Talk:Induction

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Hi, just a quick suggestion - I scrolled through this page without reading it, and it seems to me that the page might be more readable if you indented or centered your equations. Just something so that the whole thing wasn't all left-aligned. - Kate 16:37, 25 June 2012 (EDT)

Ok thanks! I'll look at the layout! Jorin 16:59, 25 June 2012 (EDT)


--Cwang3 09:52, 20 July 2012 (EDT)
Hey, Jorin.

  • In the first two sentences in the first paragraph, ("Induction is a useful method of mathematical proof ordinarily used for statements that involve a sequence of natural numbers. The principle of induction first involves proving a claim for a simple case, the first one in the sequence."), there are two "involve"s and I think you can get rid of/change the second one.
  • Don't forget about the formatting issue in the examples section. All the summation notations on the left side of the equal signs are kind of packed.

Besides these two, I think the page is ready for final review. Nice work!

Chris 10:56, 6 July 2012 (EDT)

Basic Method

Outline for an induction proof:

  • 1: Remove the explanation sentence "This just means..." Your statement can stand alone.
  • 2: Move "P(1)" from after "that is true" to before it.
  • 3: I don't understand the last sentence in the first paragraph beginning with "We do not only..."

Consider adding information about what kinds of equations are best suited for the induction method, which would make an effective transition into the example section.

Example

You have two examples, so I would call this section "Examples" and then put numers and subheadings for the two examples.

Peter, 7/17/12: I agree with Chris. As is, the horizontal bar after the first example doesn't do enough to indicate you are moving on to a whole new example.

Example 1:

  • P1 If you haven't added the info in the transition section above, consider changing the first sentence to: "The equation for the sum of the first n natural numbers is an excellent example of one that uses induction as its proof method."
  • P2 Remove "This formula is the arbitrary proposition." Change "Now" to "First"
  • P3 Remove "We have proved the base case." Change "Now" to "Next"
  • Nice job on the proof.
  • Right below the proof, S2: Remove "the truth of" before "P(k) implies P(k+1).
  • P2 below the proof: I think that your proof elegantly demonstrates these concepts and does not need the explanation. I'd delete this paragraph.
  • P3: Remove "Second,"

Example 2:

  • Remove sentence 1: "Here is another..." and "In arithmatic"(sic)
  • I would reorganize this section and put the general distributive property in summation notation second.
  • P7: "We are going to...controls how big the sum is." Change to "controls the number of terms in the sum."
  • In the induction proof, step 3: "Now let... = a" Does the sum really equal a?

Chris 08:31, 9 July 2012 (EDT)

Variants

P1S4 Remove: "Now, if we showed that P(k) implies P(k+2) with two base cases" Begin the sentence with "P(1) would imply..."

P2S3 Remove "to" between "then we need" and "the base cases"

Induction on Negative Numbers

Remove S6: "Thus, we also have induction for negative numbers."

Strong Induction

P1S1 After the word "imply", do you mean "that" instead of "the"?

P1S4 I would use "common" instead of "weak" for the form that is not called "strong".

More Examples

Nice idea to make this interactive and have the reader try it out for themselves.

1. The first two sentences in the answer are not necessary. Simply begin with "We can use either..."

Have the inductive case equal b, not a.

2. P2S1: "Just as a fun fact..." This is more than a fun fact; this is what turns out to be Nicomachus's theorem. It IS lots of fun and I like your enthusiasm (as I enjoy it throughout the page.

Step 2 of the inductive case: The last number in the numerator should be 1, not 4. (Just before the closed parenthesis).

Paragraph below the first four steps of the inductive case proof, S2: Change "sane" to "same."

Second part of proof. I think you can eliminate the line where it is written by itself, since that is repeated in the next line before the changes begin. Let's talk about how best to format this (whether you need to repeat the left side each time, etc.).

3. No suggestions.

4. S1: Add "as" between "0" and "counts."

Below the 4 base cases, S2: Add "is true" after "then P(k+4)" S3: Change "that case for k+4" to "the case for k+4." S4: Change to "This method is discussed in the 'Variations' section."


Chris 14:10, 17 July 2012 (EDT)

Here's a shot at the opening few sentences:

Induction is a method of mathematical proof typically used for statements that involve a sequence of natural numbers. First, a conjecture (or claim) is proved for a simple example, called the base case. Next, it is proven that if any statement is true, the next one in the sequence must be true.

Chris 13:31, 18 July 2012 (EDT) I think your opening paragraph is now fine. Do consider whether you want to talk about its typical application to sequences of natural numbers. I would remove "all of the other" from sentence 2, leaving just "showing that the more complicated cases follow."

Hey Chris. I really like those opening sentences you posted. Maybe I'll try to work some of those ideas in. Jorin 13:58, 18 July 2012 (EDT)

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