In this video, we're going to talk about 
how computations are formed on computers. 
And, in particular, how numbers are 
represented on by computers. 
Turns out this is going to be very 
important for us to figure out how 
computers function so we can convert from 
analog signals to digital signals, 
understand the nature of the errors that 
are involved, and also the nature of the 
errors that are involved in when you do 
computations on digital computers. 
So, the first thing we need to talk about 
is what's called Positional Notation. 
This amounts to something we're very 
familiar with. And it turns out, how we 
deal with numbers is very similar to how 
computers deal with numbers, at least in 
a conceptual way. 
We'll talk about Base-2, which is the 
number system that computers use, and I'm 
going to emphasize that knowing your 
powers of two is going to be very, very 
useful skill to have. 
So, let's talk about positional notation. 
Now, I claim that one of the critical 
discoveries in mathematics was positional 
notation. 
So, what positional notation means is 
that we have a number that we would 
write, 257. 
What that means is that the 2 means 
2*100. 
So, 2*100 + 5*10 + 7*1. And, the 
so-called base of the number 
representation has to do with the powers 
they're involved here. 
And the word position comes from the fact 
that if you start counting from the, the 
right end, the position tells you what 
the exponent is. 
That's implicit in the positional 
representation. 
So, we know that the 7 is, is the number 
the 1's, the 5 corresponds to the number 
of 10's, etc. 
And so, this is called base 10, that's 
what decimal means is base 10. 
And you write that with this little 
subscript over here to tell you the base. 
Also in base 10 of every digit in each 
position is an integer between 0 and 9, 
and this 9 is important, it's 1 less than 
the base. 
So, if you add two numbers together and 
one of the positions adds up to being a 
number bigger than 9, 
you know what you do is you do a carry to 
the next position to the left and adjust 
the number accordingly so it fits in the 
range between 0 and 9. 
So, that's all we need to know about base 
10. 
The, I've got to point out that 
positional notations really important. A 
alternate representation of numbers is 
Roman numerals, and 257 happens to be 
this in Roman numerals. 
I think you might be hard for us to 
multiply this by 2, 
which I'm going to write again in Roman 
numerals without using positional 
notation. 
If you tried to stay in the Roman numeral 
world, it's very, very hard to multiply. 
not as hard but pretty difficult to add 
and I would not even think about dividing 
in Roman numerals, that's going to be a 
disaster. 
So, 
positional notation when it was 
discovered, makes all the simple 
arithmetic operations very, very easy and 
it's quite concise in representing very 
big numbers and that's a very useful 
thing. 
Well, in a computer, numbers are 
represented in what we call binary and 
all that is, is base 2 is still 
positional notation. So here, I have a 
number 10000001, and the little subscript 
here 2 tells me it's page 2. 
So, I know that the, the position 
furthest to the right represents the 1's 
position, which is 2 to the 0, the next 
position over represents 2^1, and the 
next position to the left from that, 2^2, 
etc. 
And, I know my powers of 2. 
So I know that this, of course is 1, 
that's 2, that's 4, 8, 16, etc. 
So, this number this red one corresponds 
to the 2^8 position. 
So, our number here in binary is 2^8+2^0 
or 1. 
I happen to know that 2^8 is 256. So, 
this turns out to be our old number that 
we've been talking about that represented 
in binary. 
So, everything's the same in base 2 or in 
base 10. 
In base 2, you, all the numbers are 
either 0 or 1 in each position because 
that's 1 less than the base. 
Carries work the same way it's all the 
same except it's in base 2. 
Like I said, it's because voltage is 
either being 0 or some positive voltage 
make it easy to represent numbers 
electricly in base 2. 
Now, here are the powers of 2, 
which I suggest you learn. 
It's very important to know them. 
I know that 2^5 is 32, for example. 
one of the more important ones, though, 
is the one at the bottom. 
And that is very interesting and very 
useful result to know. 
And that, and what that means is 2^10, 
it's about equal to 10^3. 
In fact, on some of the computers when 
they say a k, it usually means 1000, 1k 
and something. 
1k on some computer systems actually 
means 1024 because it's very close to 
being 1,000. 
now, how do you represent numbers? 
And what are some of the various 
conventions for this positional notation? 
So, computers are often arranged in terms 
of what they call words. 
And here, I show a word that's 8-bits 
long, 
bit means binary digit. You may not know 
that but it means binary digit. 
And so, this word is 8-bits long here and 
turns out that it is what is known in the 
trade as a byte. 
And you, the most computers these days 
make devote many more bits to an integer 
than 8. usually though, they're multiples 
of 8 so you can have 64-bit. 
When we're talking about 64-bit 
computers, that means integers can be 
represented by 8 bytes etc. 
Let's talk about the simpler easy to 
manage a 8-bit manager. 
And notice, the word here on this is 
unsigned. So, this is exactly positional 
notation. 
The subscript here refers to the power of 
2. 
So, in 8-bits, the powers of 2 go from 0 
to 7. And that means I can represent 
numbers equal to 0 to 255, and 255 is 
2^8-1 because there is no eighth 
exponent. 
When you turn on every one of these 
digits, 
you get 255 which is 1 less than 2^8. 
Now, we'd also like to be able to talk 
about integers with the sign. And what is 
done is they take the same 8-bits that 
you had before, and you you use them. 
You have to devote one bit here to the 
sign. And if it's a 1, that tells you 
it's negative, 
if it's 0, it's positive. But now, you 
only have 7 bits to represent numbers. 
When all is said and done that the range 
numbers that's represented by a signed 
8-bit integer ranges from -128 to 127. 
And, you might wonder why there's more to 
the negative side than another. But it 
turns out you, in some sense, get an 
extra number because 0+0 is the same as 
-0. 
So, what is normally done is they 
allocate the extra, if you will to the 
negative side for all kinds of reasons 
that aren't worth getting into. 
Well, that's very nice. But, how about 
anything with a fraction? What are we 
going to do about that? And that's where 
a floating point comes in. 
So, in floating point, the representation 
number is m, so called mantissa, 
times 2^e where e is the exponent. 
So, what you do is you take a very long 
word, not 8-bits, much longer than that. 
You devote one part of it to the 
exponent, including a sign, so it's a 
signed integer. 
And you devote another part of the word 
to being assigned an integer, which 
stands at the mantissa, 
except it's not an integer. 
It turns out to be what we will call a 
floating point fraction. 
So, when you divide this up into 
individual bits, 
this is the 2^-1 bit, 
this one is the 2^-2 bit, etc. 
Okay? 
And this restriction that m has to be 
between the 1/2 of one means, this bit is 
the first bit is always on and the rest 
are whatever a number it's required to 
fill in that range. 
So, what happens is that you get a 
number, you scale it so the mantissa is 
in the right range, and that tells you 
what the exponent is. 
And that is written as a signed integer. 
So, a 1/32, which is 2^-5, is actually 
expressed in floating point as 1/2*2^-4. 
And that means there is a one in first 
position and zero everywhere else since 
it's exactly 1/2. 
And then, I represent as a signed integer 
-4. 
So, that's how floating point handles 
numbers that actually span in very large 
range, and can be negative and can be 
fractions. 
So, you can go from minus a very small 
fraction up to plus a very big number all 
in one concise representation. 
On computers these days, you can see one 
64-bit floating point. 
You can actually see 128-bit floating 
point for some calculations. 
So, that's how numbers are represented on 
computers and how computations are done. 
Now I want to point out, that the numbers 
which we're going to relate to signal 
values are all, can be represented a 
variety of ways signed, unsigned, 
floating point or not, but always with 
discrete values. 
I want to point out a number like 1/3 is 
an infinte repeating decimal, it's also 
an infinite repeating fraction in binary. 
Since you only have a fixed number of 
bits for any wor, number on a computer, 
they have, can, 1/3 cannot be represented 
exactly. 
There's always in effect, an error in 
representing a number. 
So, we're going to have to learn how to 
live with discrete amplitude values 
represented by different numbers in a 
computer. And this also means, by the 
way, that the computations are also not 
quite exact. 
It depends on what the numbers are. 
Furthermore, numbers are always stored in 
individual memory locations. So, that may 
seem like a minor point, but that means 
we cannot store a continuum of values. 
So, we've been talking about analog 
signals, which are functions of 
continuous time and usually have a 
continuous range of values. 
So, storing an individual memory 
locations mean we have to store just a 
few values of our signal. 
We can't store them all because the 
number of values in any interval in 
continuous time is uncountably infinite, 
that's a very big number. 
Furthermore, the amplitude values cannot 
be made exactly because there's some 
numbers we cannot represent exactly. 
But also, even beyond that, we can't 
represent all possible numbers, let's say 
between -1 and 1. 
That's impossible. 
So, we're going to have to learn about 
quantization and time, and quantization 
and amplitude. 
And that's going to be the subject of our 
next video.