In this video will be talking
about the floating point

notation, which is designed to
address the problem that we

talked about in one of the
previous videos, where we noted

that the radix point was jumping
around and I mentioned that it

can cause a bit of a confusion
for the computer. Throughout

this video I will be talking
about the 8 bit floating point

notation which you will find out
very soon. It's a really small

system, so it doesn't allow you
to work with huge numbers.

However, it's convenient enough
to understand and work with on a

paper, and you can just extend
the general rules to any

systems. So let's see what is
this floating point notation

looks like. Well, the floating
point notation is, 8 bits are

broken up into three parts. The
first part is just one single

digits telling you what is the
sign of your number. The next 3

digits. We called the exponent,
which will tell the computer

where to move the decimal point
from here and the last four

digits. Called the mantissa. Now
the mantissa is the normalized

version of your binary number
into the 0.122 format. We're

going to go through it in in
details. Now. The exponent bit

in this format is just three
bits, so you can only move your

decimal point to three in to the
right or to the left, which

straight away as it as you that
you can't really cover great

ranges, but it's good enough.

And the exponent is expressed as
a 3 bits 2's complement

notation. So the sign bit if
we have a positive number be

going to use zero if we have a
negative number, we're going to

use one. As I said, the exponent
is 3 bits 2's complement.

And the mantissa, which is
always 4 bits in this particular

number, we're going to have it

normalized. So when you
have got a negative

number, something like
minus 3 1/2.

How are you going to convert
this into an 8 bit floating

point notation while step one

will be?

Identify.
The sign bit.

That this is going to be the
easiest one for negative #0 for

positive number. Step two will
be is convert into binary.

And normalize

To get to them on this so.

Step

3.
Is find the exponent.

Which will be something
to do with the month. So

how you normalize your
number and express it.

As three bits.

These complement notation and
the last step.

Is just to pull everything
together and put it into the

format of the sign followed by
three of the exponent digits and

followed by four of the

mantissa. OK, it sounds quite a
long and probably a little bit

alien in theory, but I think
what we are going to help us if

we go through a couple of simple

examples. So let's look at minus

one. And a quarter.

OK sign bit going to
be one because this

is a negative number.

Not we're going to find the
mantissa for that first. What we

need to do? We need to convert 1
1/4 into binary.

So. Let's again bring

in the. Place values Now this is
the radix point. Here is 124 and

the rest of them. The whole
numbers and after comes up half

a quarter on 8 etc etc. Now I've
chosen deliberately choosing a

simple example so to write 1 1/4
in binary or we need to do is 1.

Radix Point followed with 01. So
this is our binary number, but

we need to normalize it into the

mantissa. So the normalized
the normalized monthly start

then will be 0.101. And
remember the amount Esther

needs to be 4 bits, so I need
to put an extra 0 into here.

Now what happens in this case?
Now I found my mantissa.

I found mine on Monday so now,
but I still need my exponent.

The exponent the easier way to
think about it is to try to

think about the exponent from
the computer's point of view,

because once the computer read

the. Sign me and the exponent
wait, then the computer will

know where to start from and how
many places to move that Radix

point. Now from this radix point
here in them from the mantissa

to get back to the original
number which remember was minus

one and a quarter to get back to
the original number. The

computer will need to move one
places into this direction and

this is the positive direction.
So I'm moving one place too.

The right that's a positive.
It's like multiplying by 10

in normal numbers in normal
circumstances, so I'm moving

the decimal point by one
places into the positive

direction, so my exponent.

Will be positive one, but what
does positive one look like in 3

bits? This compliment?

So remember I've only got 3
bits. It's a positive number, so

the 1st digit will be 0 and it's
just positive one, so it's 001.

So pulling all these things
together and putting it into the

sign, exponent, exponent,

mantissa. Format the sign
negative number will be one. The

exponent is positive one, so it
will be 001 and the months are

now because the computer knows
that the mantissa is designed to

be zero point something
something so that zero point. We

can always forget and I can just
write down the following 4

digits 1010. So this is
equivalent to minus 1 1/4. So

this is the 8 bit.

Floating point notation 4 -- 1

1/4. Now you might spot here
what happens if I had to put.

Let's see after normalization, I
end up with five 6 or 7 digits.

Well with this Ed bits floating
point, we would have to cut off

at the first 4 bits, so we would
have trunking error. Obviously

the computer works with bigger
length of digits, therefore that

trunking error doesn't come up
as much. So in practical

situation when you have to carry
out the calculations, trunking

errors do occur.

In real situations, when the
computer does it, it's very

very minimized.

Let's look at another example.

Minus two and two 8.

Now when I look at this number.

Well, I can see here straight
away that I can cancel this one

down, and this is actually minus
two end of quarter.

OK, so step one.

Sign and in this case the sign
will be equal to 1 because I'm

talking about a negative number.

Step
2

convert.

And find the mantissa.

So 2 1/4.
Let's convert that

into binary. Ring in
the place values again, this is

your radix .124 half or quarter
and an 8. So for two to build

up the whole part we need 1 zero
and the radix point for 1/4. We

don't need any half and we use
1/4 so this is our ordinary

binary which is equal to 2 1/4.
But remember now we need to

normalize it. And the
normalization process goes as.

Bring it into the zero point.

One and the rest of the digit
would follow, so if I normalize

it, what would means I would
need to move this decimal point

into the front in here, so I

would have. The point moved into

here. And then
what would follow

now is 0.1001.

OK, and remember that this part
will be our mantissa, the one

that follows after the point.

Then step three is
we need to identify

or. Express the

exponent. Now remember the
easiest way to think about the

exponent is how many places
you would need to move the

decimal point. If you were the
computer. So you were reading

this number and the origonal
number was one 0.01, so you

need to tell the computer via
the exponent that he needs to

get to this number from this
number. So what would you need

to do in here? You would need
to move the decimal point.

Two places into this direction,
and this is the positive

direction, so the exponent would
have to be positive too, and the

positive two in three bits 2's
complement. Remember positive

number is zero and two is 1
zero, so this is our exponent

now. And Lastly, pull all these
bits together. The sign bit was

one. The three bits of the
exponent is 010, and the four

bits of the mantissa.

Is 1001, which is
this bit here.

So this is
equal to minus

two and two
eighths.

Now look at the last example
and the locks. That last example

will be minus 116th sign bit
which is. First step is again

one because it's a negative
number, Step 2.

Now I need to convert.

And find the mantissa.

So place values. I don't have
any whole numbers, so all I'm

going to do, I'm going to
express the fractional place

values or half a quarter on

AIDS. And the 16 and
again because of the way

I've chosen this number,
this will be 0.0001.

OK that remember that your
monthly site has to be 0.1

something. So how can I get from
this number into this format

now? The only way I can get
to here is by moving the decimal

.123 places so.

The exponent, which is
our Step 3.

I moved my decimal .3 places.

But have I moved it into
positive or negative direction?

Let's look at that again. So a
normalized it. But remember that

mantis and needs to be 4 bits
long, so I need to put.

Three more zeros in here and
now. The computer comes in and

you telling the computer that
from this number you want to go

back to the original number.

OK, so from here to get back
to the origin number of 0.3

zeros followed by the one you
have to tell the computer to

move the radix point.

Into the opposite direction done
before, so this exponent will be

negative 3. Now let's see
if we can express negative 3IN.

3 bits, two complements.

Remember how we did that is
compliment. If it comes to

negative numbers, just Express 3
as a 3 bit binary number,

positive three and positive
three again using the place

values is 124. Remember, we
don't have anything else in

here. So it's 011 and to make
it into negative three you need

to copy the digits until you
copy the one which is in our

case is the very first step and
then invert everything else swap

once for zeros and zeros for
once for 01. So this is your

negative three in here.

So your exponent.

He is 101.

Put the whole thing
together. Sign bit was one,

the exponent is 101 and a
month is soft.

Is 1 followed by 3 zero, so
this is the equivalent of minus

1 / 16.

Not just to recap, with the
computer would actually do in

here. The computer would come
and start to read the digits OK

with this digit. Here you
telling the computer that he

needs to think about a negative
number. With this digit you

telling the computer how many
places in what direction he

needs to move the radix point
that starts from here and then.

That way it will be able to re
normalize the mantissa. Find out

what number hides behind this

binary code. I hope you not have
a clear idea of how to convert

numbers into 8 bits floating
point notation on the following

pages you will have some
properties questions and then

you will find the answers to

these. So these are the
practice questions.

And here are the answers.