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Er ... So but before we go to Taylor series and you complain about complexity of that ...
... I wanted to talk about numbers.
About all the usual things you are dealing with on daily basis.
So er ...
For the numbers we use decimal presentation, right? And that's what everybody is used to.
Er ... so the number e ...
[...] just to take something ...
... unusual.
So the number e is familiar, right? To everybody. So ...
Two point what?
Seven.
That's it?
Is that how familiar that number is?
[...].
Don't worry about the arrow [...].
So er ...
Well that's something I memorized five minutes ago.
Er ... I don't [...] in high school but ...
The point is not to remember a lot of digits.
The point is that er ...
The meaning of all that is very, very precise.
And the meaning is in terms of powers.
And that's where this ten comes from the ...
Base ten for our system.
So it is two times ...
... ten to some power.
Right? And I mean this two.
And that power is zero.
Plus seven, the next digit ...
... times ten to the power minus one.
Plus that next digit one times ten to the power negative two.
Plus eight times ten to the power negative three.
And we go on with decreasing powers of ten.
Or increasing powers of one over ten.
Right? And if you go left ...
... if the numbers greater than ...
... well has more digits on the left ...
... you will go by increasing powers of ten.
So we can go both ways.
And that is something similar to Taylor series.
Right? In Taylor series you go by powers of x minus x zero.
And that can be thought of as base for Taylor series.
And then what we rarely do in real life is we rarely change the base.
Do you ever go to the different base?
Does anybody do binary numbers?
Yes, computer science yes. I understand that.
So the binary ...
So what is that about?
Does anybody even know what is about -- binary presentation of numbers?
How would you present e binary?
What can you use to write e in binary presentation?
Student: [...]. NB: Zeros and ones.
Right? That's the only digits that you can use.
Binary means two, right?
Two allowable digits.
And that means you have to write e ...
... using powers of two.
And the digits used are going to be either zero or one.
So it should be something ...
... times two to the power ...
Well two to the power two is a little bit greater than e so you don't use that.
But just in case we can write that.
Plus something times two to the power one.
Plus something times two to the power zero.
Plus something times two to the power minus one.
And so on.
Now what should be the digit in front of two to power two?
Well that's clearly zero, right?
So ...
That's the first digit.
Well not the first one because you can imagine a lot of higher powers of two on the left.
But you usually skip all those zeros to the left.
So it begins with two.
Right? So two has ...
... a digit there.
And then two to the power zero.
Which digit should we put there?
We already have two here.
And we have to make that number.
How many ones should we add?
Well there is no such a digit as point seven, right?
So we have to add zero once.
And how many halves?
Exactly one. Right?
And that will make two point five.
And then how many quarters?
Because the next one is going to be ...
... something times two to the power negative two.
So that's about quarters.
How many quarters should I add ...
... to two point five ...
... to get closer to e?
Zero or one?
If I add one I will get to two point seventy five.
And that's already greater than what I want.
So I should not add any quarters.
But then I'll probably add eighth.
Right? And this is how I would make binary presentation. So the ...
Well, it'll be in blue.
So the decimal point should be where?
Because we usually put decimal point somewhere.
Right? Well, one of course.
Where I should go in sequence I have?
One, zero, one, zero ...
... probably one.
Where should I put decimal point?
Student: [...].
After first one?
Student: After first zero.
Well that's important, isn't it?
Isn't it important to know where decimal goes?
So where I should ... How do you know it goes there?
Can you decide by looking at this expression?
How do you decide?
Student: [...].
So when the power goes to the negative, right?
Right before that you put the point.
So when the power goes to the negative right before that one you put the point.
So e in binary is ten point one zero one [...].
Right? So ...