 At the end of our lectures, we've seen a lot of writing tips that have been trying to give us. And I should mention that while some of these have to do with grammar, right? You should need to be grammatically correct. Most of the suggestions we're making are really just that, suggestions. There are situations where you might violate them with good reasons. We should try to avoid it. They're good rules of thumb. But be aware, these things are suggestions. I wanna offer another suggestion. And that suggestion is when you write in mathematics, use the active voice as opposed to the passive voice. Writing in the active voice typically makes your writing more lively. It feels more active, hence why we call it that. Also, I should mention, it typically makes things feel briefer. You can often say the same thing shorter in the active voice as opposed to the passive voice. But be aware that this is not a hard rule. It's just a strong suggestion. I mean, cause on the other hand, sometimes the passive voice can be the best composition of a sentence. But its usage should be minimized when possible. Did you see what I just did there? I used the passive voice in my sentence. There's gonna be times where you do use the passive voice, by all means. Try to use the active voice, but honestly the rule that you should be going for is clarity, okay? When you're deciding how to write something, it should be clear, right? Clarity and composition are the gold standard for mathematical writing. If you think breaking one of the rules that I've been talking about, like the active voice, actually makes your writing more clear. Like if using the passive voice, you actually believe makes your writing more clear, then break this rule and use the passive voice. These are general rules that I would suggest to you. Let clarity be your guide. What is best for your reader? So let's look at an example here. Let's take the sentence, the value x equals three is obtained through the division of both sides by five. This is an example of writing the sentence in passive voice. If we were to rewrite it in active voice, you end up with dividing both sides by five, we get x equals three. We definitely can all agree that this second sentence is shorter than the first one, much longer there on the screen. It says the exact same information. We have an equation, we divide both sides by five, we got the answer is x equals three. So we got the solution to an equation, we justified how we got it from the previous step division by five, but it's much shorter and much clearer. So this is definitely an example where I think the active voice definitely improves the clarity of your writing. Is that always gonna be the case? No, but strive to write with the active voice because generally that'll make your writing better in the long run. And so that brings us to the end of lecture 24. Thanks for watching. If you learned anything in this lecture or any of the videos you've seen, please like these videos, subscribe to the channel to see more videos like this in the future. If you have friends or colleagues who might benefit from these videos, feel free to share them with them, just share the videos with them. They will love them if you love them too. And as always, if you have any questions, feel free to post them in the comments below and I'll be glad to answer them as soon as I can.