Solution to Rishi's problem

bw wrote:"sqrt(x)" is a function and a function returns only one value for any input "x" which, in this case, is the positive root of "x". a function cannot return two values. that's that.

bw wrote:"sqrt(x)" is a function and a function returns only one value for any input "x" which, in this case, is the positive root of "x". a function cannot return two values. that's that.

Seva Lamberdar wrote:

bw wrote:"sqrt(x)" is a function and a function returns only one value for any input "x" which, in this case, is the positive root of "x". a function cannot return two values. that's that.

This is a discussion in mathematics on square roots and not some half-brained imbecilic use of square root function (subroutine) on computers.

MaxEntropy_Man wrote:

Seva Lamberdar wrote:

MaxEntropy_Man wrote:

Seva Lamberdar wrote:
What does the satement "The real solutions of x^2 = a are written as +,- sqrt(a)" mean to you?

it means that √a is a positive number, and the attaching of a negative sign in front of it changes its sign. the attaching of a positive sign by itself is redundant although not incorrect, but when referring to both roots together, one writes ±√a. the very fact that you have to write ±√a implies that √a cannot be the symbol that refers to both roots as you claimed originally.

No. You are the one who first made a statement "as x -> - inf, sqrt(1+x^2) goes to abs(x), since x^2 >> 1" using abs(x), instead of just x (or +,- x), which led to this entire discussion.

here is what i said, now that i can type in math symbols:

√(1+x²) which is a positive number, tends to √x² also a positive number, and thus equal to |x|, when x is large and negative. i maintain that when you leave the root unsigned as in √x², the implication is that you are referring to the positive root.

Seva Lamberdar wrote:

MaxEntropy_Man wrote:

Seva Lamberdar wrote:

MaxEntropy_Man wrote:

Seva Lamberdar wrote:
What does the satement "The real solutions of x^2 = a are written as +,- sqrt(a)" mean to you?

it means that √a is a positive number, and the attaching of a negative sign in front of it changes its sign. the attaching of a positive sign by itself is redundant although not incorrect, but when referring to both roots together, one writes ±√a. the very fact that you have to write ±√a implies that √a cannot be the symbol that refers to both roots as you claimed originally.

No. You are the one who first made a statement "as x -> - inf, sqrt(1+x^2) goes to abs(x), since x^2 >> 1" using abs(x), instead of just x (or +,- x), which led to this entire discussion.

here is what i said, now that i can type in math symbols:

√(1+x²) which is a positive number, tends to √x² also a positive number, and thus equal to |x|, when x is large and negative. i maintain that when you leave the root unsigned as in √x², the implication is that you are referring to the positive root.

Here is what you said in the beginning "as x -> - inf, sqrt(1+x^2) goes to abs(x), since x^2 >> 1" and spent the whole time defending in it the arbitrary use of "abs".

bw wrote:

Seva Lamberdar wrote:

bw wrote:"sqrt(x)" is a function and a function returns only one value for any input "x" which, in this case, is the positive root of "x". a function cannot return two values. that's that.

This is a discussion in mathematics on square roots and not some half-brained imbecilic use of square root function (subroutine) on computers.

In mathematics, a function[1] is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

are you questioning this, seva?

Seva Lamberdar wrote:

bw wrote:

Seva Lamberdar wrote:

bw wrote:"sqrt(x)" is a function and a function returns only one value for any input "x" which, in this case, is the positive root of "x". a function cannot return two values. that's that.

This is a discussion in mathematics on square roots and not some half-brained imbecilic use of square root function (subroutine) on computers.

In mathematics, a function[1] is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

are you questioning this, seva?

If you really want to get on this discussion BW, go through all the comments on this thread right from the beginning. I don't think you have a clue about what exactly is being discussed and why?

bw wrote:

Seva Lamberdar wrote:

bw wrote:

Seva Lamberdar wrote:

bw wrote:"sqrt(x)" is a function and a function returns only one value for any input "x" which, in this case, is the positive root of "x". a function cannot return two values. that's that.

This is a discussion in mathematics on square roots and not some half-brained imbecilic use of square root function (subroutine) on computers.

In mathematics, a function[1] is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

are you questioning this, seva?

If you really want to get on this discussion BW, go through all the comments on this thread right from the beginning. I don't think you have a clue about what exactly is being discussed and why?

you are claiming that the function sqrt(x) does not represent just the positive root. mathematics, definition of functions, max and some others disagree with you. what else is there to understand?

by the way, abs(x) and sqrt(x^2) are the same.

MaxEntropy_Man wrote:

Seva Lamberdar wrote:

MaxEntropy_Man wrote:

Seva Lamberdar wrote:

MaxEntropy_Man wrote:

it means that √a is a positive number, and the attaching of a negative sign in front of it changes its sign. the attaching of a positive sign by itself is redundant although not incorrect, but when referring to both roots together, one writes ±√a. the very fact that you have to write ±√a implies that √a cannot be the symbol that refers to both roots as you claimed originally.

No. You are the one who first made a statement "as x -> - inf, sqrt(1+x^2) goes to abs(x), since x^2 >> 1" using abs(x), instead of just x (or +,- x), which led to this entire discussion.

here is what i said, now that i can type in math symbols:

√(1+x²) which is a positive number, tends to √x² also a positive number, and thus equal to |x|, when x is large and negative. i maintain that when you leave the root unsigned as in √x², the implication is that you are referring to the positive root.

Here is what you said in the beginning "as x -> - inf, sqrt(1+x^2) goes to abs(x), since x^2 >> 1" and spent the whole time defending in it the arbitrary use of "abs".

i still stand by that and it does not contradict anything else i have said. i am done with this.

aanjaneya wrote:In general, it is okay to view the sqrt as a (discrete) set-valued mapping, that maps each (strictly) positive real number to a pair of real numbers that differ only in sign. The case of zero is a trivial case.
E.g. the positive number 25 gets mapped to the set {+5, -5}.
If the context requires a point-valued map (i.e. what is commonly understood as a function), then it is convention to choose the positive root, when tacit/unstated.

Also, note that sqrt(x^2) = x is ambiguous, if you want a point-valued map.
Because, it implicitly requires you to solve for a point-value x, whose square is the input argument (which is some positive scalar with value x^2).
Which in turn requires you to make a choice between two real numbers that differ only in sign.
And that choice is unspecified.

Which is why it is a sensible convention is to take the positive root, and use sqrt(x^2) = abs(x).
Note that the "abs" operator essentially reduces the set of two real numbers that differ in sign, into a single number, and thus it will resolve the ambiguity.

Lastly, the limit question here would get quite a bit complicated if you insist
on interpreting sqrt as a set-valued mapping. In fact, it would not make
any sense unless you also "order" the set. And even then, it needs the notion of point-wise convergence, and some specification of set arithmetic however obvious, and so on.
Definitely not what a Calculus 1 type of question is intending to ask.

aanjaneya wrote:In general, it is okay to view the sqrt as a (discrete) set-valued mapping, that maps each (strictly) positive real number to a pair of real numbers that differ only in sign. The case of zero is a trivial case.
E.g. the positive number 25 gets mapped to the set {+5, -5}.
If the context requires a point-valued map (i.e. what is commonly understood as a function), then it is convention to choose the positive root, when tacit/unstated.

Also, note that sqrt(x^2) = x is ambiguous, if you want a point-valued map.
Because, it implicitly requires you to solve for a point-value x, whose square is the input argument (which is some positive scalar with value x^2).
Which in turn requires you to make a choice between two real numbers that differ only in sign.
And that choice is unspecified.

Which is why it is a sensible convention is to take the positive root, and use sqrt(x^2) = abs(x).
Note that the "abs" operator essentially reduces the set of two real numbers that differ in sign, into a single number, and thus it will resolve the ambiguity.

Lastly, the limit question here would get quite a bit complicated if you insist
on interpreting sqrt as a set-valued mapping. In fact, it would not make
any sense unless you also "order" the set. And even then, it needs the notion of point-wise convergence, and some specification of set arithmetic however obvious, and so on.
Definitely not what a Calculus 1 type of question is intending to ask.

Seva Lamberdar wrote:
You are wrong on most of the ponts in the above, especially "it is a sensible convention is to take the positive root, and use sqrt(x^2) = abs(x)."

MaxEntropy_Man wrote:

Seva Lamberdar wrote:
You are wrong on most of the ponts in the above, especially "it is a sensible convention is to take the positive root, and use sqrt(x^2) = abs(x)."

you might also be interested in looking up words such as "convention" and "definition" in the context of mathematics and also the difference between a "definition" and "physical law". you seem very confused.

Seva Lamberdar wrote:

aanjaneya wrote:In general, it is okay to view the sqrt as a (discrete) set-valued mapping, that maps each (strictly) positive real number to a pair of real numbers that differ only in sign. The case of zero is a trivial case.
E.g. the positive number 25 gets mapped to the set {+5, -5}.
If the context requires a point-valued map (i.e. what is commonly understood as a function), then it is convention to choose the positive root, when tacit/unstated.

Also, note that sqrt(x^2) = x is ambiguous, if you want a point-valued map.
Because, it implicitly requires you to solve for a point-value x, whose square is the input argument (which is some positive scalar with value x^2).
Which in turn requires you to make a choice between two real numbers that differ only in sign.
And that choice is unspecified.

Which is why it is a sensible convention is to take the positive root, and use sqrt(x^2) = abs(x).
Note that the "abs" operator essentially reduces the set of two real numbers that differ in sign, into a single number, and thus it will resolve the ambiguity.

Lastly, the limit question here would get quite a bit complicated if you insist
on interpreting sqrt as a set-valued mapping. In fact, it would not make
any sense unless you also "order" the set. And even then, it needs the notion of point-wise convergence, and some specification of set arithmetic however obvious, and so on.
Definitely not what a Calculus 1 type of question is intending to ask.

You are wrong on most of the ponts in the above, especially "it is a sensible convention is to take the positive root, and use sqrt(x^2) = abs(x)."

bw wrote:

Seva Lamberdar wrote:

aanjaneya wrote:In general, it is okay to view the sqrt as a (discrete) set-valued mapping, that maps each (strictly) positive real number to a pair of real numbers that differ only in sign. The case of zero is a trivial case.
E.g. the positive number 25 gets mapped to the set {+5, -5}.
If the context requires a point-valued map (i.e. what is commonly understood as a function), then it is convention to choose the positive root, when tacit/unstated.

Also, note that sqrt(x^2) = x is ambiguous, if you want a point-valued map.
Because, it implicitly requires you to solve for a point-value x, whose square is the input argument (which is some positive scalar with value x^2).
Which in turn requires you to make a choice between two real numbers that differ only in sign.
And that choice is unspecified.

Which is why it is a sensible convention is to take the positive root, and use sqrt(x^2) = abs(x).
Note that the "abs" operator essentially reduces the set of two real numbers that differ in sign, into a single number, and thus it will resolve the ambiguity.

Lastly, the limit question here would get quite a bit complicated if you insist
on interpreting sqrt as a set-valued mapping. In fact, it would not make
any sense unless you also "order" the set. And even then, it needs the notion of point-wise convergence, and some specification of set arithmetic however obvious, and so on.
Definitely not what a Calculus 1 type of question is intending to ask.

You are wrong on most of the ponts in the above, especially "it is a sensible convention is to take the positive root, and use sqrt(x^2) = abs(x)."

ouch!

Marathadi-Saamiyaar wrote:

bw wrote:

Seva Lamberdar wrote:
You are wrong on most of the ponts in the above, especially "it is a sensible convention is to take the positive root, and use sqrt(x^2) = abs(x)."

ouch!

All this for sqrt(X^2)???

At least if it is for sqrt (-X^2) I can understand.