Talk:Difference Tables

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Active Comments

Great things about this page

  • Really good scope -- you give just enough examples to explain the ideas and make it clear why difference tables are useful, without making the page overwhelming
  • Nice job walking step-by-step through the examples, and good structure for the page overall
Abram, 6/21

Archived Comments

Clean up a few places where the language is slightly imprecise or inaccurate

  • For Newton's interpolation formula, be explicit somewhere that the original sequence counts as the 0th difference.
User:Iris I made the change (6/22)
Great. (Abram, 6/22)
  • The mouse-over definition of polynomial sequence implies that sequences begin with term number 1, while everywhere else in the page, the implication is that they begin with term number 0. I don't quite know what to do about this, because it's not like there's any standard in the world at large. We can talk about this in person, if you'd like.
User:Iris I added x=0, and said that it can start from x=0 or x=1 depending on the author. (6/22)
Great. (Abram, 6/22)
Abram, 6/21 (let me know if you want help with any of this wording or have any questions about what I'm talking about)
  • Be a bit more precise with the line "The differences are found by subtracting the earlier term from the later term of the sequence." Which earlier term? Which later term? It also may help to change the language so that it's clear that the first differences (which is what this sentence is about) is itself a sequence.
IrisI made some changes to clarify what I mean. I'm not sure if I did a good job clarifying (6/22)
The new phrasing is more precise because you define what a "sequence of differences" is later in the page. The only problem is, well, that it's later in the page. Readers see the phrase "sequence of differences", but it isn't defined for another paragraph. See if you can reorganize things a bit to define "sequence of differences" before, or as, you start using it (6/22) (Abram, 6/22).
Iris(6/25) I made some changes, but I'm not sure if I understand your point. I don't see why the term 'sequence of differences' in the first place should be defined, and I'm can't find the definition of the 'sequence of differences' in another paragraph. Maybe we should talk about this
I completely agree with you. I have no idea what I was saying. The only problem I see at this point is that if you look at your current definition of the sequence of first differences, the wording doesn't require the nth term of the sequence of first differences to be the difference specifically between the nth term and the (n+1)th term of the original sequence. (Abram, 7/1)
Iris (7/6) I added another sentence to clarify this. see if it makes sense.
The new sentence is totally accurate and unambiguous. It's just a nightmare to read. I don't have any good solution, though. See if you can come up with wording you like better. If not, your options are: 1) leave this sentence as is, 2) re-write it using n and n + 1, 3) drop this sentence, even though I told you before to add it. (Abram, 7/8)
Iris(7/12) I just deleted the sentence. Yes it was really confusing.
Looks great. (Abram, 7/12)
  • Be a bit more precise with the line "can include higher-order differences that list the differences of the sequence of first differences." In fact, you have just defined the second-order differences, not "higher-order differences" as your sentence claims.
Iris I changed to second-order differences(6/22)
Great. You still refer to higher-order differences, which is a good idea. They just aren't defined now. Add a phrase defining them. (Abram, 6/22)
Iris I defined "higher-order differences".
The current mouse-over definition doesn't actually define "third-order sequence", "fourth-order sequence", etc. The mouse-over definition could instead say, "Other higher-order differences include the third-order sequences, which is the sequence of differences of the second-order differences, and so on, up to any order you'd like". (Abram, 7/1)
Iris (7/6)I added this part.
Looks good. (Abram, 7/9)
  • The sentence "Only polynomials eventually reduce to 0's" is only kind of true. Technically, you can write non-polynomial expressions that eventually reduce to 0's in their difference tables. Consider the expression  \cos(2 \pi x) as an example. It's just that given such a non-polynomial expression, there will always be a polynomial expression that gives the same values if a natural number (1, 2, 3, 4, ...) is plugged in. You can remove this ambiguity by saying instead, "Every sequence that eventually reduces to 0's can be written as a polynomial sequence."
Iris I plan to ask pf. Maurer about this today. (6/22)
OK. I am sure what I'm saying is right, so if you don't get a chance to talk to pr. Maurer, not a big deal. But if you do, great. (Abram, 6/22)
Abram, 6/21 (let me know if you want help with any of this wording or have any questions about what I'm talking about)
Iris We fixed this
Agreed. (Abram, 7/1)
  • The table showing the perfect squares sequence is currently missing the number "4" in the original sequence. (Abram, 7/1)
Iris(7/6) I fixed this
Looks good. (Abram, 7/9)
  • In the sentence "Thus, we can find the polynomial expression for sequence0, 1, 4, 9, 16... to be x^2 for x=0, 1, 2, 3, 4.", should the final period be three dots instead? (Abram, 7/8)

IRis (7/12) I fixed this

Looks good. (Abram, 7/12)
  • The current definition of a difference table allows it to show only a finite number of terms. Either this definition should be changed, or, more likely, when you explain how polynomial sequences reduce to 0's, point out that this row of zeroes is infinite. (Abram, 7/8)
IRis(7/12) I fixed this
Hmm, maybe we should say something like it *would* continue infinitely if we made the table infinite, or something. My wording is terrible, and maybe it's fine the way it is, but see if you can come up with something. (Abram, 7/12)
Iris(7/19) I can't think of a better wording. I'll just leave this as it is
OK. (Abram, 7/19)

Clean up one place where the wording is confusing

Replace "If the original sequence (the first row of the difference table) corresponds to the x values 0,1,2,..., and does come from a polynomial" with "If the original sequence (the top row of the difference table) is a polynomial sequence..." Instead of including the clarification of what x represents in this sentence, put it right after you state the the formula. You could something like, "... x_k is the kth falling factorial, and x represents the term number (i.e. first term, second term, etc, but starting with zero)."

Iris I made some changes, except I didn't include the last part about what x represents. I don't think including that is necessary
Nice job, and I agree. (Abram, 6/22)

Add a couple of animations

I know this is a helper page, but if it's easy to add an animation showing, for example, the first differences being generated for the perfect squares sequence, that could be really helpful. If it takes a long time, don't worry about it. (Abram, 6/21) Iris We decided not to do this (6/22)

Missing definition

Maybe provide a mouse-over definition of sequence (just something informal like a list of numbers, where the first one is called the "first term"). (Abram, 7/8)

Iris (7/12) I provided a mouse-over definition of sequence
Great. You could probably drop the phrase "depending on the order of the term" and just say "etc.", if you wanted. (Abram, 7/12)
Iris(7/19) I fixed this
Looks great. (Abram, 7/19)

Grammar police

  • Somewhere you have "lets" instead of "let's".
  • Somewhere else you have the line "The row of second difference gives a constant value", but it should be the row of second differences.

(Abram, 7/12)

Iris(7/19) I fixed these
Looks good. (Abram, 7/19)

Provide a couple of extra links

It's probably a good idea to link to the Fibonacci numbers page the first time you mention the Fibonacci sequence (you can also keep the link you already have). You may want to consider other places to include a link, as well. For instance, you could like to the Ulam Spiral page with the polynomial sequences, explaining that this is an example of finite differences in action. If that feels too forced or unnatural, though, don't worry about it. I feel on the fence about the Ulam sprial (the Fibonacci link is almost certainly a good idea). (Abram, 7/12)

Iris(7/19) I added a link for Fibonacci numbers
Looks good. (Abram, 7/19)

A couple of typos

  • Moving the explanation of how differences can be negative to the beginning of the page was great. The only problem is that you still have the sentence that makes that point in the Fibonacci example as well.
Iris(7/6) I took this out from the Fibonacci example
Looks good. (Abram, 7/9)
  • We've got that problem of text that says, "To learn more about finding a specific polynomial, click hide."
Iris(7/6) I fixed this
Looks good. (Abram, 7/9)

(Abram, 7/1)

Give an example or two for the perfect squares sequence

Choose a couple of the entries in that table you set up for the perfect squares sequence and explain them (e.g. the "3" in the first difference tables comes from subtracting 4 - 1). (Abram, 7/1)

Iris(7/6) I did this
Looks good. (Abram, 7/9)

One small inconsistency

You first present the sequence of perfect squares as beginning with 1, and then in the table and in your subsequent polynomial derivation, it begins with 0. (Abram, 7/19)

Iris(7/19) fixed this
Looks good. (Abram, 7/19)

One place to use simpler definitions

The definition of a perfect square could replace "integer" with "whole number", which might be a good idea, because lots of people forget what integers are. (Abram, 7/12)

Iris(7/19) I don't think using 'whole number' is a good idea, since square numbers can also refer to numbers that are squares of a negative number.
We agreed to replace "integer" with "whole number or negative whole number". (Abram, 7/19)
Iris(7/19) I made the change
Looks good. (Abram, 7/19)

Slightly edit the Newton's interpolation formula

Instead of stating in the first line that all remaining terms are 0, it might be helpful to say  p(x)=\frac{{{\Delta}^0}p_0}{0!} x_{(0)}+\frac{{{\Delta}^1}p_0}{1!} x_{(1)}+\frac{{{\Delta}^2}p_0}{2!} x_{(2)} + \frac{{{\Delta}^3}p_0}{3!} x_{(3)} + \dots

and then point out in the *next* line that all those terms are 0. That way, the first line is consistent in its "goal" of simply rewriting the general formula in expanded form, and the next line is consistent in its goal of plugging in specific values. (Abram, 7/12)

Iris(7/19) I fixed this
If you wanted, you could leave the first line as is, make the second line read
=\frac{0}{0!}+\frac{1}{1!}x+\frac{2}{2!}x(x-1)   [All further terms are 0]
and remove what is currently the third line. Or you can leave it as is. (Abram, 7/19)
Iris(7/19) I just decided to leave it as it is. It's easier to follow
OK. (Abram, 7/19)