 welcome back. And in this video, we're going to finish the process of writing up a proof for a theorem, a result that we now know to be true. We're going to be converting from a no show table to a paragraph form style proof. Now, why is this important? Why can't we just stop at a decent no show table, which like we saw, here's the no show table from the previous video where we were proving that if n is even then n cubed plus n squared plus n plus one is odd. Why can't we just stop with this? Well, people do not necessarily read tables. And if you handed this table off to appear, there would be a lot of questions. Okay, so when we communicate in mathematics, which after all is the title of this course, communicating in mathematics, we want to do it in a way that is makes it as easy as possible for the reader to understand what we're getting at here. So we want to convert this into a format that is a little that is as familiar as possible and leads to the fewest questions that would be more like a paragraph form where the reader can start at the beginning continue through through the middle and end with the conclusion. So the way we're going to do this, this is a little awkwardly set up on the screen here. First of all, it's a really big screen. So what I would suggest technically is to go down to the lower right of your YouTube screen and select the highest possible resolution for this video, which ought to be 720p resolution and then make it full screen. That way you should be able to capture all the action and make everything legible here. So I'm also I want to mention that I'm using a straight text editor over here on the right. I'm not using latex or Microsoft Word equation editor to do any formatting at this point. We'll deal with the formatting of variables and equations and so forth when we get there. Alright, so in your book, you read about six basic mathematical writing guidelines. We're going to be adding to those as we go through the course. But we'll see each of these six things play out as we just write up this simple proof. The first guideline is to begin with a carefully worded statement of the theorem or result to be proven. So what was it we were trying to prove? It's not totally obvious from the table unless you look at the first line and the last line. Okay, so what we were proving here and we're going to call this a theorem now a theorem is just a fact that has a proof is that if n is even then n cubed plus n squared plus n plus one is odd. Now in formatting this in a text in a in latex or in a word processor, I would probably boldface and underline the word theorem to say, look heads up, this is important. One of the other guidelines, the fourth guideline you read about in your book is that whenever you see a variable, you should use italics in the word processor. So in a word processor, you should italicize the n here. I'll do that here at the very beginning of this theorem. But I won't continue this through the end just to save some time here. So that's what the theorem should look like. And also the exponents up here should really look like exponents. But this gets into some type setting issues I don't necessarily want to tackle right now. I just want to worry about the content. So we'll kind of play it fast and loose with with math formatting for right now. Anyhow, what we do have here is a very clearly stated statement we're trying to prove if n is even then n cubed plus n squared plus n plus one is odd. Okay, so then we're going to begin the proof and I will say a good way to begin a proof is to say proof colon. Now it's bolded. We're going to begin the proof with a statement of our assumptions. Okay, so our assumptions in this case that happened in the very first line. So we're going to say assume that n is even done. Okay, so the reader now knows exactly what's on the table here. We're assuming that n is even. And there's nothing else that's hidden. Now as we flow through the proof, we didn't write this proof over here on the left from top to bottom. If you remember, we wrote the first line, then the last line, and then we kind of bounced around making forwards and backwards steps to complete the proof. We did not complete it in the top down linear progression. That's for sure. However, when we write it up, it does need to be a top down linear progression that makes sense. We don't expect the reader to start at the first line and at the last line and kind of read in between the lines. We want to kind of flow from one point to the next. It would be like a story. A story has a beginning, a middle and an end. And we've got the beginning down in front of us. So we've begun with a carefully worded statement of the theorem, and we have stated our assumptions. So now we're going to progress through the proof and just kind of explain what's going on here. So the next line says n equals 2k for some integer k. It's because of the definition of even. So the way we might phrase this, there's certainly more than one way to do this, is to say that since n is even, there exists an integer k such that n equals 2k. So we don't just say n equals 2k. We say, we set it up. Notice we're putting the reason first and then the statement that we're gonna make here. So we basically phrased line p1 in the table as a sentence that's well constructed, has correct grammatical structure, punctuation and so forth, and gets the idea across both the statement and its justification. Very important. Every time you make a statement and approve, it must be justified. We've got the justification built into the reason column in our table, but that has to translate over into the paragraph as well. Now let's proceed. So what happens next in the no show table, there are three lines of equations here. So the way we might set this up, and this is another guideline, the fifth one that you read about in your book, and that is display important equations and mathematical expressions. We could make an argument that we should have displayed the equation n equals 2k, set it off by itself. And I didn't do that, that was just a judgment call. But I think it is important to set off the following line of equations. We might say something like then we have, and then I'm going to make a break here and in latex or in Microsoft equation editor, this might be a centered displayed math stack of equations here, I'll do the best I can with the straight text editor. Okay, then we have n cube plus n squared plus n plus one equals 2k cubed plus 2k squared, oops, plus 2k plus one. And then I'll just kind of artificially put some space in here to line up the equation. This is important to lining up your equations along the equal signs and do the math here 8k cubed plus 4k squared plus 2k plus one. And then I'll do another calculation here. And that is to pull out the two that was line P four in our table. So 4k cubed plus 2k squared plus k plus one. And I'll put a period at the end because this is like a sentence that I'm writing, and I'm done with it. A couple of things to note here, we have offset and displayed these important equations. This is a progression of mathematical statements that looks better and reads better when it's offset than what if I tried to string it all out over multiple lines within the middle of a paragraph. And also notice that we use the pronoun we, okay, then we have, okay, this is a standard mathematical writing practice and it helps to include the author in what's happening and it's just good writing style. Now let's try to finish this off here. I get back to the first line here. And we will say, by the closure of the set of integers under addition and multiplication, we know that, and I'm just going to pull this out as a copy and paste, we know that this expression is an integer. And note that I'm being very careful and very detailed about what I mean by closure. Over in the table we just said quote unquote closure. But what I really mean to make this totally precise is closure of this set under this operation and this operation. Alright, now let's continue. Therefore, we're starting to wind it up here, wind it down to the end. So therefore, we know that n, let me say that. Therefore, we know that n cubed plus n squared plus n plus one equals two l plus one for some integer l. Namely, l equals the expression that we saw a minute ago. Therefore, n cubed plus n squared plus n plus one is odd by definition. Okay, so I hope you're following through the prepositions and therefore is and so forth. We just want to make it flow and make it match the flow of the no show table but in a human readable way. The final thing we need to do here in our writing guidelines is to tell the reader when the proof has been completed. We can do that in a couple of ways. We could say we really we have done this right here in the last line. And then it's often common practice to put some sort of a symbol at the end of a proof like a little square. You'll see this in your book or maybe like a couple of double hash marks or something like that. Can't do too much in a text editor but you might put a little drop square to say the proof is over. Okay, so there's our theorem and our proof and we've abided by all the guidelines. And the most important thing here to do is to practice doing this process on your own and in class. Thanks for watching.