Article by Dr p

[Saumyojit]

I read this well written article [ https://www.nctm.org/tmf/library/drmath/view/72802.html ] by drp

The only thing that could go wrong, really, is if
you can't perform the operation at all (e.g. if you want to take the
square root of both sides but one or both may be negative). This
becomes a domain issue, if you are familiar with functions

One or both may be negative means ? suppose there are two nos -4=-4 (both are negative) then i cannot take square root of both the numbers as they will be extended to complex no but why did he said one may be negative the equation with one negative be like this 4 =-4 (which is not valid in the first place) .
What is the domain issue?
Then he says

you just have to
determine that it is well-defined (has one value) and that its domain
includes the values to which it is being applied

What does this mean?

Although it is true that
lthough it is true that
if a = b, then sqrt(a) = sqrt(b), and in fact these equations are
equivalent if you ignore domain issues

These 2 equations are equivalent from both sides .
eg : if 4=4 then sqrt(4)=sqrt(4) -> 2=2
or
if sqrt(4)=sqrt(4) then a = b i.e 4=4
I did not ignore anything but still proved both equivalent then what he is saying by ignoring domain issues?

 

[Dr.Peterson]

One or both may be negative means ? suppose there are two nos -4=-4 (both are negative) then i cannot take square root of both the numbers as they will be extended to complex no but why did he said one may be negative the equation with one negative be like this 4 =-4 (which is not valid in the first place) .
What is the domain issue?

"One or both may be negative" means that it may be true that one side is negative, or that both sides are negative.

I'm thinking there of solving something like x^2 = -4 by taking square roots, yielding sqrt(x^2) = sqrt(-4). If the domain of the problem is real numbers (that is, x can't be imaginary), then there is no solution. And the reason there would be no solution is that -4 is not in the domain of the square root function (again, assuming complex numbers are not allowed).

And I said "One or both may be negative" because (in solving an equation) you don't necessarily know ahead of time. I wasn't saying that it is possible that a = b if one is negative and the other is not.

you just have to​

determine that it is well-defined (has one value) and that its domain​

includes the values to which it is being applied​

What does this mean?

If you have an equation "a = b" and want to apply some function or operation f to it, yielding "f(a) = f(b)", you need to know that f is a well-defined function, which essentially means "single-valued" (that is, it is really a function), and that a and b are in the domain of f (that is, f(a) and f(b) are defined). Of course, you can't know ahead of time, in general, that this is true; that will be part of the check of your solution.

For example (a silly one), you couldn't apply the "function" f(x) = \(\pm\)x to solve an equation, because that has two values. That is, if a = b, it is not true that \(\pm\)a = \(\pm\)b for all choices of the signs.

Although it is true that​

if a = b, then sqrt(a) = sqrt(b), and in fact these equations are​

equivalent if you ignore domain issues​

These 2 equations are equivalent from both sides .
eg : if 4=4 then sqrt(4)=sqrt(4) -> 2=2
or
if sqrt(4)=sqrt(4) then a = b i.e 4=4
I did not ignore anything but still proved both equivalent then what he is saying by ignoring domain issues?

Of course you ignored domain issues! And you proved nothing -- you just gave one example, which was in the domain, and ignored other possibilities!

The equations a = b and sqrt(a) = sqrt(b) are not equivalent, because the first can be true for any real numbers that are equal, while the second is only true for non-negative numbers that are equal. The domains of the equations are different.

An example would be the equations x^2 - 2= 3x + 2 and sqrt(x^2 - 2) = sqrt(3x + 2). The former has solutions -1 and 4, the latter only 4 (again, if we are working only with real numbers).

Now, there are lots of subtleties in a discussion like this; that's why I am reluctant to engage your attempts to find universal rules. It is very easy to miss cases, or to make a general statement without looking for specific examples to confirm them (I had to work to find the example in the last paragraph), so there are probably errors of some sort in what I wrote here or elsewhere.

 

[Saumyojit]

both may be negative"

If the equation is both sides negative
-(x^2)= -4
_____ ____
Then √-(x^2) =√ -4 (here same domain issue)

But here -(x^2)= -4
we can also write this as
x^2=4 (taking out minus)
then we can take from both sides square roots of postive no giving us no problemn.
so is there a problem really with
both sides negative??

If you have an equation "a = b" and want to apply some function or operation f to it, yielding "f(a) = f(b)", you need to know that f is a well-defined function, which essentially means "single-valued" (that is, it is really a function), and that a and b are in the domain of f (that is, f(a) and f(b) are defined).

Understood.

you can't know ahead of time, in general, that this is true; that will be part of the check of your solution.

Are you saying that suppose I took a valid and well defined function which is closed under real domain i.e square function then you say that

If you have an equation "a = b" and want to apply some function or operation f to it, yielding "f(a) = f(b)"

Which i felt meant
a=b implies f(a)=f(b)

Might not hold true but I need to check from the solutions.(extraneous)

I took a radical equation
√x =2x-6
Squaring both sides
X= 4x^2 + 36 -24x
X=2.25 or X =4 ( two roots)

Now with 2.25 as a root I was checking
√x=2x-6 -> (√x)^2 =(2x-6)^2
first proposition is false as 2.25 is not a valid root of √x=2x-6 but working in the second equation (proposition) i.e 2.25 substituted in place of x gives both side equality .
So this is a case of false and true which doesn't affect the implication.


Now with 4 as a root I was checking

√x=2x-6 -> (√x)^2 =(2x-6)^2
Both equation is true.
So my implication a =b -> a^2=b^2 does hold true.

But if we Reverse the antecedent and consequent
Now with 2.25 as a root I was checking

(√x)^2 =(2x-6)^2 -> √x=2x-6

first proposition is true but second equation is false as 2.25 is not a valid root of √x=2x-6
So my implication a^2 =b^2 -> a=b doesn't hold true.

So what are you actually saying by

you can't know ahead of time, in general, that this is true.

What is " this".
This meant a=b implies f(a)=f(b)
I think I pretty much misinterpreted this paragraph meaning.
Sorry

For example (a silly one), you couldn't apply the "function" f(x) = ±±\pmx to solve an equation, because that has two values. That is, if a = b, it is not true that ±±\pma = ±±\pmb for all choices of the signs.

suppose my equation is
a=b
Now f(X) = +_ X
now you are saying that "+_ a =+_ b has different choices of signs" .

So when I apply the function f(X)
On a =b
F(a) =F(b)

4 arrangements arises
a=b
a= -b
-a = -b
-a = b
Now first and third same
As -a = -b if we take minus Common then we get a=b
And forth and second are same .
Its just the reverse
so I have two unique equations
a=b and a= -b
So this is wrong you are saying as 'a' has two values b and -b .Then b cannot be equal to -b which I agree by this logic!

But when we actually try to find out the solution of a quadratic equation
X^2 =4
X =+_√4
X=2 or X = -2 right??
but this solution of X are not at the same time right?
Its the context that matters.

The equations a = b and sqrt(a) = sqrt(b) are not equivalent, because the first can be true for any real numbers that are equal, while the second is only true for non-negative numbers that are equal. The domains of the equations are different.

An example would be the equations x^2 - 2= 3x + 2 and sqrt(x^2 - 2) = sqrt(3x + 2). The former has solutions -1 and 4, the latter

Thanks.

sqrt(x^2 - 2) = sqrt(3x + 2) -> x^2 - 2= 3x + 2
This implication holds true if I take X=4 as it is the only solution of first equation .
Also if I take X= -1 the first equation is not true but the second equation is true.
Although False true does not harm the implication.

But reversimg the antecedent and consequent the implication is false.

 

[JeffM]

For goodness sake, you asked me this same question in a private message. You got an answer. Now you are rehashing it in the public forum! Why do I even bother to read your private messages. Here is what I responded before. Dr. Peterson can then say where he and I seem to disagree.

First, I am not sure that the author is this site's Dr. Peterson or his brother.

Second, I am not sure I 100% agree with the explanation given. Or perhaps I should say I would not give EXACTLY the same explanation while agreeing on all the mathematics.

(Your thinking that discrepancies in words used must signify some important disagreement causes you unnecessary perplexity. There are very few disagreements in mathematics. Some people think it is more useful to include 0 as the least natural number; others think it is more useful to exclude 0 from the natural numbers and have 1 be the least of the natural numbers. The different definitions are a matter of convenience in certain abstruse branches of mathematics, but for everyday algebra, calculus, and geometry, the difference is irrelevant. Even in the abstruse branches of mathematics where the definition of the natural numbers matters, what differs because of the different definitions is the road you travel, not the final destination. You work yourself into knots on slight disagreements that make absolutely no difference to what you want to know. And this particularly goes for different ways of explaining things. There are frequently many different ways to explain the same thing.)

Third, the basic operations of arithmetic are FUNCTIONS. For a given set of arguments, a function gives exactly one result. That is what makes functions so important. The basic operations of arithmetic are functions with two numeric arguments. 3 * 5 is always 15. It is not 17 on Tuesday, 18 on Wednesday, and 15 on Thursday through Monday.
The statement a = b entails that ac = bc is a generalization of
3 * 15 = 3 * 15 to extend to all numbers.
Why then do they not say ac = ac?
Because what they want you to understand is that if a = 7 - 4 and b = 72 / 24
so that in fact 7 - 4 = 3 = 72 / 24, then
5 * (7 - 4) = 35 - 20 = 15 = 5 * 72 / 24 = 360 / 24 = 30 / 2 = 15.
If two expressions indicate the same number, then multiplying EACH of those two expressions by a second number results in two expressions that both indicate a third number. It is necessarily true because we have defined the operation of multiplication to be a function. The rule
a = b entails a * c = b * c simply says that multiplying two numbers together gives a unique result, no matter how we express the numbers.

You with me up to here? These are my words, but they mean essentially what were meant by your cited text from "the so-called addition" through "any other operation" EXCEPT I would end with "any basic arithmetic operation except division." I would then point out that there is an exception to division by zero as an introduction to the next part of the text.

Fourth, a function's definition must include a description of what numbers that specific function pertains to, those numbers being called the function's domain. An attempted use of the function with any number not in the domain is simply meaningless. But we can say that the logic that applied to multiplication, whose domain includes all real numbers, applies to any function with respect to all numbers within that function's domain even if the domain is more restricted than all real numbers. That is, if a = b and a is within the domain of the function f(x), then f(a) = f(b). And so what is described as a property of equality is better called a property of functions. But a property of functions is meaningless with respect to a number outside a function's domain. That is what the text means by "domain issues." It is highly pertinent to why discussing things in terms of a property of functions is technically better than discussing them as properties of equality.

Still with me?

Quite frankly, I would have ended the text on that point, but, to defer to the text's author, let's keep going. To show that we are talking about a property of functions, the text gives a counter example to the so-called property of equality.

It is a fact that, for any given positive real number, there are two distinct numbers that, when squared, result in that given number. And suppose we had some transformation routine, call it routine 1, that gives both both numbers. For example, routine 1 on 4 gives 2 and - 2. THAT is not a function. It does not give a unique answer. Let's see what happens if we apply routine 1 to the following valid equation

4 = 4.

We get four equations.

2 = 2 Valid
2 = - 2 Invalid
- 2 = - 2 Valid
- 2 = 2 Invalid.

Because we were not working with a function, the so-called property of equality did not consistently give a valid result. This is, to people trying to teach mathematics, an important revelation. It fully persuades me to drop the term "properties of equality." Whether it is too subtle for beginning students is something that the text's author and I may disagree on.

And this brings me full circle. Your so-called "doubts" frequently arise because the topic being discussed is not what you think it is. The text's author and I agree on all the mathematics. Where we perhaps disagree is how best to teach mathematics to beginning students and what words to use when teaching, issues of pedagogy rather than mathematics.

 

[Dr.Peterson]

@Saumyojit: Please reread my last paragraph:

Now, there are lots of subtleties in a discussion like this; that's why I am reluctant to engage your attempts to find universal rules. It is very easy to miss cases, or to make a general statement without looking for specific examples to confirm them (I had to work to find the example in the last paragraph), so there are probably errors of some sort in what I wrote here or elsewhere.

What I was saying there is that I make mistakes, especially in trying to answer questions like this that ask for broad generalizations in simple answers, and have to hit "send" without taking forever to be sure everything is perfect. I don't agree 100% with everything I write myself! In particular, the part about "one or both may be negative" is awkward and I wish I hadn't said it quite that way. Please don't try to pick it apart.

Are you saying that suppose I took a valid and well defined function which is closed under real domain i.e square function then you say that

If you have an equation "a = b" and want to apply some function or operation f to it, yielding "f(a) = f(b)"​

Which i felt meant
a=b implies f(a)=f(b)

Might not hold true but I need to check from the solutions.(extraneous)

I said nothing of the kind! Under the stated conditions it is TRUE that a=b implies f(a)=f(b); it is FALSE that f(x)=f(b) implies that a=b if f is not one-to-one, and THAT (in addition to the domain) is why you need to check.

So what are you actually saying by

you can't know ahead of time, in general, that this is true.​

What is " this".
This meant a=b implies f(a)=f(b)
I think I pretty much misinterpreted this paragraph meaning.
Sorry

"This" is what I had just stated as a condition: "that a and b are in the domain of f (that is, f(a) and f(b) are defined)". And the reason we can't know this is that we are applying some function to expressions whose values are not yet known.

The ± example is, as I said, silly, because no one would ever do that; so you don't need to analyze it in detail. I was just illustrating what "no well-defined" might mean.

 

[Saumyojit]

An example would be the equations x^2 - 2= 3x + 2 and sqrt(x^2 - 2) = sqrt(3x + 2).

Really this eg actually made a sense in understanding "you can't know ahead of time, in general, that this is true."
Although the function is single valued but due to domain issues working with that Eg creates problem.

what is described as a property of equality is better called a property of functions.

Ok that's the whole point of article.

It is a fact that, for any given positive real number, there are two distinct numbers that, when squared, result in that given number. And suppose we had some transformation routine, call it routine 1, that gives both both numbers. For example, routine 1 on 4 gives 2 and - 2. THAT is not a function. It does not give a unique answer. Let's see what happens if we apply routine 1 to the following valid equation

4 = 4.

We get four equations.

2 = 2 Valid
2 = - 2 Invalid
- 2 = - 2 Valid
- 2 = 2 Invalid.

Great explanation.
Thanks

 

[JeffM]

Ok that's the whole point of article.

I would not say that.

It is the main point of a brief note.

I think that it is an important point, and one that had not previously occurred to me. I am very grateful to Dr. Peterson for making me think this through.

From now on the only properties of equality that I shall ever talk about are reflexivity, symmetry, and transitivity.

[MATH]a = a \implies a = a;\\ a = b \iff b = a; \text { and}\\ (a = b) \land (b = c) \implies a = c.[/MATH]I have been educated.

There are several other points in the note, one about an aspect of invertibility that I had never consciously articulated.

 

[Cubist]

This topic is interesting. I think the source article might be wrong in the following section...

  (-1)^2 = 1^2

Taking the square root of each side by just canceling out the
squares, you'd get

  -1 = 1

which is not true!

Taking the square root of both sides you obtain

((-1)^2) ^ (1/2) = (1^2) ^ (1/2)

Here the LHS can't be simplified by the "power of power" rule - if you believe the set of constraints in this post(and the one after). Therefore the only way to proceed on the LHS is to calculate the inner square first...

LHS
=((-1)^2) ^ (1/2)
=(1) ^ (1/2)
=1

I'm very open to being corrected! I can't reference any "formally accepted" definition of the power rule that includes these constraints. I came up with them myself based purely on plugging lots of numbers in.

--

Perhaps on a related note, I personally believe that (-8)^(1/3) has only one value, that being 1+i*sqrt(3). I've always thought that exponentiation, sqrt, and nth-root always return only the principal root.

This doesn't contradict with the fact that x^3 = -8 has 3 solutions (three possible values for x that satisfy the equation). These values are 2, 1+i*sqrt(3), 1-i*sqrt(3). To determine these values I think we have to use De Moivre.

 

[Saumyojit]

I hope my doubts at complex post which drp and jeffm disagreed gets resolved a bit .
Dr p quotation from complex post

when you allow negative bases, and then you have to take fractional powers as multiple-valued


41/241/2\displaystyle 4^{1/2} is not identical to √44\displaystyle \sqrt{4}, and in particular the rules for exponents applied to the former don't work unless you treat it as double-valued. There are different ways to look at the exact cause of the problem, which is why I said, "that's a big part of it" rather than saying that was exactly the issue.

The radical is defined as the principal root, that is, the non-negative one. The fractional power is (at least sometimes) taken that way too, but that breaks some of the "laws".

@Cubist

 

[Cubist]

I hope my doubts at complex post which drp and jeffm disagreed gets resolved a bit .
Dr p quotation from complex post
@Cubist

I've read some of the other thread - it's quite long! I completely understand Dr. Peterson not wanting to continue that discussion. There's a lot of agreement in there. However I see there's some disagreement about the value, or values, that expressions like (-8)^(1/3) and \(\sqrt[3]{-8}\) return. I can't provide an authoritative answer myself. I've told you my own thinking which is different from Dr. Peterson's and JeffM's. But I doubt that our different viewpoints would make any difference on any exam paper that I remember because it's always been very clear from the context if more than one root is required in a question. It's a bit frustrating that there isn't a single agreed international mathematics institute with a published in-depth definition of each symbol's meanings. (Perhaps Wikipedia will become this one day?) In the meantime you can always write your assumptions or intentions in words by the side of your work if you're unsure. But I advise that you try to move on from this line of discussion now.

 

[Saumyojit]

Does (a^m)^n Always Equal a^(mn)?

Is the exponent law (a^m)^n = a^(mn) always true? Maybe not! A look at some cases where the law needs some careful interpretation.

www.nctm.org

@Cubist Did u read this .
Have you drawn any conclusion other than what I have drawn

The rule (a^b)^c = a^(b*c) should be
interpreted as: Every value of a^(b*c) is one of the values of
(a^b)^c. (It doesn't even work in the other direction, since
sometimes some of the values of (a^b)^c aren't any of the values of
a^(b*c).

this.
In wiki it is said that

If exponentiation is considered as a multivalued function then the possible values of (−1 ⋅ −1)^1/2 are {1, −1}. The identity holds

That means 4^1/2 gives 2 or -2
√4 = +_2 i.e (2 or -2 ) .

dr p also said some of the laws of Exponent will hold only if we consider x^1/2 or √x as double valued
( comment 31

Complex no

why does -3^2 translate to -9 . I thought -3^2 meant (-3)^2 which gives me 9 . Is there any connection of Bedmas with the evaluation of this expression. First, BEDMAS is a simplifying acronym designed to help beginning students. Second, the minus sign means has at least three different...

www.freemathhelp.com

)
Which agrees with Wikipedia.

Perhaps you should have noticed Dr p said in this article (https://www.nctm.org/tmf/library/drmath/view/72802.html)

if you forget that the radical symbol means only the positive root

I am not sure but this line means what I have written above to see "square root or to the power half " as a mutlivalued function (it's actually not a function!) .

U can also see JeffM comment in this post .
Comment 4 .
But I think there he didn't considered 'Routine 1' as a multivalued function but Routine1 is consisting of two different functions
Routine 1 =+_ √x which consists of F1(X) & F2(X)
F1(X) = √x , (√x >_ 0)
F2(X)= - √x , (- √x < _ 0)

I am putting a lot of effort in writing .
My back hurts but still writing coz I also want to know what going on.

 

[JeffM]

IBut I doubt that our different viewpoints would make any difference on any exam paper that I remember because it's always been very clear from the context if more than one root is required in a question... But I advise that you try to move on from this line of discussion now.

Exactly. I could not agree more with both points. This is not about fundamentals. It is about POSSIBLE slight differences in the meaning of notation. I have admitted that I may be wrong, but it is not important to understanding anything.

And no, I did not consider "routine 1" a multi-valued function, a concept I will not touch, but as an algorithm, a procedure.

 

[Dr.Peterson]

It's a bit frustrating that there isn't a single agreed international mathematics institute with a published in-depth definition of each symbol's meanings. (Perhaps Wikipedia will become this one day?)

Well, the very nature of Wikipedia implies that it can never be authoritative!

I was sent this link recently, which makes that even clearer than previous discussions here: https://www.theringer.com/2021/1/15/22232667/wikipedia-lamest-edit-wars

But I doubt that our different viewpoints would make any difference on any exam paper that I remember because it's always been very clear from the context if more than one root is required in a question.

But I advise that you try to move on from this line of discussion now.

Fully agreed. It's already been stated over and over that different contexts result in different perspectives, so any attempt at a single view of the whole thing is a total waste of time. It's not just that people disagree; it's that different mathematics require different definitions. There is no one right answer.

 

[Cubist]

I was sent this link recently, which makes that even clearer than previous discussions here: https://www.theringer.com/2021/1/15/22232667/wikipedia-lamest-edit-wars

That's a nice link, very funny! I didn't know about these "edit wars".

 

[Saumyojit]

Well, the very nature of Wikipedia implies that it can never be authoritative!

I was sent this link recently, which makes that even clearer than previous discussions here: https://www.theringer.com/2021/1/15/22232667/wikipedia-lamest-edit-wars


Fully agreed. It's already been stated over and over that different contexts result in different perspectives, so any attempt at a single view of the whole thing is a total waste of time. It's not just that people disagree; it's that different mathematics require different definitions. There is no one right answer.

this was a reply to what .
The evaluation of "power of power" rule.
That cubist posted above in post 8 and the different meanings of the notation to the power half or root sometimes considered double valued
Right.

 

[Saumyojit]

Let's see what happens if we apply routine 1 to the following valid equation

4 = 4.

We get four equations.

2 = 2 Valid
2 = - 2 Invalid
- 2 = - 2 Valid
- 2 = 2 Invalid.

Routine 1 is--> f(x) =+_ √x
And Yes this is not a function!!

what if my equation is this
x^2 = 4 instead of (4=4)
Then I apply routine 1 on this
+- √x^2 = +- √4
I will get four solutions
x= 2

x= -2

-x = -2

-x =2

Is this inconsistent in terms of results.
What I feel is that first and third are same
and second and fourth are same.
But first and second are contradicting
Right??

So its inconsistent.

@JeffM

 

[Dr.Peterson]

Is this inconsistent in terms of results.
What I feel is that first and third are same
and second and fourth are same.
But first and second are contradicting
Right??

So its inconsistent.

Among the four equations are two distinct solutions, as you recognize. There's nothing inconsistent about that; you already know this equation has two solutions, don't you? It would be inconsistent only if you said that x=2 and x=-2 simultaneously.

Are you just deliberately trying to make things difficult???

 

[JeffM]

Routine 1 is--> f(x) =+_ √x
And Yes this is not a function!!

what if my equation is this
x^2 = 4 instead of (4=4)
Then I apply routine 1 on this
+- √x^2 = +- √4
I will get four solutions
x= 2

x= -2

-x = -2

-x =2

Is this inconsistent in terms of results.
What I feel is that first and third are same
and second and fourth are same.
But first and second are contradicting
Right??

So its inconsistent.

@JeffM

Yes, you have what I was trying to convey with my routine 1 fantasy.

If you go [MATH]x^2 = 4 \implies \pm \sqrt{x} = \pm \sqrt{4} \implies \\ \text {I } x = 2 \text { and}\\ \text {II } x = -2 \text { and} \\ \text {III } - x = 2 \text { and} \\ \text {IV } - x = - 2.[/MATH]I and III are consistent as is II and IV, but I is inconsistent with II and IV and III is inconsistent with II and IV.

In short, it is a mess.

The simple way to understand the ideas and notation is to go

[MATH]x^2 = 4 \implies \sqrt{x^2} = \pm \sqrt{4} \implies x = \pm 2[/MATH]
which means

[MATH]\{ x = 2 \text { and } - x = - 2 \} \text { OR ELSE } \{x = - 2 \text { and } - x = 2\}.[/MATH]
I think this answers your private message to me as well.

Others may prefer to explain it differently or to use different words, but I do not think there is any disagreement whatsoever on the substance.

 

[Saumyojit]

which means

{x=2 and −x=2} OR ELSE {x=−2 and −x=2}.

x= 2 and - X = - 2
Or
X= -2 and -X = 2
Right??

I and III are consistent as is II and IV, but I is inconsistent with II and IV and III is inconsistent with II and IV.

I feel that
1 and 2 are consistent using "Or"
1 and 4 are consistent using "And"
2 and 3 are consistent using " And"
1 and 3 are consistent using "or"

So, when we apply routine 1 on
x^2 = y^2 (eg taken from the original article)
we get
x= y or X= -Y or -X = y or -X = -Y
So, here if we say
X= Y or X = -Y ( consistent) Then its correct .

What Dr p was trying to say that in the original article
X^2 = Y^2 doesn't imply only that X=Y as one can be positive and other negative.

And he actually brought the concept of "using a function as double valued" just to prove the above implication .
((-1)^2)^1/2 =(1^2)^1/2
here we first square the Value inside,
1^ 1/2 = 1^1/2
Now to only way we can bring -1 on the lhs if we consider "wrongly" that sqrt gives 2 value for one input.
So, -1 = 1 ( which is Wrong).

An alternative method is to cancel the square by taking square root which also gives us -1 .

 

[JeffM]

x= 2 and - X = - 2
Or
X= -2 and -X = 2
Right??

Yes, except keep x and X consistent.

And just plain disregard what I said about I, I, III, and IV. I'll ask Subhotish if I can fix it. I was getting myself mixed up with all the options from that goofy routine 1. This is exactly the reason we do not use things like Routine 1. We use functions if we want to remain sane, which these threads are making me doubt whether I still qualify. The whole thing about routine 1 was to explain why what was being called a property of equality should be called a property of functions. It is not really a thing in mathematics that is worth discussing except as an example in that particular discussion, which is not about proper notation at all.

1 and 2 are consistent using "Or"

Yes, I and II are consistent using "or" but inconsistent using "and."

1 and 4 are consistent using "And"

I and IV are consistent using "and" or "or." Of course, "and" is stronger.

2 and 3 are consistent using " And"

II and III are consistent using "and" or "or." Of course, "and" is stronger.

1 and 3 are consistent using "or"

Yes, I and III are consistent using "or" but inconsistent using "and."

So, when we apply routine 1 on
x^2 = y^2 (eg taken from the original article)
we get
x= y or X= -Y or -X = y or -X = -Y

Yes if we apply routine 1 to the original equation, we get
(x = y) or (x = -y) or (-x = y) or (- x = - y)
But no one works with routine 1. Forget it.

So, here if we say
X= Y or X = -Y ( consistent) Then its correct .

Now please, stop this.