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In this video, we're going to talk about 
how computations are formed on computers. 

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And, in particular, how numbers are 
represented on by computers. 

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Turns out this is going to be very 
important for us to figure out how 

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computers function so we can convert from 
analog signals to digital signals, 

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understand the nature of the errors that 
are involved, and also the nature of the 

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errors that are involved in when you do 
computations on digital computers. 

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So, the first thing we need to talk about 
is what's called Positional Notation. 

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This amounts to something we're very 
familiar with. And it turns out, how we 

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deal with numbers is very similar to how 
computers deal with numbers, at least in 

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a conceptual way. 
We'll talk about Base-2, which is the 

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number system that computers use, and I'm 
going to emphasize that knowing your 

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powers of two is going to be very, very 
useful skill to have. 

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So, let's talk about positional notation. 
Now, I claim that one of the critical 

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discoveries in mathematics was positional 
notation. 

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So, what positional notation means is 
that we have a number that we would 

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write, 257. 
What that means is that the 2 means 

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2*100. 
So, 2*100 + 5*10 + 7*1. And, the 

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so-called base of the number 
representation has to do with the powers 

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they're involved here. 
And the word position comes from the fact 

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that if you start counting from the, the 
right end, the position tells you what 

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the exponent is. 
That's implicit in the positional 

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representation. 
So, we know that the 7 is, is the number 

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the 1's, the 5 corresponds to the number 
of 10's, etc. 

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And so, this is called base 10, that's 
what decimal means is base 10. 

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And you write that with this little 
subscript over here to tell you the base. 

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Also in base 10 of every digit in each 
position is an integer between 0 and 9, 

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and this 9 is important, it's 1 less than 
the base. 

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So, if you add two numbers together and 
one of the positions adds up to being a 

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number bigger than 9, 
you know what you do is you do a carry to 

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the next position to the left and adjust 
the number accordingly so it fits in the 

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range between 0 and 9. 
So, that's all we need to know about base 

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10. 
The, I've got to point out that 

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positional notations really important. A 
alternate representation of numbers is 

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Roman numerals, and 257 happens to be 
this in Roman numerals. 

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I think you might be hard for us to 
multiply this by 2, 

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which I'm going to write again in Roman 
numerals without using positional 

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notation. 
If you tried to stay in the Roman numeral 

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world, it's very, very hard to multiply. 
not as hard but pretty difficult to add 

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and I would not even think about dividing 
in Roman numerals, that's going to be a 

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disaster. 
So, 

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positional notation when it was 
discovered, makes all the simple 

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arithmetic operations very, very easy and 
it's quite concise in representing very 

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big numbers and that's a very useful 
thing. 

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Well, in a computer, numbers are 
represented in what we call binary and 

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all that is, is base 2 is still 
positional notation. So here, I have a 

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number 10000001, and the little subscript 
here 2 tells me it's page 2. 

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So, I know that the, the position 
furthest to the right represents the 1's 

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position, which is 2 to the 0, the next 
position over represents 2^1, and the 

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next position to the left from that, 2^2, 
etc. 

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And, I know my powers of 2. 
So I know that this, of course is 1, 

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that's 2, that's 4, 8, 16, etc. 
So, this number this red one corresponds 

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to the 2^8 position. 
So, our number here in binary is 2^8+2^0 

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or 1. 
I happen to know that 2^8 is 256. So, 

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this turns out to be our old number that 
we've been talking about that represented 

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in binary. 
So, everything's the same in base 2 or in 

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base 10. 
In base 2, you, all the numbers are 

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either 0 or 1 in each position because 
that's 1 less than the base. 

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Carries work the same way it's all the 
same except it's in base 2. 

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Like I said, it's because voltage is 
either being 0 or some positive voltage 

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make it easy to represent numbers 
electricly in base 2. 

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Now, here are the powers of 2, 
which I suggest you learn. 

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It's very important to know them. 
I know that 2^5 is 32, for example. 

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one of the more important ones, though, 
is the one at the bottom. 

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And that is very interesting and very 
useful result to know. 

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And that, and what that means is 2^10, 
it's about equal to 10^3. 

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In fact, on some of the computers when 
they say a k, it usually means 1000, 1k 

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and something. 
1k on some computer systems actually 

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means 1024 because it's very close to 
being 1,000. 

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now, how do you represent numbers? 
And what are some of the various 

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conventions for this positional notation? 
So, computers are often arranged in terms 

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of what they call words. 
And here, I show a word that's 8-bits 

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long, 
bit means binary digit. You may not know 

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that but it means binary digit. 
And so, this word is 8-bits long here and 

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turns out that it is what is known in the 
trade as a byte. 

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And you, the most computers these days 
make devote many more bits to an integer 

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than 8. usually though, they're multiples 
of 8 so you can have 64-bit. 

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When we're talking about 64-bit 
computers, that means integers can be 

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represented by 8 bytes etc. 
Let's talk about the simpler easy to 

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manage a 8-bit manager. 
And notice, the word here on this is 

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unsigned. So, this is exactly positional 
notation. 

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The subscript here refers to the power of 
2. 

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So, in 8-bits, the powers of 2 go from 0 
to 7. And that means I can represent 

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numbers equal to 0 to 255, and 255 is 
2^8-1 because there is no eighth 

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exponent. 
When you turn on every one of these 

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digits, 
you get 255 which is 1 less than 2^8. 

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Now, we'd also like to be able to talk 
about integers with the sign. And what is 

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done is they take the same 8-bits that 
you had before, and you you use them. 

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You have to devote one bit here to the 
sign. And if it's a 1, that tells you 

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it's negative, 
if it's 0, it's positive. But now, you 

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only have 7 bits to represent numbers. 
When all is said and done that the range 

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numbers that's represented by a signed 
8-bit integer ranges from -128 to 127. 

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And, you might wonder why there's more to 
the negative side than another. But it 

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turns out you, in some sense, get an 
extra number because 0+0 is the same as 

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-0. 
So, what is normally done is they 

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allocate the extra, if you will to the 
negative side for all kinds of reasons 

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that aren't worth getting into. 
Well, that's very nice. But, how about 

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anything with a fraction? What are we 
going to do about that? And that's where 

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a floating point comes in. 
So, in floating point, the representation 

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number is m, so called mantissa, 
times 2^e where e is the exponent. 

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So, what you do is you take a very long 
word, not 8-bits, much longer than that. 

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You devote one part of it to the 
exponent, including a sign, so it's a 

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signed integer. 
And you devote another part of the word 

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to being assigned an integer, which 
stands at the mantissa, 

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except it's not an integer. 
It turns out to be what we will call a 

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floating point fraction. 
So, when you divide this up into 

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individual bits, 
this is the 2^-1 bit, 

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this one is the 2^-2 bit, etc. 
Okay? 

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And this restriction that m has to be 
between the 1/2 of one means, this bit is 

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the first bit is always on and the rest 
are whatever a number it's required to 

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fill in that range. 
So, what happens is that you get a 

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number, you scale it so the mantissa is 
in the right range, and that tells you 

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what the exponent is. 
And that is written as a signed integer. 

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So, a 1/32, which is 2^-5, is actually 
expressed in floating point as 1/2*2^-4. 

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And that means there is a one in first 
position and zero everywhere else since 

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it's exactly 1/2. 
And then, I represent as a signed integer 

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-4. 
So, that's how floating point handles 

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numbers that actually span in very large 
range, and can be negative and can be 

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fractions. 
So, you can go from minus a very small 

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fraction up to plus a very big number all 
in one concise representation. 

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On computers these days, you can see one 
64-bit floating point. 

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You can actually see 128-bit floating 
point for some calculations. 

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So, that's how numbers are represented on 
computers and how computations are done. 

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Now I want to point out, that the numbers 
which we're going to relate to signal 

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values are all, can be represented a 
variety of ways signed, unsigned, 

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floating point or not, but always with 
discrete values. 

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I want to point out a number like 1/3 is 
an infinte repeating decimal, it's also 

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an infinite repeating fraction in binary. 
Since you only have a fixed number of 

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bits for any wor, number on a computer, 
they have, can, 1/3 cannot be represented 

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exactly. 
There's always in effect, an error in 

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representing a number. 
So, we're going to have to learn how to 

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live with discrete amplitude values 
represented by different numbers in a 

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computer. And this also means, by the 
way, that the computations are also not 

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quite exact. 
It depends on what the numbers are. 

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Furthermore, numbers are always stored in 
individual memory locations. So, that may 

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seem like a minor point, but that means 
we cannot store a continuum of values. 

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So, we've been talking about analog 
signals, which are functions of 

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continuous time and usually have a 
continuous range of values. 

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So, storing an individual memory 
locations mean we have to store just a 

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few values of our signal. 
We can't store them all because the 

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number of values in any interval in 
continuous time is uncountably infinite, 

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that's a very big number. 
Furthermore, the amplitude values cannot 

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be made exactly because there's some 
numbers we cannot represent exactly. 

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But also, even beyond that, we can't 
represent all possible numbers, let's say 

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between -1 and 1. 
That's impossible. 

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So, we're going to have to learn about 
quantization and time, and quantization 

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and amplitude. 
And that's going to be the subject of our 

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next video.