage is
hardly a pressing concern.
IV. Recommended Usage
To indicate the multivalued square-root relation, write either
(+ or -)Vx or
(+ or -)x^(1/2). Not only is this common practice, but it also
has the advantage that it cannot reasonably be misunderstood!
[At worst, someone might claim that (+ or -) is redundant.]
To indicate the principal-valued square-root function, simply
write Vx or x^(1/2); to indicate the other (i.e., nonpositive)
square-root function, write -Vx or -x^(1/2). This recommendation
is based on the fact that this usage is clearly the most common.
5^(1/2) can be read, most obviously, as "five to the power one-
half", for example. V5 can be read properly as "the principal
square root of 5", or similarly with "positive" or "nonnegative"
instead of "principal". [However, to be technically precise, we
should not read V5 simply as "the square root of 5" if we are
using the most common definition of "square root", according to
which a positive number has two square roots. Nonetheless,
technically precise or not, "the square root" of a positive
number is often heard and should always be taken to indicate the
principal square root.]
In computer arithmetic, SQRT always denotes the principal square
root. Outside that context, abbreviations such as sqrt and Sqrt
should ideally either be avoided or else explained by the writer.
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Owen Holden
For the natural numbers: sqrt(x)=(the y:y^2=x), eg. sqrt(4)=2
In the integers: +sqrt(x)=(the y:y^2=x & y>0), eg. +sqrt(+4)=+2
and -sqrt(x)=(the y:y^2=x & y<0), eg. -sqrt(+4)=-2
sqrt(x), (the y:y^2=x), does not exist in the integers. It is not unique.
If x>0 then y has no integer value unless y^2 is an integer, but then it has
two solutions.
If x<0 then y has no integer value. In no instance does 'the' y exist.
A square root of x, (a y:y^2=x) does exist, ie. (some y)(y^2=x)
eg. +2=(a y:y^2=+4) ie.(+2)^2=+4 and -2=(a y:y^2=+4), ie.(-2)^2=+4
For example, in the integers:
sqrt(x-9)+6=2, has no solution.
+sqrt(x-9)+6=2, has no solution.
-sqrt(x-9)+6=2, has 'the' solution x=25.
-4+6=2
(a sqrt)(x-9)+6=2, is true.
Owen
David W. Cantrell <dwcantrel...@aol.com.invalid> wrote in message
news:006d6b28...@usw-ex0102-084.remarq.com...
David W. Cantrell
Holden" <oho...@idirect.com> wrote:
>IMO,
>For the natural numbers: sqrt(x)=(the y:y^2=x), eg. sqrt(4)=2
>
>In the integers: +sqrt(x)=(the y:y^2=x & y>0), eg. +sqrt(+4)=+2
>and -sqrt(x)=(the y:y^2=x & y<0), eg. -sqrt(+4)=-2
Well, that's certainly the most common usage, as I said.
>sqrt(x), (the y:y^2=x), does not exist in the integers. It is
>not unique.
The way you have defined sqrt(x) above does not fit any current
usage (of anyone else)! Thus, I do not take it seriously.
>If x>0 then y has no integer value unless y^2 is an integer,
>but then it has two solutions.
I suppose I know what you meant to say.
>If x<0 then y has no integer value. In no instance does 'the' y
>exist.
Last time I checked, zero was considered to be an integer. That
sounds like an instance.
>A square root of x, (a y:y^2=x) does exist, ie. (some y)(y^2=x)
>eg. +2=(a y:y^2=+4) ie.(+2)^2=+4 and -2=(a y:y^2=+4),
>ie.(-2)^2=+4
Yes.
>For example, in the integers:
>sqrt(x-9)+6=2, has no solution.
If you are imploying your own special definition of sqrt(x-9),
by which it does not exist for any x other than 9, then who
cares?
>+sqrt(x-9)+6=2, has no solution.
>-sqrt(x-9)+6=2, has 'the' solution x=25.
> -4+6=2
Yes. These statements are correct according to a commonly
accepted usage.
>(a sqrt)(x-9)+6=2, is true.
Sure.
David Cantrell
Neil W Rickert
>Confusion often arises in sci.math about usage related to square
>roots. In the spirit of an answer to a FAQ, I have written the
>following usage survey. Suggestions for improvements will be
>appreciated.
> David Cantrell
>______________________
>SQUARE ROOTS: A USAGE SURVEY
>I. What is a square root?
>As most often defined, a _square_root_ of a number x is any
>number y such that
>y^2 = x. For example, according to this definition, since both
>(+3)^2 and (-3)^2 equal 9, both +3 and -3 are square roots of 9.
>In general, a positive number has both a unique positive square
>root and a unique negative square root. Then, by convention, the
>_principal_ square root of a nonnegative number is taken to be
>its nonnegative square root. For example, the principal square
>root of 9 is +3.
So far, so good.
>It should be noted that some mathematicians apparently do not
>use the general definition of square root as given above.
>Instead, for them, in the context of real analysis, the
>phrase "square root" itself indicates specifically a nonnegative
>value (and thus it would be superfluous to modify "square root"
>by "principal"). According to this alternative definition, +3
>would be the only square root of 9. This alternative definition
>is relatively uncommon, and so we will hereafter use the
>definition of square root as originally stated (according to
>which both +3 and -3 are square roots of 9).
I have to disagree with this one. The problem is that your "general
definition of square root" is not a general definition of square
root. What you defined was "_a_ square root". But you have failed
to define "_the_ square root", which is almost universally taken to
be the principal value (the non-negative square root). There is an
important need to have _the_ square root, so that we can use
functional notation and so that we can write down unambiguous
mathematical expressions.
David W. Cantrell
<ricke...@cs.niu.edu> wrote:
>I have to disagree with this one. The problem is that
>your "general definition of square root" is not a general
>definition of square root. What you defined was "_a_ square
>root". But you have failed to define "_the_ square root",
>which is almost universally taken to be the principal value
>(the non-negative square root). There is an
>important need to have _the_ square root, so that we can use
>functional notation and so that we can write down unambiguous
>mathematical expressions.
Did you read my _entire_ post? If not, we have nothing to
discuss as yet.
Regards,
David Cantrell
Hugo van der Sanden
> 5^(1/2) can be read, most obviously, as "five to the power one-
> half", for example. V5 can be read properly as "the principal
> square root of 5", or similarly with "positive" or "nonnegative"
> instead of "principal".
I assume this is the primary justification for what I am used to, that
V5 represents the principal square root while 5^(1/2) represents both.
I think it is a problem here on sci.math more often than elsewhere,
since an initial question often involves ambiguous use without
sufficient context to guess the intent. Which takes us back to the
recent thread about the importance of clarifying definitions.
Hugo
Neil W Rickert
>In article <8dsft6$j...@ux.cs.niu.edu>, Neil W Rickert
><ricke...@cs.niu.edu> wrote:
>>I have to disagree with this one. The problem is that
>>your "general definition of square root" is not a general
>>definition of square root. What you defined was "_a_ square
>>root". But you have failed to define "_the_ square root",
>>which is almost universally taken to be the principal value
>>(the non-negative square root). There is an
>>important need to have _the_ square root, so that we can use
>>functional notation and so that we can write down unambiguous
>>mathematical expressions.
>Did you read my _entire_ post? If not, we have nothing to
>discuss as yet.
Clearly we have nothing to discuss. You are dishonest. You quote my
paragraph with out the context (the paragraph of yours to which it
responded. Then you uncharitably and falsely accuse me of not having
read your whole article.
Doug Norris
>Then you uncharitably and falsely accuse me of not having
>read your whole article.
Well, *did* you? David defines the things you're complaining about later
in the same article.
Doug
David W. Cantrell
writes:
>David W. Cantrell <dwcantrel...@aol.com.invalid> writes:
>
>>In article <8dsft6$j...@ux.cs.niu.edu>, Neil W Rickert
>><ricke...@cs.niu.edu> wrote:
>
>>>I have to disagree with this one. The problem is that
>>>your "general definition of square root" is not a general
>>>definition of square root. What you defined was "_a_ square
>>>root". But you have failed to define "_the_ square root",
>>>which is almost universally taken to be the principal value
>>>(the non-negative square root). There is an
>>>important need to have _the_ square root, so that we can use
>>>functional notation and so that we can write down unambiguous
>>>mathematical expressions.
>
>>Did you read my _entire_ post? If not, we have nothing to
>>discuss as yet.
>
>Clearly we have nothing to discuss. You are dishonest.
I don't understand how you justify such a strong accusation. Rest assured that
I was not attempting to deceive in any respect.
>You quote my paragraph with out the context (the paragraph of yours to which
it
>responded.
Yes. And you quoted my opening paragraphs out of the context of my entire
article. Your having done so would have been perfectly understandable to me,
except for the fact that the point you wish to address was discussed later in
my article, in a section which you did not quote. (Of course, that is why I
thought that _perhaps_ you had not read the entire article.)
>Then you uncharitably and falsely accuse me of not having read your whole
article.
No, I most certainly did not. I am no telepathist. I had no way of knowing
whether you had read the entire article or not. That is why I _asked_ whether
you had read it in its entirety! It was not a rhetorical question. And that is
why I went on to say "If not,..." Had I thought with assurance (and apparently,
falsely) that you had not read the entire article, I would have bluntly said
"Read the entire article! The point which you have raised is discussed in a
later section."
Since apparently you did read the entire article, perhaps I did a poor job of
addressing the point which you've raised. I'll now go back to your original
post (snipping nothing!) and do my best to address your objection.
David Cantrell
David W. Cantrell
<ricke...@cs.niu.edu> wrote:
>David W. Cantrell <dwcantrel...@aol.com.invalid> writes:
>
>>square roots. In the spirit of an answer to a FAQ, I have
>>written the following usage survey. Suggestions for
>>improvements will be appreciated.
>
>> David Cantrell
>>______________________
>
>>SQUARE ROOTS: A USAGE SURVEY
>
>>I. What is a square root?
>
>>As most often defined, a _square_root_ of a number x is any
>>number y such that y^2 = x. For example, according to this
>>definition, since both (+3)^2 and (-3)^2 equal 9, both +3 and
>>-3 are square roots of 9.
>>In general, a positive number has both a unique positive square
>>root and a unique negative square root. Then, by convention,
>>the _principal_ square root of a nonnegative number is taken
>>to be its nonnegative square root. For example, the principal
>>square root of 9 is +3.
>
>
>>It should be noted that some mathematicians apparently do not
>>use the general definition of square root as given above.
>>Instead, for them, in the context of real analysis, the
>>phrase "square root" itself indicates specifically a
>>nonnegative
>>value (and thus it would be superfluous to modify "square root"
>>by "principal"). According to this alternative definition, +3
>>would be the only square root of 9. This alternative definition
>>is relatively uncommon, and so we will hereafter use the
>>definition of square root as originally stated (according to
>>which both +3 and -3 are square roots of 9).
>
>your "general definition of square root" is not a general
>definition of square root.
I don't understand why you say that it is "not a general
definition".
>What you defined was "_a_ square root".
Yes. But I also defined "the _principal_ square root".
>But you have failed to define "_the_ square root",
And for very good reason! Later in my article, in section IV.
Recommended Usage, I said:
To indicate the principal-valued square-root function, simply
write Vx or x^(1/2); to indicate the other (i.e., nonpositive)
square-root function, write -Vx or -x^(1/2). This recommendation
is based on the fact that this usage is clearly the most common.
5^(1/2) can be read, most obviously, as "five to the power one-
half", for example. V5 can be read properly as "the principal
square root of 5", or similarly with "positive" or "nonnegative"
instead of "principal". [However, to be technically precise, we
should not read V5 simply as "the square root of 5" if we are
using the most common definition of "square root", according to
which a positive number has two square roots. Nonetheless,
technically precise or not, "the square root" of a positive
number is often heard and should always be taken to indicate the
principal square root.]
(End quotation from article.)
I thought that this comment would have addressed your concern,
Neil. Why does it not do so?
>which is almost universally taken to be the principal value
>(the non-negative square root). There is an
>important need to have _the_ square root, so that we can use
>functional notation and so that we can write down unambiguous
>mathematical expressions.
Your "_the_ square root" is the _principal_ square root.
David Cantrell
Neil W Rickert
>>> David Cantrell
>>>______________________
>>So far, so good.
Your subject line implied that you were discussing usage. In
ordinary usage, mathematicians rarely talk about "the _principal_
square root." They often talk about "the square root," intending
that to refer to the principal value. And it is not just "some
mathematicians" but most mathematicians who use this terminology.
>>But you have failed to define "_the_ square root",
>And for very good reason! Later in my article, in section IV.
>Recommended Usage, I said:
>To indicate the principal-valued square-root function, simply
>write Vx or x^(1/2); to indicate the other (i.e., nonpositive)
>square-root function, write -Vx or -x^(1/2). This recommendation
>is based on the fact that this usage is clearly the most common.
>5^(1/2) can be read, most obviously, as "five to the power one-
>half", for example. V5 can be read properly as "the principal
>square root of 5", or similarly with "positive" or "nonnegative"
>instead of "principal". [However, to be technically precise, we
>should not read V5 simply as "the square root of 5" if we are
>using the most common definition of "square root", according to
>which a positive number has two square roots. Nonetheless,
>technically precise or not, "the square root" of a positive
>number is often heard and should always be taken to indicate the
>principal square root.]
>(End quotation from article.)
>I thought that this comment would have addressed your concern,
>Neil. Why does it not do so?
Perhaps I made the mistake of assuming that your "usage survey" was a
survey of usage, and that your "recommended usage" was your
recommendation for your preferred usage. Many (most?) mathematicians
believe they are being technically precise using "the square root of
5" to mean only the positive square root.
>Your "_the_ square root" is the _principal_ square root.
Are you talking about usage, or about your particular preferred
definitions?
David W. Cantrell
Indeed. And I would suggest that it would be difficult to find
another single source which discusses all the variations in
usage which I did.
>In ordinary usage, mathematicians rarely talk about "the
>_principal_ square root." They often talk about "the square
>root," intending that to refer to the principal value. And it
>is not just "some mathematicians" but most mathematicians who
>use this terminology.
I agree completely with your statements above.
Yes. That was certainly my intent.
>Many (most?) mathematicians believe they are being technically
>precise using "the square root of 5" to mean only the positive
>square root.
Most mathematicians also hold that 5 has two square roots, one
positive and one negative.
>>Your "_the_ square root" is the _principal_ square root.
>
>Are you talking about usage, or about your particular preferred
>definitions?
Huh? Surely you agree that in common usage "_the_ square root"
refers to the _principal_ square root.
Did I not say "Nonetheless, technically precise or not, 'the
square root' of a positive number is often heard and should
always be taken to indicate the principal square root."? Do you
disagree with this?
Hans Steih
<dwcantrel...@aol.com.invalid> schreibt:
>
>Most mathematicians also hold that 5 has two square roots, one
>positive and one negative.
>
>
1) "Most mathematicians"???? - How do you know?
Have you any references? A bibliography?
The mathematicians _I know_ define sqrt(5) as the _non-negative_ number which
square is 5.
By this it follows that the _equation_ x^2 = 5 has two roots:
{x el IR| x^2 = 5} ={ - sqrt(5) ; sqrt(5) }
And there are no problems with this definition!
2) But _your_ definition (and as a former poster pointed out: it's a problem of
definition, it's not a problem of "usage"!) has the consistency of a
plum-pudding:
A) I say that the equation x^2 - 2x - 5 = 0 has two roots: 1 + sqrt(5) and
1-sqrt(5), and I know that 1-sqrt(5) is a negative number, that 1+sqrt(5) is a
positive one.
And I can write: 1-sqrt(5) < 1 + sqrt(5) ! This is the consequence of sqrt(5)
being defined as _unique_ number !
And what a bothertrouble in this case with the definition "sqrt(5) has two
solutions"!?!
B) The function f: x -> x^2 + sqrt(5) has a positive range, hasn't it?
What do you say?- "It depends on...If you mean the principal...."??!
"5 has two square roots!"?? - No, thx!
BR
Hans
--
Hans Steih || HSt...@aol.com
D-47533 Kleve, Germany
"Ich hoffe, es wird niemanden befremden, dass ich den Homer und Virgil zu
Asymptoten gemacht habe" (Lichtenberg, Vom Nutzen der Mathematik)
Hans Steih
Sorry for this lack of concentration :-(
Corrected:
"the equation x^2 - 2x - 4 = 0 has two roots: 1 + sqrt(5) and 1-sqrt(5)"
David W. Cantrell
hst...@aol.com (Hans Steih) wrote:
>Im Artikel <049cd9a9...@usw-ex0102-016.remarq.com>, David
W. Cantrell
><dwcantrel...@aol.com.invalid> schreibt:
>
>>
>>Most mathematicians also hold that 5 has two square roots, one
>>positive and one negative.
>>
>
>1) "Most mathematicians"???? - How do you know?
>Have you any references? A bibliography?
Of course I have references, one of which was mentioned in my
article which originated this thread. I would be interested to
see if you can find any references which state clearly that a
positive number has only one square root. (Precise quotations
would be appreciated.) Although I suspect that you will be able
to find such references, I would be willing to wager that, for
every reference which you find which supports "a positive number
has a unique square root", you will encounter at least five
other references which support "a positive number has two square
roots".
>The mathematicians _I know_ define sqrt(5) as the _non-
>negative_ number which square is 5.
Sure.
>By this it follows that the _equation_ x^2 = 5 has two roots:
>{x el IR| x^2 = 5} ={ - sqrt(5) ; sqrt(5) }
>And there are no problems with this definition!
True. I never said otherwise.
>2) But _your_ definition (and as a former poster pointed out:
>it's a problem of definition, it's not a problem of "usage"!)
>has the consistency of a plum-pudding:
Which of my definitions? Several were discussed in my article.
>A) I say that the equation x^2 - 2x - 5 = 0 has two roots: 1 +
>sqrt(5) and 1-sqrt(5), and I know that 1-sqrt(5) is a negative
>number, that 1+sqrt(5) is a positive one. And I can write: 1-
>sqrt(5) < 1 + sqrt(5) ! This is the consequence of sqrt(5)
>being defined as _unique_ number !
That's perfectly fine (assuming the appropriate correction to
the equation as mentioned in your later post).
>And what a bothertrouble in this case with the definition
>"sqrt(5) has two solutions"!?!
Mercifully, I have never seen any mathematician claim that
"sqrt(5) has two solutions"! Certainly, I have never said such.
However, some would claim that sqrt(5) represents either of two
values.
>B) The function f: x -> x^2 + sqrt(5) has a positive range,
>hasn't it? What do you say?- "It depends on...If you mean the
>principal...."??!
Clearly for you, "sqrt(5)" itself denotes the principal root.
There is absolutely nothing wrong with that. It is a very common
usage. According to that usage, your f has a positive range.
Furthermore, according to any other usage, your f wouldn't be a
function at all!
>"5 has two square roots!"?? - No, thx!
You are welcome to your opinion. I do not doubt that there are
some other mathematicians who agree with you.
Hans Steih
<dwcantrel...@aol.com.invalid> schreibt:
>
>Of course I have references, one of which was mentioned in my
>article which originated this thread. I would be interested to
>see if you can find any references which state clearly that a
>positive number has only one square root. (Precise quotations
>would be appreciated.) Although I suspect that you will be able
>to find such references, I would be willing to wager that, for
>every reference which you find which supports "a positive number
>has a unique square root", you will encounter at least five
>other references which support "a positive number has two square
>roots".
>
>
Neil W Rickert
>>>>> David Cantrell
>>>>>______________________
>>>>So far, so good.
Then either your original statement was wrong, or was written badly
enough to be confusing. For it seemed to contradict this.
Therefore what is said in tha latter section (on recommended usage)
is on a different topic than the paragraph I criticized (on observed
usage).
>>Many (most?) mathematicians believe they are being technically
>>precise using "the square root of 5" to mean only the positive
>>square root.
>Most mathematicians also hold that 5 has two square roots, one
>positive and one negative.
I suspect that the answer they would give to this is highly
context sensitive.
>>>Your "_the_ square root" is the _principal_ square root.
>>Are you talking about usage, or about your particular preferred
>>definitions?
>Huh? Surely you agree that in common usage "_the_ square root"
>refers to the _principal_ square root.
I agree that "the square root" from common usage means the same
as "the principal square root" in your usage.
>Did I not say "Nonetheless, technically precise or not, 'the
>square root' of a positive number is often heard and should
>always be taken to indicate the principal square root."? Do you
>disagree with this?
I disagree with the implication that the term "the square root" might
not be technically precise.
Ronald Bruck
:Im Artikel <049cd9a9...@usw-ex0102-016.remarq.com>, David W.
:Cantrell
:<dwcantrel...@aol.com.invalid> schreibt:
:
:>
:>Most mathematicians also hold that 5 has two square roots, one
:>positive and one negative.
:>
:>
:
:1) "Most mathematicians"???? - How do you know?
:Have you any references? A bibliography?
:
:The mathematicians _I know_ define sqrt(5) as the _non-negative_ number
:which
:square is 5.
:By this it follows that the _equation_ x^2 = 5 has two roots:
:{x el IR| x^2 = 5} ={ - sqrt(5) ; sqrt(5) }
:And there are no problems with this definition!
I don't know ANY mathematicians who define sqrt(5) AT ALL. Programmers,
and computer scientists, but not mathematicians. Mathematicians do use
it in informal posts to Usenet to avoid scaring TeX newbies.
Mathematicians will almost universally agree that $\sqrt{5}$--this is
the correct TeX markup for the traditional radical-and-vinculum--is
positive.
These are three different terms: "square root", "sqrt", and
"$\sqrt{}$". Whether "square root" is synonymous with "$\sqrt{}$"
depends on the article used or implied:
"THE square root of 5" is 2.236... (definite article)
"5 has two square roots..." is common usage (implied indefinite
article).
"0 has two square roots, counting multiplicities..." is also common,
especially in complex analysis (although it's a sloppy way of
saying it; "z^2 = 0 has two solutions, counting multiplicities"
would be more common).
"sqrt(4) = 2" is correct, by the morphing of the programming function
into the common vocabulary. (In a programming language it would be
incorrect, since the function should appear only on the right side
in an assignment statement.)
The point is, these are DISTINCT ideas.
--Ron Bruck
--
Due to University fiscal constraints, .sigs may not be exceed one
line.
David W. Cantrell
I disagree, but you are welcome to your opinion that my
statement was poorly written. In the unlikely event that I will
ever repost such an article, I will attempt to improve its
clarity.
>>>>>But you have failed to define "_the_ square root",
>
>>>>And for very good reason! Later in my article, in section IV.
>>>>Recommended Usage, I said:
>
>>>>To indicate the principal-valued square-root function, simply
>>>>write Vx or x^(1/2); to indicate the other (i.e.,
>>>>nonpositive) square-root function, write -Vx or -x^(1/2).
>>>>This recommendation is based on the fact that this usage is
>>>>clearly the most common.
>>>>5^(1/2) can be read, most obviously, as "five to the power
>>>>one-half", for example. V5 can be read properly as "the
What if they were teaching students about square roots for the
first time? I have *never* seen a text appropriate to such a
context in which a positive number was considered to have a
unique square root.
>>>>Your "_the_ square root" is the _principal_ square root.
>
>>>Are you talking about usage, or about your particular
>>>preferred definitions?
>
>>Huh? Surely you agree that in common usage "_the_ square root"
>>refers to the _principal_ square root.
>
>I agree that "the square root" from common usage means the same
>as "the principal square root" in your usage.
I don't know why you say "in your usage". Surely you know of no
usage in which "principal square root" of a positive number
means anything other what I gave. To the best of my
knowledge, "principal square root" is (thankfully!) completely
unambiguous.
>>Did I not say "Nonetheless, technically precise or not, 'the
>>square root' of a positive number is often heard and should
>>always be taken to indicate the principal square root."? Do you
>>disagree with this?
>
>I disagree with the implication that the term "the square root"
>might not be technically precise.
Recall that the sentence with which you disagree was preceeded
by "However, to be technically precise, we should not read V5
simply as 'the square root of 5' if we are using the most common
definition of 'square root', according to which a positive
number has two square roots." Note the "if" clause! But of
course, were we using the relatively uncommon definition
according to which a positive number has only one square root,
reading V5 as "the square root of 5" would be perfectly correct.
My only objection in this regard is to those who would
simultaneously (1) claim that a positive number has two square
roots, (2) speak of "the square root" of a positive number, and
(3) think that they were then being technically correct.
Hans Steih
<dwcantrel...@aol.com.invalid> schreibt:
>
>Of course I have references, one of which was mentioned in my
>article which originated this thread. I would be interested to
>see if you can find any references which state clearly that a
>positive number has only one square root. (Precise quotations
>would be appreciated.) Although I suspect that you will be able
>to find such references, I would be willing to wager that, for
>every reference which you find which supports "a positive number
>has a unique square root", you will encounter at least five
>other references which support "a positive number has two square
>roots".
>
>
May I quote what Mike Ecker posted in the thread "Square root"?!
Begin of the quotation:
DEFINITION:
Sqrt(x) = the nonnegative number whose square is x. This is by definition/
convention. You may not like it or you may not have learned it, but denying it
does not make it otherwise.
So, sqrt(25) = 5, not -5, as 5 is nonnegative but -5 is negative. This is the
principal-root convention, cited above.
Nobody disputes that (5)(5) and (-5)(-5) each equal 25. That is not the issue.
If sqrt(25) were either or both 5 and -5 (as you seem to think), here are some
of the unpleasant consequences:
1) Square root would not be a function.
2) The quadratic formula would not need that "plus or minus" before the root.
3) The article "the" in English connotes uniqueness, so why would people say
"the square root of..."?
4) Sqrt(x^2) would not equal the absolute value of x. In turn, this would mean
that the distance formula for two points in the plane would not reduce to
ordinary distance along a line (x-axis) if the two points happened to be on the
line.
5) Distance could be negative if square root per se were negative as well as
positive, in view of the distance formula.
End of the quotation
Maybe I didn't understand you completely: Is the _unique_ definition "sqrt(25)
= 5" compatible with your definition or isn't it ?
BR
Hans
David W. Cantrell
hst...@aol.com (Hans Steih) wrote:
>Maybe I didn't understand you completely: Is the _unique_
>definition "sqrt(25)= 5" compatible with your definition or
>isn't it ?
Please remember that my survey included several *different*
usages. Sqrt(25) = 5 (uniquely) is certainly compatible with one
of the definitions I discussed. (That definition also happens to
be my personally preferred usage, not that that should be of any
real significance!) According to the usage which is most
generally accepted and which I therefore recommended, V25 = 5
(uniquely).
Regards,
Neil W Rickert
I will cut this short by responding to only one paragraph. I am
snipping the rest.
>My only objection in this regard is to those who would
>simultaneously (1) claim that a positive number has two square
>roots, (2) speak of "the square root" of a positive number, and
>(3) think that they were then being technically correct.
I am saying that, on the basis of the above assertion, your objection
applies to most mathematicians.
David W. Cantrell
>snipping the rest.
Thanks!
>>My only objection in this regard is to those who would
>>simultaneously (1) claim that a positive number has two square
>>roots, (2) speak of "the square root" of a positive number, and
>>(3) think that they were then being technically correct.
>
>I am saying that, on the basis of the above assertion, your
>objection applies to most mathematicians.
You may be correct.
But can you give any reasonable way in which a person (who knows
how to use English properly) can be "self consistent" while
doing (1) and (2) simultaneously? Suppose that mathematician M
asserts that 9 has two square roots. Then by merely uttering the
phrase "the square root of 9", M implies uniqueness, i.e., that
9 has only one square root, despite the fact that M asserts
otherwise. Thus, M is being self contradictory. Please, Neil,
show me that I am wrong! I'd really love for you to do so. I'm
not joking!
Sincerely,
Fred W. Helenius
>In article <8dvmvp$r...@ux.cs.niu.edu>, Neil W Rickert
>>I am saying that, on the basis of the above assertion, your
>>objection applies to most mathematicians.
>You may be correct.
>But can you give any reasonable way in which a person (who knows
>how to use English properly) can be "self consistent" while
>doing (1) and (2) simultaneously? Suppose that mathematician M
>asserts that 9 has two square roots. Then by merely uttering the
>phrase "the square root of 9", M implies uniqueness, i.e., that
>9 has only one square root, despite the fact that M asserts
>otherwise. Thus, M is being self contradictory. Please, Neil,
>show me that I am wrong! I'd really love for you to do so. I'm
>not joking!
Usage of a definite article does not imply uniqueness; it implies
"definiteness". There are many examples of the distinction, in
both ordinary and mathematical English.
Ordinary English: "Are you familiar with the works of the Bard?"
This question refers to Shakespeare, who, while certainly not
unique in being a bard, was certainly remarkable, notable or
distinguished -- and these qualities make usage of "the"
possible. Similarly, some classical music aficionados refer
to "the symphonies" without any overt reference to Beethoven.
Another example: "To Sherlock Holmes she is always the woman."
(Sir Arthur Conan Doyle, "A Scandal in Bohemia", first sentence.)
This means that, to Holmes, Irene Adler is distinguished from
all other women, not that she is the unique woman in existence.
In mathematics: "The ring Z/mZ has phi(m) units. The unit is the
class containing 1, i.e., 1 + mZ, which we write as \bar{1}." A
ring may have many units, but one of them is distinguished (by
being the multiplicative identity) and is called "the unit".
--
Fred W. Helenius <fr...@ix.netcom.com>
Neil W Rickert
>In article <8dvmvp$r...@ux.cs.niu.edu>, Neil W Rickert
><ricke...@cs.niu.edu> wrote:
>>David W. Cantrell <dwcantrel...@aol.com.invalid> writes:
>>I will cut this short by responding to only one paragraph. I am
>>snipping the rest.
>Thanks!
>>>My only objection in this regard is to those who would
>>>simultaneously (1) claim that a positive number has two square
>>>roots, (2) speak of "the square root" of a positive number, and
>>>(3) think that they were then being technically correct.
>>I am saying that, on the basis of the above assertion, your
>>objection applies to most mathematicians.
>You may be correct.
>But can you give any reasonable way in which a person (who knows
>how to use English properly) can be "self consistent" while
>doing (1) and (2) simultaneously?
There is no inconsistency there. The English language (or any
natural language) is not a formal language. You cannot sensibly
apply standards of logic to a natural language. The meaning of
"square root" changes between (1) and (2), with the use of the
definite article being an indicator of this change.
> Suppose that mathematician M
>asserts that 9 has two square roots. Then by merely uttering the
>phrase "the square root of 9", M implies uniqueness, i.e., that
>9 has only one square root, despite the fact that M asserts
>otherwise. Thus, M is being self contradictory. Please, Neil,
>show me that I am wrong! I'd really love for you to do so. I'm
>not joking!
You are taking a far too formalistic view of English.
John Savard
<dwcantrel...@aol.com.invalid> wrote, in part:
>There are three different ways in which square roots may be
>indicated using mathematical symbols:
> (1) using the common radical sign, which will be approximated
>here by typing "V",
> (2) using an exponent of 1/2, and
> (3) using some abbreviation of "square root".
>Examples of such notations are V9, 9^(1/2), and sqrt(9),
>respectively.
>In computer arithmetic, SQRT always denotes the principal square
>root. Outside that context, abbreviations such as sqrt and Sqrt
>should ideally either be avoided or else explained by the writer.
Using an exponent of 1/2 is a valid notation for square root, and if
that notation were abolished, then using other exponents would be hard
to do. The radical sign exists additionally to give a shorter and more
convenient notation for the square root, as square roots are often
used in some areas of mathematics. (Thus, the question of giving up
one of those two possibilities, if not for mathematicians who prefer
the other one, doesn't really come up.)
SQRT (in FORTRAN) or SQR (in BASIC) are not _notations_ for the square
root; they are requests that a square root be calculated, and, as a
result, must return a single value - the principal value. (Languages
like Mathematica could, of course, do otherwise.)
John Savard (teneerf <-)
Virgil
Helenius <fr...@ix.netcom.com> wrote:
>David W. Cantrell <dwcantrel...@aol.com.invalid> wrote:
>
>>In article <8dvmvp$r...@ux.cs.niu.edu>, Neil W Rickert
>>>I am saying that, on the basis of the above assertion, your
>>>objection applies to most mathematicians.
>
>>You may be correct.
>>But can you give any reasonable way in which a person (who knows
>>how to use English properly) can be "self consistent" while
>>doing (1) and (2) simultaneously? Suppose that mathematician M
>>asserts that 9 has two square roots. Then by merely uttering the
>>phrase "the square root of 9", M implies uniqueness, i.e., that
>>9 has only one square root, despite the fact that M asserts
>>otherwise. Thus, M is being self contradictory. Please, Neil,
>>show me that I am wrong! I'd really love for you to do so. I'm
>>not joking!
>
>Usage of a definite article does not imply uniqueness; it implies
>"definiteness". There are many examples of the distinction, in
>both ordinary and mathematical English.
>
>Ordinary English: "Are you familiar with the works of the Bard?"
>This question refers to Shakespeare, who, while certainly not
>unique in being a bard, was certainly remarkable, notable or
>distinguished -- and these qualities make usage of "the"
>possible.
Bad example. "The Bard" unquestionably implies uniqueness where "The
bard" need not.
Similarly, some classical music aficionados refer
>to "the symphonies" without any overt reference to Beethoven.
Beethoven may be the only composer of symphonies to you, but not to me
nor to many classical music aficionados. Haydn is often credited with
being the first. Mozart did one or two of note. And there are many
others. I would not understand a reference to "the symphonies" without
at least an implicit reference to Beethoven necessarily to refer to his
symphonies at all.
--
Virgil
vm...@frii.com
David W. Cantrell
Fred W. Helenius <fr...@ix.netcom.com> wrote:
> David W. Cantrell <dwcantrel...@aol.com.invalid> wrote:
>
> >In article <8dvmvp$r...@ux.cs.niu.edu>, Neil W Rickert
> >>objection applies to most mathematicians.
>
> >You may be correct.
> >But can you give any reasonable way in which a person (who knows
> >how to use English properly) can be "self consistent" while
> >doing (1) and (2) simultaneously? Suppose that mathematician M
> >asserts that 9 has two square roots. Then by merely uttering the
> >phrase "the square root of 9", M implies uniqueness, i.e., that
> >9 has only one square root, despite the fact that M asserts
> >otherwise. Thus, M is being self contradictory. Please, Neil,
> >show me that I am wrong! I'd really love for you to do so. I'm
> >not joking!
>
> Usage of a definite article does not imply uniqueness; it implies
> "definiteness". There are many examples of the distinction, in
> both ordinary and mathematical English.
Use of the definite article with a *singular* noun implies uniqueness.
> Ordinary English: "Are you familiar with the works of the Bard?"
> This question refers to Shakespeare, who, while certainly not
> unique in being a bard, was certainly remarkable, notable or
> distinguished -- and these qualities make usage of "the"
> possible.
Shakespeare is the unique bard (in the English language, at least) who
is distinguished by capitalization: "Bard", rather than merely "bard".
There are many examples of such: "the Renaissance", "the Calculus",...
Similarly, in mathematics, I happen to prefer the convention according
to which a principal-valued function is distinguished from the
corresponding multivalued relation by capitalization. For example,
Atan(1)=pi/4 while atan(1)=(4n+1)pi/4 for integral n.
> Similarly, some classical music aficionados refer
> to "the symphonies" without any overt reference to Beethoven.
Speaking as a semi-professional musician, let me say that I have never
heard such. But in any event, if Beethoven's symphonies are to be
distinguished in this fashion, they should be referred to as "the
Symphonies", rather than just "the symphonies".
> Another example: "To Sherlock Holmes she is always the woman."
> (Sir Arthur Conan Doyle, "A Scandal in Bohemia", first sentence.)
> This means that, to Holmes, Irene Adler is distinguished from
> all other women, not that she is the unique woman in existence.
"To Sherlock Holmes" establishes a special context. Within an
established context, something can easily be unique which, outside that
context, would not be unique. For example, it would be correct
(although rather strange sounding perhaps) to say "In the context of
principal values, the square root of 9 is +3." Thus, there is nothing
wrong with the phrase "the square root of 9" if it has been made clear
that only principal values are under discussion.
David Cantrell
Sent via Deja.com http://www.deja.com/
Before you buy.
David W. Cantrell
<ricke...@cs.niu.edu> wrote:
>David W. Cantrell <dwcantrel...@aol.com.invalid> writes:
>>In article <8dvmvp$r...@ux.cs.niu.edu>, Neil W Rickert
>><ricke...@cs.niu.edu> wrote:
>>>David W. Cantrell <dwcantrel...@aol.com.invalid> writes:
>
>>>>My only objection in this regard is to those who would
>>>>simultaneously (1) claim that a positive number has two
>>>>square roots, (2) speak of "the square root" of a positive
>>>>number, and (3) think that they were then being technically
>>>>correct.
>
>>>I am saying that, on the basis of the above assertion, your
>>>objection applies to most mathematicians.
>
>>You may be correct.
>>But can you give any reasonable way in which a person (who
>>knows how to use English properly) can be "self consistent"
>>while doing (1) and (2) simultaneously?
>
That is your opinion. In my opinion, there is inconsistency.
Furthermore, I have today had the opportunity to ask the opinions
of three English teachers and of one lawyer (who also holds
degrees in mathematics and computer science) on this matter. All
of them said that (2) is inconsistent with (1).
>The English language (or any natural language) is not a formal
>language.
Sure.
>You cannot sensibly apply standards of logic to a natural
>language.
Certainly you can't expect _all_ aspects of a natural language to
be logical. But if no standards of logic apply to a natural
language, then we had better quit using natural languages in any
contexts where precision of meaning is required. Thankfully
however, there are ways (almost always, at least) to be
adequately precise using natural languages.
>The meaning of "square root" changes between (1) and (2), with
>the use of the definite article being an indicator of this
>change.
It sounds strange to say that "The meaning of 'square root'
changes..."! Surely it is merely the context which is appropriate
which changes. Anyway, it's not the definite article by itself
which indicates the change. For example, "The square roots of 9
are +3 and -3." is perfectly fine.
A better claim would be that the definite article followed by a
singular noun indicates that the context appropriate for (2) must
be strictly that of principal values. That is why I said in my
original article that "the square root" of a positive number
should always be taken to indicate the principal one.
Perhaps I should be happy to regard this "better claim" above as
resolving the problem. But I am uncomfortable about generally
accepting "automatic context shifts", much as I am uncomfortable
about generally accepting "automatic extension by limit" for
functions with removable discontinuities.
>>Suppose that mathematician M asserts that 9 has two square
>>roots. Then by merely uttering the phrase "the square root of
>>9", M implies uniqueness, i.e., that 9 has only one square
>>root, despite the fact that M asserts otherwise. Thus, M is
>>being self contradictory. Please, Neil, show me that I am
>>wrong! I'd really love for you to do so. I'm not joking!
>
>You are taking a far too formalistic view of English.
You're certainly entitled to your opinion.
Regards,
Neil W Rickert
>In article <8e09uv$r...@ux.cs.niu.edu>, Neil W Rickert
><ricke...@cs.niu.edu> wrote:
>>David W. Cantrell <dwcantrel...@aol.com.invalid> writes:
>>>In article <8dvmvp$r...@ux.cs.niu.edu>, Neil W Rickert
>>><ricke...@cs.niu.edu> wrote:
>>>>David W. Cantrell <dwcantrel...@aol.com.invalid> writes:
>>>>>My only objection in this regard is to those who would
>>>>>simultaneously (1) claim that a positive number has two
>>>>>square roots, (2) speak of "the square root" of a positive
>>>>>number, and (3) think that they were then being technically
>>>>>correct.
>>>>I am saying that, on the basis of the above assertion, your
>>>>objection applies to most mathematicians.
>>>You may be correct.
>>>But can you give any reasonable way in which a person (who
>>>knows how to use English properly) can be "self consistent"
>>>while doing (1) and (2) simultaneously?
>>There is no inconsistency there.
>That is your opinion. In my opinion, there is inconsistency.
>Furthermore, I have today had the opportunity to ask the opinions
>of three English teachers and of one lawyer (who also holds
>degrees in mathematics and computer science) on this matter. All
>of them said that (2) is inconsistent with (1).
ROTFL.
Next you will be saying that you consulted three Math teachers, who
said that Shakespeare's plays don't add up.
>>The English language (or any natural language) is not a formal
>>language.
>Sure.
>>You cannot sensibly apply standards of logic to a natural
>>language.
>Certainly you can't expect _all_ aspects of a natural language to
>be logical. But if no standards of logic apply to a natural
>language, then we had better quit using natural languages in any
>contexts where precision of meaning is required.
It is quite sufficient to use precise notation, perhaps
quite sparingly, where it is needed to remove ambiguity.
I suggest that you take your original article in this thread, and
attempt to rewrite it using only strict first order predicate
calculus. I think you will find that it cannot be done. Natural
languages have an expressiveness that is not found in formal
languages. Mathematicians need this expressiveness.
>>The meaning of "square root" changes between (1) and (2), with
>>the use of the definite article being an indicator of this
>>change.
>It sounds strange to say that "The meaning of 'square root'
>changes..."! Surely it is merely the context which is appropriate
>which changes. Anyway, it's not the definite article by itself
>which indicates the change. For example, "The square roots of 9
>are +3 and -3." is perfectly fine.
My comment was admittely sloppy. Strictly speaking, meaning should
apply to complete sentences (or sometimes paragraphs, or chapters),
rather than to individual words or phrases.
>A better claim would be that the definite article followed by a
>singular noun indicates that the context appropriate for (2) must
>be strictly that of principal values. That is why I said in my
>original article that "the square root" of a positive number
>should always be taken to indicate the principal one.
But you are sounding too formalistic. All that matters is that
mathematicians properly communicate what they mean. Where there
might otherwise be ambiguity, they will often use symbolic notation.
We can do without rigid rules of language usage, for such rules would
reduce the expressiveness of language. We do need to be precise in
defining our symbolic notation.