modulo division with negative power ?

[fivemack]

Originally Posted by smslca

Here i am concerned only with the remainder that it generates x may be any value and if we consider values of

then can i do the modulo operation on above example.

What do you mean by 'x may be any value'? You're talking about polynomial arithmetic, 'x' is a symbol.

To do this sort of generalisation, you do need to be quite careful about what the objects are that you're talking about; it's the better part of a one-year undergraduate class, full of terminology like 'a representative of the equivalence class containing 'x' of polynomials with coefficients in Q under the equivalence relation "A==B iff (A-B) is exactly divisible over Q[x] by x^2-3', to get the notation absolutely straight.

To get some idea of what your intuition of how these things behave goes, what would you say the remainder on division by (x^2-3) of (pi*x^2 + e*x + sqrt(37)) would be? How about x^2-2? How about x^2/3?

[R.D. Silverman]

Originally Posted by fivemack

What do you mean by 'x may be any value'? You're talking about polynomial arithmetic, 'x' is a symbol.

I presume he means 'x is any value within the domain'. The difficulty is
that the domain is unspecified.

To do this sort of generalisation, you do need to be quite careful about what the objects are that you're talking about; it's the better part of a one-year undergraduate class, full of terminology like 'a representative of the equivalence class containing 'x' of polynomials with coefficients in Q under the equivalence relation "A==B iff (A-B) is exactly divisible over Q[x] by x^2-3', to get the notation absolutely straight.

This confirms what I said: there is no "short introduction". One can't learn
this stuff "on the cheap", and there are LOTS of definitions to learn.
Contrary to what others have implied, I don't say these things to be mean.

[Arkadiusz]

Originally Posted by R.D. Silverman

x^(-29) is not a polynomial! This is stuff that should have been covered in high school.
It's a reflection of the sad state of secondary education.

In answer to your question, you first need to define the DOMAIN over
which you are working. Then you need to define what you mean by
x^(-29). It is not a polynomial. Normally when you talk about f(x) mod g(x),
f and g are polynomials are they not?? The fact that x^(-29) is not
a polynomial should have been covered in secondary school.

Your question can be answered, but the answer depends on some stuff
that is not normally covered in high school. Some honors pre-calc
classes might cover it, but I would expect this to be rare.

The answer depends on the RING over which you are working. Discussion
of rings is most often a college level topic. Any good book on modern
algebra will cover it. I can recommend some if you like.

I still remember math I learned in ELEMENTARY school.

[R.D. Silverman]

Originally Posted by Arkadiusz

I still remember math I learned in ELEMENTARY school.

Actually, noone is taught math in elementary school. We are taught
arithmetic. i.e. how to plug and chug numbers. This is not math.

[CRGreathouse]

x^(-29) is a rational function, not a polynomial. Polynomials don't allow negative (or fractional) exponents.

[wblipp]

This thread stumbles towards deep mathematical topics through seemingly simple questions illuminated by subtle masters. I fear the acolytes have gotten lost in the false starts and misdirectional banter and the masters are losing interest. Here is what the clever acolyte should see in the thread.

Peeling away the dross, this thread is like the question asked Deep Thought; the real answer is that you don't understand your question, and understanding your question is much harder than answering it.

The original, and unknowingly naive question was

Originally Posted by smslca

if f(x)modg(x) is valid(means , if it yield a remainder) then , can there be negative powers of x in f(x)?

for example
is (x^-29)mod(x² - 3) possible ?
can we do modulo division like this or is it strictly defined only for positive powers of x?

The first answer starts

Originally Posted by R.D. Silverman

x^(-29) is not a polynomial!

So MUCH is wrapped up in these few words! The blind move quickly on, but true understanding should see so much here.

The first thing to see is the word polynomial. The original poster (OP) didn't mention polynomials, so why has Master Bob mentioned them? The acolyte should ponder this. His wide ranging thoughts should at some time include "Should I have asked about polynomials instead of functions?" and "Is the answer easier for polynomials than functions?" and "How are polynomials different from functions?" The last question would have been a particularly helpful pondering because some of the false start dead ends in the thread have come from confusing polynomials and functions. The clear eyed acolyte would have spotted that functions are mappings - rules to associate one object with another one - often one real number with another one. Polynomials are expressions in and of themselves. Polynomials can be interpreted as functions through the process of evaluating the polynomial, but that's a different mathematical topic than the properties of polynomials.

The second thing to see is Deep Thought's message - you don't understand your question. This is the most important learning from the ruminations on the first topic.

Next in this first response is a discourse on the state of American education.

Originally Posted by R.D. Silverman

It's a reflection of the sad state of secondary education.

This is a recurrent theme in Master Bob's postings, but why has he inserted it here? The thoughtful acolyte will recognize this as a warning the truth is subtle and will require things his previous instructors may have failed to teach him. The lucky acolyte will recognize this and rebound to the first sentence, discovering the wide ruminations he missed the first time. The unprepared acolyte will see an insult or an invitation to unhelpful side conversations.

Next Master Bob is uncharacteristically direct. He is well known for exhorting questioners to mathematical sophistication by mentioning complicated text books they could study. But here he explicitly lays out what essential information is missing in the problem formulation.

Originally Posted by R.D. Silverman

you first need to define the DOMAIN over which you are working.

Master Bob then goes to a nearly unprecedented level of helpfulness by describing a likely domain and why the selection of this domain renders the question nonsensical:

Originally Posted by R.D. Silverman

Normally when you talk about f(x) mod g(x), f and g are polynomials are they not?? ... x^(-29) is not a polynomial

Continuing in this untypical effusion of helpfulness, Master Bob then names the area mathematics necessary to render the question sensical:

Originally Posted by R.D. Silverman

The answer depends on the RING over which you are working.

Now the genuine acolyte should be asking "What the heck is a ring?" and "What are my choices for the ring?" He should have learned from his friend google-san that a ring is a set with two operations, usually called addition and multiplication. Addition behaves much as the word leads us to expect - closure, associative, commutative, existence of an identity and an inverse. Multiplication is not required to match expectations so closely. It must be closed and associative, but commutative is optional and an identity is optional. Addition and multiplications are required to be distributive.

Given all these hints, the acolyte should be pondering "What ring includes both the polynomials and x^-29?"

While this learning should be going on, Master Tom worried about the acolyte missing another issue. Recognizing the subtle Socratic style initiated by Master Bob in this thread, he provided direction with this:

Originally Posted by fivemack

What would you say ((x^2)/3) was equal to modulo (x^2-3) ?

Continuing his forthcomingness, Master Bob bantered back both a hint an a reminder that the acolyte should be thinking about rings:

Originally Posted by R.D. Silverman

First define 1/3 for the ring

Next the acolyte demonstrated the ability to consult google-san and the understanding that the question was nonsensical if the ring under discussion was polynomials.

Originally Posted by smslca

I read it in wikipedia .... the above example i have given is not possible.

The acolyte then stumbled onto one of the answers - he found the minimum ring that incudes both polynomials and x^-29. Although not expressed with mathematical sophistication, he said

Originally Posted by smslca

cant we continue the division for negative powers and display the remainder involving negative powers.?

I don't know if there is a standard name for this ring, but it would be finite sums of coefficient * x^n, n taken from integers (negative as well as positive).

I'm surprised the masters have not taken up this comment. Perhaps they are waiting for a more complete response. Master Tom's point seems to have been completely missed. When the ensuing side discussions and dead ends continue to ignore his hint, he returned with a less subtle hint

Originally Posted by fivemack

what would you say the remainder on division by (x^2-3) of (pi*x^2 + e*x + sqrt(37)) would be?

This should steer the acolyte back to Master Tom's original teaching. From these two hints, the acolyte should have noticed that "ring of polynomials" is incomplete because the permitted coefficients have not been identified. The first question pointed to the possibility of rational coefficients rather than integer coefficients; this second question points to real coefficients as another possibility.

Master Tom then leads towards even deeper mysteries with

Originally Posted by fivemack

How about x^2-2? How about x^2/3?

The thread then seems to be filled with side discussions and dead ends. I'm unclear if the masters are waiting the acolyte to demonstate additional learnings or if the acolyte is waiting to the masters to provide additional subtle direction. I'm fearful the side discussions and dead ends will drown the sophisticated Socratic tutoring. I'm hopeful that this summary will reignite and retrack the mathematical discussion.

[LaurV]

@wblipp: Genial! I also saw the topic in this light, but you put it impeccable on words.

[R.D. Silverman]

Originally Posted by wblipp

The thread then seems to be filled with side discussions and dead ends. I'm unclear if the masters are waiting the acolyte to demonstate additional learnings

I got the (perhaps mistaken) impression that the O.P. had lost interest.

The next question he/she should be asking is: "What is a ring and why do
they matter"? I hinted at this when I mentioned the word "field" and discussed one of its properties.

And a lot of what you call "side discussions" consists of gibberish. I'm sorry,
but the people presenting this gibberish should clear out.

[R.D. Silverman]

Originally Posted by R.D. Silverman

I got the (perhaps mistaken) impression that the O.P. had lost interest.

The next question he/she should be asking is: "What is a ring and why do
they matter"? I hinted at this when I mentioned the word "field" and discussed one of its properties.

And a lot of what you call "side discussions" consists of gibberish. I'm sorry,
but the people presenting this gibberish should clear out.

There is a lot of stuff that might be discussed here in an elementary way.
For example, "Why did I mention polynomials"? "Why do they matter"?
"Why not other functions"? "Why does the domain matter so much"?
"What is the role of 'closure' and why does it matter"?

The question of "What is f(x) mod g(x)" also gets us into questions
such as "What is division, really?" "When is division defined"? etc.

Much of this should be covered in secondary school algebra.
Unfortunately, too much of what is taught is just "rote manipulation"
and solution of "canned" problems. Discussing the reasons behind
the mathematics that is taught is almost never taught. The result is
that we get a very high percentage of students who enter college but
are unprepared for college level math.

I ask the O.P. to consider the following (deliberately somewhat vague)
question: Can you tell us the fundamental difference(s) between a
polynomial function and a function such as sin(x)??? Why are they
so different? Why do we not see questions such as "What is sin(x) mod
(x^2-3)".

And would someone please confine sm88 to posting his "stuff" to
the misc. math. thread?? He isn't helping at all, and will confuse the
accolyte.

[R.D. Silverman]

Originally Posted by wblipp

Next in this first response is a discourse on the state of American education.

This is a recurrent theme in Master Bob's postings, but why has he inserted it here? The thoughtful acolyte will recognize this as a warning the truth is subtle and will require things his previous instructors may have failed to teach him. The lucky acolyte will recognize this and rebound to the first sentence, discovering the wide ruminations he missed the first time. The unprepared acolyte will see an insult or an invitation to unhelpful side conversations..

In fact, I have already been accused of insulting the O.P.
Someone sent me a private message to that effect, even though I
was quite careful (or so I thought) not to make any personal remarks
in my response. Lamenting secondary school education is NOT a
reflection on the O.P., but of course someone chose to interpret it
in exactly that way. This illustrates what I have been saying --> some
people seem to go out of their way to find offense when none is intended.

[xilman]

Originally Posted by wblipp

While this learning should be going on, Master Tom worried about the acolyte missing another issue. Recognizing the subtle Socratic style initiated by Master Bob in this thread, he provided direction with this:

Herein lies a problem, at least as far as I see it. You, I, Bob, Tom and doubtless others reading this thread recognize the Socratic style when presented and we use it when attempting to educate others. Unfortunately, Socratic teaching is not that widespread and it is almost unused in primary and secondary educational establishments. At least, that is my observation.

A novice unused to this technique seems to interpret questions which are intended to indicate a course of enquiry and self-education as if they are belittling the novice in the eyes of on-lookers. The impedance mismatch then tends to cause more heat than illumination.

Perhaps some of us --- Bob and myself, amongst others --- should try to keep this phenomenon in mind and to tailor our use of the Socratic method of education to our likely audience. That's not to say we should avoid it --- certainly not --- but to recognize its limitations when the subtlety is beyond the intended audience's present level of sophistication.

Paul