square roots - no negative answer
Terry Louie
square roots, but she is teaching that the square root of 4 is either 2
or -2!!
The math textbook material is simple square roots; not algebra, not
imaginary numbers.
In algebra, x-squared has two possible solutions; x and -x. Imaginary
numbers are the square root of negative numbers. But the teacher is
teaching basic positive integer square roots, and told the class that
because 2 squared and -2 squared are both equal to 4, so therefore
the square root of 4 is either 2 or minus 2 !!!
How can I explain the illogic of this? I looked up the definition of
square root in a big dictionary, but that was of no help.
Thanks!
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Brian Harvey
>My son's 7th grade math teacher is teaching basic positive integer
>square roots, but she is teaching that the square root of 4 is either 2
>or -2!!
>The math textbook material is simple square roots; not algebra, not
>imaginary numbers.
I think your discussion of imaginary numbers is a red herring here; the
teacher is not proposing to take square roots OF negative numbers, but
to allow negative square roots of POSITIVE numbers.
It's a convention that "the square root" means "the positive square root,"
and so this teacher is, technically, wrong because of that convention.
But what she's saying isn't crazy or illogical. In fact, I think it's
a good first step. That is, first you have to understand that there are
two numbers which, when squared, give 4. THEN you can learn that we
adopt the convention that "THE square root" means the positive one.
But it's important to understand that that's only a convention, because
later on you'll need to remember that in order to understand why you
can't just take the square of both sides of an equation, the way you
can divide both sides by 2, for example. It's because one of those
operations has a unique inverse, and the other doesn't -- except by
convention.
Don Specht
<ttl...@discover.earthlink.net> wrote:
>My son's 7th grade math teacher is teaching basic positive integer
>square roots, but she is teaching that the square root of 4 is either 2
>or -2!!
Semantics. The principal square root of 4 is 2.
>The math textbook material is simple square roots; not algebra, not
>imaginary numbers.
>In algebra, x-squared has two possible solutions; x and -x. Imaginary
>numbers are the square root of negative numbers. But the teacher is
>teaching basic positive integer square roots, and told the class that
>because 2 squared and -2 squared are both equal to 4, so therefore
>the square root of 4 is either 2 or minus 2 !!!
>
>How can I explain the illogic of this?
One question first: Is your goal to correct what you perceive as a
mathematical error, or to undermine the teacher. I would agree that the
former is worthy, but the latter may make it extremely difficult for the
class to progress. Just something to think about.
>I looked up the definition of
>square root in a big dictionary, but that was of no help.
Not surprising.
Expressions are groups of symbols that form meaningful assertions. By
their very nature numeric expressions must be unique, that is they may
have only one value. Hence the principal root rule, which essentially
says that when you have two choices, keep the positive one.
sqrt(9)=3, even though 3^2=9 and (-3)^2=9
cubert(8)=2, because 2^3=8
fourthrt(-625) is not a real number
fifthrt(-32)=-2, because (-2)^5=-32
Statements are expressions that are true or false, not both. Open
sentences are statements containing variables. The solution set of an
open sentence is the subset of the domain of the variable that makes the
open sentence true.
Assuming the domain is the set of reals:
x^2=4, x=-2 or 2
x^3=27, x=3
x^4=-81, no real solutions
x^5=-32, x=-2
In other words, when you simplify an expression, there will be only one
acceptable replacement value. When solving an equation you are looking
for all possible roots.
--
dspecht at ix dot netcom dot com
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SEBarnett
In article <365b4a73....@news.wenet.net>, Terry Louie
<ttl...@discover.earthlink.net> writes:
>My son's 7th grade math teacher is teaching basic positive integer
>square roots, but she is teaching that the square root of 4 is either 2
>or -2!!
The square roots of 4 _are_ 2 and -2. But 2 is the _principal_ square
root--that indicated by the radical sign (with no index) around the 4. One
wishing to indicate the other square root of 4 would need to write a negative
sign in front of the radical.
Hope this helps,
Scott Barnett
Sheldon Ackerman
>teaching basic positive integer square roots, and told the class that
>because 2 squared and -2 squared are both equal to 4, so therefore
>the square root of 4 is either 2 or minus 2 !!!
>
>square root in a big dictionary, but that was of no help.
>
both positive and negative two.
When the 4 is in the radical sign then the answer is only the positive value
because by convention that radical sign is understood to be asking for only
the positive value.
Someone please correct me if I am in error.
Sheila King
<365b4a73....@news.wenet.net>:
:My son's 7th grade math teacher is teaching basic positive integer
:square roots, but she is teaching that the square root of 4 is either 2
:or -2!!
This is correct. Most Algebra II books will state this in the beginning
of the Chapter on evaluating square roots.
However, the principle square root of 4 is only positive two, and it is
this root that is normally assumed unless otherwise indicated by a
negative sign. That does not take away from the fact that the number 4
has two square roots.
:The math textbook material is simple square roots; not algebra, not
:imaginary numbers.
:In algebra, x-squared has two possible solutions; x and -x. Imaginary
:numbers are the square root of negative numbers. But the teacher is
:teaching basic positive integer square roots, and told the class that
:because 2 squared and -2 squared are both equal to 4, so therefore
:the square root of 4 is either 2 or minus 2 !!!
:
:How can I explain the illogic of this? I looked up the definition of
:square root in a big dictionary, but that was of no help.
There's nothing illogical about it. It is exactly correct.
Sheila King
It's all fun and games until someone loses an eye
Then it's all fun and games,
but without depth perception
William L. Bahn
Terry Louie wrote in message <365b4a73....@news.wenet.net>...
>My son's 7th grade math teacher is teaching basic positive integer
>square roots, but she is teaching that the square root of 4 is either 2
>or -2!!
>
I think that this is appropriate for this level of material. The concept of
"principal" square roots is something that is best taught only after the
student comprehends the concept of "square roots" since it is, after all, a
refining augmentation of a more basic concept.
A very common cause for error - both in mathematics on paper and in using
mathematics to design things with the potential for killing someone if they
are not designed correctly - is blindly assuming that the principal square
root is the only square root. In a physical system, this ignored solution
often has physical implications - the system might go into resonance there
or the control algorithms might go singular and lock up or behave
unpredictably or a multitude of other sins with potentially disasterous
consequences.
Consider the quadratic formula:
a x^2 + b x + c = 0
Find the values of x that satisfy this equation.
We know that:
(x+d)^2 = x^2 + 2d x + d^2
So we cast the original equation in this form:
x^2 + (b/a) x + (c/a) = 0
x^2 + 2 (b/(2a)) x + (c/a) = 0
So d = b/(2a) and d^2 = [b^2/(4a^2)] therefore:
x^2 + 2 (b/(2a)) x + [b^2/(4a^2)] = [b^2/(4a^2)] - (c/a)
[ x + b/(2a) ] ^2 = (b^2-4ac)/(4a^2)
We now need to take the square root of both sides - but we HAVE to
acknowledge that nothing we have done to this point imposes any constraint
on those roots. To be more explicit, any value that, when squared, yields
x + b/(2a) ] ^2 on the left and (b^2-4ac)/(4a^2) on the right IS a valid
solution of the original equation.
If I use the convention that sqrt(y) is the principal square root of y, then
I have to write my next step as:
(+/-) sqrt( [ x + b/(2a) ] ^2 ) = (+/-) sqrt( (b^2-4ac)/(4a^2) )
Now, (+/-) A = (+/-) B reduces to A = (+/-) B, so I immediately have:
x + b/(2a) = (+/-) sqrt(b^2-4ac) / (2a)
x = -b/(2a) + (+/-) sqrt(b^2-4ac) / (2a)
x = [ -b + (+/-) sqrt(b^2-4ac) ] / (2a)
Which is, of course, the familiar quadratic equation.
To recap, the point of this exercise is to ask WHERE the (+/-) sign comes
from in this formula? It comes from the fact that when asking what the
square root of a quantity is, in general there are TWO possible answers
neither of which can be ignored.
Therefore, I think it is entirely appropriate to introduce students to the
concept of square roots with the understanding that there are sometimes
multiple values that answer the same question.
>The math textbook material is simple square roots; not algebra, not
>imaginary numbers.
>In algebra, x-squared has two possible solutions; x and -x.
And what if x=4?
> Imaginary
>numbers are the square root of negative numbers.
And your point would be? I fail to see where imaginary numbers enter into
this discussion at all.
> But the teacher is
>teaching basic positive integer square roots, and told the class that
>because 2 squared and -2 squared are both equal to 4, so therefore
>the square root of 4 is either 2 or minus 2 !!!
Yes. In its general meaning, the square root of a number is any value that,
when squared, is equal to the original number. 2 and -2 both satisfy this
condition if the original number is 4.
The point to keep in mind is the the principal square root is NOT the same
concept as the square root - otherwise why do we bother to even have an
concept called the "principal square root"? Saying it another way, If we
have square roots that are principal square roots, doesn't it make sense
that we probably have square roots that are not principal square roots?
Therefore, if we simply ask for the square root of something, shouldn't we
be looking for ALL of the square roots, principal or not?
>
>How can I explain the illogic of this? I looked up the definition of
>square root in a big dictionary, but that was of no help.
I would suggest that you NOT try to use "normal" dictionaries, no matter HOW
big they might be, to decide the meanings of technical terms. These
references are intended to relay the general meanings of words as used in
common language - they are not meant to relate the detailed and specific
concepts embodied by those words when used in a technical discussion.
Instead, use a glossary or a dictionary that is specifically written for the
purpose of defining technical terms when used in a technical context.
math...@hotmail.com
My gosh man! She's correct! Every positive real
number has two square roots --one positive and one negative.
It is "conventional" to use the standard square root
symbol to represent the positive root.
If you go to complex numbers then all real numbers have 3 cube roots
4 fourth roots, etc.
In article <365b4a73....@news.wenet.net>,
Terry Louie <ttl...@discover.earthlink.net> wrote:
> My son's 7th grade math teacher is teaching basic positive integer
> square roots, but she is teaching that the square root of 4 is either 2
> or -2!!
>
> The math textbook material is simple square roots; not algebra, not
> imaginary numbers.
> In algebra, x-squared has two possible solutions; x and -x. Imaginary
> numbers are the square root of negative numbers. But the teacher is
> teaching basic positive integer square roots, and told the class that
> because 2 squared and -2 squared are both equal to 4, so therefore
> the square root of 4 is either 2 or minus 2 !!!
>
> How can I explain the illogic of this? I looked up the definition of
> square root in a big dictionary, but that was of no help.
>
> Thanks!
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Rob Morewood
: My son's 7th grade math teacher is teaching basic positive integer
: square roots, but she is teaching that the square root of 4 is either 2
: or -2!!
Your son's teacher is quite correct. Every positive integer has two
square roots, one negative and one positive. (And negative numbers
have no real square roots while zero has exactly one square root.)
Check any textbook for the definition of "square root".
It should read something like:
"A number whose square is the original quantity."
There IS a convention (for real numbers) that:
__
-/x = the Positive square root of x
__
- -/x = the Negative square root of x
but they are BOTH square roots of x.
_
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