That's usually not the case in a real bank.

Periods per year

Periods

But I'm just going to keep it a simple example, compounding annually.

We get r is equal to 0.23, or 23 .

Compounded for 64 years.

In the last video, I hopefully showed you that if I borrowed P dollars, and I borrow it for a year, and you were to charge me an interest rate of r, or you could say 10r , then at the end and we were to compound continuously.

So t is equal to three years.

I borrow 50, and let's say it's continuously compounded at some rate r, and let's say it is continuously compounded for 10 years.

I'm just multiplying this times itself.

That equaled 2.613.

Just to clarify.

And we raise that to the n power.

At the end of 10 years, I owe 500.

I don't know why I put brackets around that.

We said it's essentially 100 interest, but we're dividing it by 12 per month, but we're compounding it 12 times.

166 effective interest rate.

It seems to be approaching, right?

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

So if I do plus 1.000114 to the 8,760th power 2.71443.

You're actually compounding continuously.

100 divided by the number of times you wanted to compound in the year.

And then once you type in all of these things, the monthly mortgage payment is calculated.

So you see, depending on what the values of these are (And we could even do more complex things.)

So now, in this situation. we are compounding the interest.

I'm just formalizing it with a couple of variables.

That equals my final payment or how much I owe once all the interest and the principal has compounded.

You're compounding every zillionth of a second.

So how much are you going to have to pay me at the end of a year?

Let's say that and I think you'll understand why I'm doing the substitution let's say that r over n and let's say I want to find the limit as n approaches infinity, as I compound continuously.

It has 5 above and 5 below, and that happens to be a 3.

A lot of people, you know, when you hear 10 interest rate or even a 25 interest rate, that no one really makes a big deal about it.

Let's say this is what we assume is how many cupcakes we sell.

By the end of 5 or 6 or 7 flips, there will probably be and later on we'll actually figure out the probability that there is someone who has a streak.

So that becomes the new P, so I could write Pe to the r, and it's going to and that new principal is going to compound for another year.

My alltime hero who won first place for 397 consecutive weeks...

And we'll focus on that a little bit later because that's a little bit more nuanced relative to the income statement.

If you have an algorithm that just guesses the answers, that algorithm is going to wrong some other time and we're not talking about the complexity of algorithms that are allowed to be wrong.

Some badly made Python interpreters might raise an exception on this, but it should not.

Specify years for the calculation in the input file.

But when you compound it, and especially when you compound it continuously, it can very quickly turn into very, very large numbers.

We can offer people the grand continuity of experience and personal investment.

So we compound every zillionth of a second, but we compound it a trillion times, however many of those intervals are in a year, that at the end of a year, I would owe you P times e to the r dollars.

Perform symbolic and numeric calculations using Maxima

My head I think implicitly made the numbers work out nicely.