What is the number "e" and where does it come from? - YouTube

Okay, you got a fact matter that yes, okay? Good. You will need it in a second now

when you hear the word exponential

There are a few different options for this, but what what's the next word [that] you expect to come in the phrase growth?

I think is the right exponential growth, and that just means something's growing really fast, right?

Well, there's a specific kind of exponential growth that there is and it comes from this

Okay now

Exponential growth doesn't just mean grow that's very very fast

Okay, it means a particular kind of growth. Let me put it to this way right, x squared it grows fast right, but

X cubed grows faster and you know x to the 300 grows faster, okay?

These all grow very fast, but [they're] all the same kind of growth. We call them

Polynomial growth none of them no matter how ridiculous this number gets?

none of them are as fast as

Exponential growth because [it is] doing something different, okay?

The idea behind exponential growth is a population right the larger population gets

The faster it can grow right now. Make sense like rabbits. You know the more that there are of them right the more quickly

They can reproduce. It's not just all they'll keep growing faster

It's by their very nature the more rabbits there are the faster

They can make new families, and that's exactly the same thing is happening with money

the more money you have

The more interest you can make which means more [money] you have right so in other words

what distinguishes the E cigarettes even it's as fast is that this is going to get ever faster and

It's proportional to how big something is so the bigger

It gets its just gonna go crazy okay now think about this with me. Just think about a particular

[rat] suppose. I just have a dollar right

I don't know if you guys [realize] this [but] if you have five dollars right, and you have some interest rates on the other time

That five dollars is going to earn exactly the same amount of interest as a dollar he a dollar here dollar he dollar here

Here, okay

Do you get?

Fired backing out over the dollar of them and just by putting them in a pile of five dollars

They don't magically throw more interest you go ahead and hand you check out the numbers exactly that's it. Okay, so therefore

I'm just going to consider the case [of] one dollar because every dollar in my bank account. They're all the same

They're all going to behave exactly the same way. It's got what's them doing the same thing

Okay, now let's suppose

[there's] [a] very very generous bank account which gives you a hundred percent interest, okay?

Now this sounds [very] generous doesn't it but why do you think in terms of the real world actually?

It's not that unusual [think] about a population right if you've got like

Rabbits they literally can double right and actually doesn't take them a year to do it okay, so this is not that unusual, right?

So let's consider one year a hundred percent interest okay, so this is number one, okay?

What will happen to my dollar what will it become?

You'll become two dollars and the reason why is because if we one plus the rate hundred percent. Which is just [1]

^ 1 so you get two

Okay, now as you know though

At least if you're dealing with money right they don't wait for the end of the year to pay your extra dollar okay?

Actually, they're calculating interest constantly and the more they calculate it your money you get right?

So let's do a bit of an experiment number two

Suppose instead of waiting for a whole year. They do it say twice a year

Semester level okay now. Here's what I'm Gonna have to do

I'm still getting a hundred percent interest per annum right so the father is not going to be this

great how must I adjust this formula there are two compounding periods

but here's something - what's going to divide by 2

[so] the [rate] instead of getting a hundred percent at the end

I'll get 50 percent and then I'll get another 50 [or] seven so I get yes that make sense, so

[what's] that equal to?

That's yeah one huh is going to be 2.25 right nine and [a] four

To a quarter, so this is better. This is good right? Well, why just do it once every semester

[why] don't you do it every month? What formula should I write?

1 plus 1 [over] 12 to the 12 okay calculators. Tell me what you get

two dollars sixty three sixty six one

[Channel] 61

Okay, now this is interesting look when I doubled [I] got an extra 25 cents

It's not bad sex with [10%] plus right for [rita] here. I've gotten more frequent but I'm still gaining more, but actually

Thanks. Good. Even more than this right. They'll do it every day for you right, okay?

It's number four yeah, they won't pay it every day

But they'll calculate every day I tend to pay monthly so what equation should I write?

One plus one on 365 and every day

They're going to go and calculate the interest she's tiny tiny time to point now. Give me some more decimal places

Okay, all right now. What's going on, right?

I would say from monthly to daily you've increased your rate of compounding dramatically

But all you've gained is like a piddly [ten] cents. All right well just for the sake of it

Maybe you think there's something wrong with their computers forget it every day. We have favor fast computers now. Let's do it every

minute okay another fine

So let's go one plus one on every day 365

How many hours 24 how many minutes 60 whatever that is okay?

365 24 60 someone's calculator

guess what spoilers it's

2.71828

blah blah blah blah

What happened?

Now here's the thing right? I didn't even get a whole set out of it right. [I] suppose [you] can round up

What happened? Well you know what like all of these these are just arbitrary?

Okay, what mathematicians do is they say forget all of this for days months hours you and your feeble units of measurement

They just say well

What happens if

you just say

infinity?

Right now in math. We call this a limit because of course you can't actually do this infinities, not number. It's just an idea

All right, but you can take some numbers. You're pretty close. You can take like

a million or something like that

Whatever large number you like I do it really angry. You did a trillion, and I'll tell you right now

You're gonna get e dollars. Okay? This number

Just like pie

I didn't say oh it better be a certain number

I better get [there] right it comes from a structure [that] exists in nature namely how things grow

when they grow in relation to their own size or if you like how things

Decay when they decay with relation to their own size

half-lives those kinds of things radioactive decay

Temperatures that come down the hottest something is the faster it loses temperature, okay?

This is this is the number of proportionality or proportional growth or proportional decay

and that's why it's so important right because that's why it appears in your calculator because

It's a number that exists in reality. Just like Pi does