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In this video will be talking
about the floating point

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notation, which is designed to
address the problem that we

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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ranges, but it's good enough.

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

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

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

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

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

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number, we're going to have it

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normalized. So when you
have got a negative

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number, something like
minus 3 1/2.

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

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point notation while step one

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will be?

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Identify.
The sign bit.

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

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

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And normalize

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To get to them on this so.

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Step

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3.
Is find the exponent.

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Which will be something
to do with the month. So

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how you normalize your
number and express it.

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As three bits.

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These complement notation and
the last step.

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Is just to pull everything
together and put it into the

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

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followed by four of the

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

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

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we go through a couple of simple

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examples. So let's look at minus

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one. And a quarter.

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OK sign bit going to
be one because this

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is a negative number.

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

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

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So. Let's again bring

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

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

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

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

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

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we need to normalize it into the

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mantissa. So the normalized
the normalized monthly start

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then will be 0.101. And
remember the amount Esther

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

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

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

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

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

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because once the computer read

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

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

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

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to get back to the original
number which remember was minus

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

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computer will need to move one
places into this direction and

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

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

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

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the decimal point by one
places into the positive

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direction, so my exponent.

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

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bits? This compliment?

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

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

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So pulling all these things
together and putting it into the

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sign, exponent, exponent,

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

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

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now because the computer knows
that the mantissa is designed to

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be zero point something
something so that zero point. We

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

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

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this is the 8 bit.

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Floating point notation 4 -- 1

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

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

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

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

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the computer works with bigger
length of digits, therefore that

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

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situation when you have to carry
out the calculations, trunking

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errors do occur.

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

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very minimized.

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Let's look at another example.

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Minus two and two 8.

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Now when I look at this number.

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

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

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OK, so step one.

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

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talking about a negative number.

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Step
2

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convert.

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And find the mantissa.

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So 2 1/4.
Let's convert that

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

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

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

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

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

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normalize it. And the
normalization process goes as.

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Bring it into the zero point.

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

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

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into the front in here, so I

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would have. The point moved into

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here. And then
what would follow

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now is 0.1001.

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

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that follows after the point.

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Then step three is
we need to identify

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or. Express the

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exponent. Now remember the
easiest way to think about the

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exponent is how many places
you would need to move the

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

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

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

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

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

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Two places into this direction,
and this is the positive

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

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

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

148
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now. And Lastly, pull all these
bits together. The sign bit was

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

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bits of the mantissa.

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Is 1001, which is
this bit here.

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So this is
equal to minus

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two and two
eighths.

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

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

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

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Now I need to convert.

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And find the mantissa.

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

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

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values or half a quarter on

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AIDS. And the 16 and
again because of the way

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

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OK that remember that your
monthly site has to be 0.1

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

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

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.123 places so.

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The exponent, which is
our Step 3.

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I moved my decimal .3 places.

170
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But have I moved it into
positive or negative direction?

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

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

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

174
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you telling the computer that
from this number you want to go

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back to the original number.

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

177
00:13:56,014 --> 00:14:00,070
zeros followed by the one you
have to tell the computer to

178
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move the radix point.

179
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Into the opposite direction done
before, so this exponent will be

180
00:14:08,878 --> 00:14:15,973
negative 3. Now let's see
if we can express negative 3IN.

181
00:14:16,670 --> 00:14:18,530
3 bits, two complements.

182
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Remember how we did that is
compliment. If it comes to

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

184
00:14:30,798 --> 00:14:35,289
positive three and positive
three again using the place

185
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values is 124. Remember, we
don't have anything else in

186
00:14:40,279 --> 00:14:46,394
here. So it's 011 and to make
it into negative three you need

187
00:14:46,394 --> 00:14:51,620
to copy the digits until you
copy the one which is in our

188
00:14:51,620 --> 00:14:56,444
case is the very first step and
then invert everything else swap

189
00:14:56,444 --> 00:15:01,670
once for zeros and zeros for
once for 01. So this is your

190
00:15:01,670 --> 00:15:03,278
negative three in here.

191
00:15:03,790 --> 00:15:05,710
So your exponent.

192
00:15:07,690 --> 00:15:11,320
He is 101.

193
00:15:11,990 --> 00:15:17,210
Put the whole thing
together. Sign bit was one,

194
00:15:17,210 --> 00:15:22,430
the exponent is 101 and a
month is soft.

195
00:15:24,280 --> 00:15:31,573
Is 1 followed by 3 zero, so
this is the equivalent of minus

196
00:15:31,573 --> 00:15:33,256
1 / 16.

197
00:15:35,140 --> 00:15:38,803
Not just to recap, with the
computer would actually do in

198
00:15:38,803 --> 00:15:42,799
here. The computer would come
and start to read the digits OK

199
00:15:42,799 --> 00:15:46,129
with this digit. Here you
telling the computer that he

200
00:15:46,129 --> 00:15:49,792
needs to think about a negative
number. With this digit you

201
00:15:49,792 --> 00:15:53,122
telling the computer how many
places in what direction he

202
00:15:53,122 --> 00:15:57,118
needs to move the radix point
that starts from here and then.

203
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That way it will be able to re
normalize the mantissa. Find out

204
00:16:01,447 --> 00:16:03,112
what number hides behind this

205
00:16:03,112 --> 00:16:09,040
binary code. I hope you not have
a clear idea of how to convert

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00:16:09,040 --> 00:16:12,540
numbers into 8 bits floating
point notation on the following

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pages you will have some
properties questions and then

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you will find the answers to

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00:16:17,790 --> 00:16:21,080
these. So these are the
practice questions.

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And here are the answers.