Hopefully what I've done here will give you an intuition of why borrowing works.

Hopefully you'll get the hang of how this borrowing or regrouping, depending on how you want to view it, actually works.

Hopefully you recognize this.

And it works out the dimensional analysis and hopefully it also makes sense to you intuitively

Hopefully, you found this helpful.

And hopefully, with this blog

Hopefully you know this person.

But hopefully this makes sense.

Let's get going with more examples of function problems, and hopefully as we keep doing this, you're going to get the idea of how all this stuff works.

Hopefully in the future I can give you a seminar on actually why this method of counting the numbers behind the decimal points actually works.

And hopefully, this doesn't confuse you.

This, hopefully, shouldn't be too difficult.

So, hopefully, you found this satisfying.

Anyway, hopefully you found this useful.

And hopefully this makes sense to you.

Anyway, hopefully you found this mildly useful.

Well anyway, hopefully you found this useful.

So hopefully, this will connect everything together.

But anyway, hopefully you found this useful.

It'll always give you the right answer, and then hopefully I'll give you a little intuition for why it works.

And here, once again, we can use you could call it a trick, but hopefully, you understand why it works.

This is, hopefully, making sense to you at this point.

So hopefully this gives you a pretty good sense. And let me just do one more, just to really make sure that we really understand how all of this carrying business works.

And hopefully, you will never forget this intuition.

Hopefully, at least you find this somewhat instructive.

And hopefully, you should be able read this.

And even this will hopefully make some intuition.

But hopefully this will give you a sense.

This should hopefully be a relatively easy question.

Now, hopefully you'll see why I'm doing this.

But hopefully you get this a little bit.

Statistically, this works.

Hopefully this gives you a sense of why this property holds.

Hopefully you see that this is the same thing as this.

Now this, hopefully you recognize this as already a perfect square.

Hopefully this one will be a little bit simpler.

Anyway, hopefully you found this rationalizing the denominator interesting.

And hopefully, you're reasonably familiar with what this represents.

Hopefully, hopefully, they do it the same way.

So this works out.

And hopefully, this video made it a little bit intuitive.

I know this is messy, but hopefully you're understanding it.

But, hopefully, this will give you a little deeper understanding.

In general and hopefully you have the intuition on this.

Hopefully people will be watching this in the next millenia.