 In proof writing, we often use the following commonly used adverbs, things like thus, hence, therefore, consequently, ergo. We might, we could throw a few other examples in there like so, also, and this list goes on and on and on and on. These adverbs are used all the time in mathematical writing. And we use them for basically two reasons. The first reason is that using these words generally improves the composition of our writing. And honestly, when you look at these words for the most part, they're synonyms. Like what does thus mean? What does hence mean? What does ergo mean? Well, ergo is a Latin phrase, a Latin word that means therefore. And so these are all basically synonyms. So we have lots of these different synonyms because by adding variety, it can change how our sentences are read and that can generally be good for composition. All right, so we have lots of synonyms to improve writing. That's why one has a thethorus sometimes saying the same thing in a different way can improve the writing, improve the composition. So that's the first reason. But the main reason that we use these adverbs in mathematical writing is that it suggests some type of implication. When we say something like thus, thus the result follows. That's a sentence that people say improves all the time, the result follows. This word thus is suggesting that there was something previous that was mentioned. And previously there was this implication that you had some statement over here that because it's true, then this one's true as well. So that these words thus hence therefore, they suggest some type of implication that P implies Q. So we could say something like thus, Q because P happened earlier. So P holds thus Q holds. This thus is suggesting the implication here. But it's not just implication here. It's also suggesting that because this thing is true, this thing is gonna hold as well. And so we use these words all the time in mathematical writing. Feel free to diversify how you phrase things. For me personally, the word therefore feels much more like conclusionary. I like to reserve therefore for like big statements like this is the main idea. And I like to use things like thus and hence for more casual ones. But you can do it however you want. It doesn't make much of a difference there. So let's consider the following example below here. What's going on here? So if you had the sentence therefore, two K plus one, this is not a really good sentence. And that's not because I don't know what comes before it. There's an implication here. So therefore suggesting that because of the things that happened before it, it must follow that K plus or two K plus one holds. But the thing is the issue here is that this isn't a statement. Two K plus one is just a mathematical expression. We can assume there's things that provide this, but whenever you have a therefore or a hence, it needs to be followed by a statement. And so, it should say something like therefore A equals two K plus one. That would in fact be a statement. That would be a possibility here. I should also mention that whenever we're using things like thus and therefore, because it's suggestive of implication, the thing that comes before should actually imply the thing that comes after it. If there's no implication between this statement and the statement beforehand, you probably shouldn't use things like therefore consequently because they're not related to each other. So be mindful of how you use these type of implication adverbs in your mathematical writing. And with that, I should mention that this is our last video with regard to our mathematical communications. As a reminder in this lecture series, every lecture contains three videos. The first one about mathematical topics. The second one about logical topics. And the third one is about mathematical communication. So while we're not at the end of our lecture series yet, we are arriving upon the end of the mathematical communication sub-series. And just so you know, we won't have any more of these going forward. That brings us to the end of lecture 29. Thanks for watching. If you learned anything or liked anything in these videos, please like them. Subscribe to the channel to see more videos like this in the future. And as always, if you have any questions, post them in the comments below and I'll be glad to answer them as soon as I can.