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Welcome to the next video in the
binary series. In the previous

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video we learned how to convert
numbers to floating point

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notation. In this video I will
show you how to do the other way

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around, so had to convert a
floating point notation number

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back into ordinary decimal
numbers. So let's say that you

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are given an 8 bits floating
point notation, something like

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1001. 1011 OK,
now the first step that you

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would do in here is break
this code into it

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constituents break this
code into its parts.

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Remember that the first bit
is the sign bit. That's

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going to tell you if it's a
positive or a negative

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number. The next 3 digits.

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Is your exponent?

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And this exponent, these 3
bits twos complement notation

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will tell you how many places
you need to move the radix

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point either in to the right
or to the left and the last

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part. The last four bits.

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Is your normalized?

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Month is self.

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This is the number that you will
have to apply the exponent two

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and move the radix point into

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the right. Place to be able to
decode the binary number so.

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Step one. Sign because the sign
equals to one. We have got a

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negative number here.

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Then Step 2.

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Exponent Our exponent is
001. Well, that's a positive

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number because it starts with
zero and it's positive one. So

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my exponent is positive one, so
I will need to move my

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normalized mantissa one place
into the positive direction that

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three. My normalized mantissa is

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0.1011. Apply the exponent
two. It move the radix .1

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in the positive direction.

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So the number the binary number
that is given to me here is

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one data .011. Now I need to
decode it back into decimal.

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This is one that's the radix
point. This is 1/2. This is 1/4

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and this is an 8. So what I have
in here is.

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1 + 1/4 plus an 8. Remember,
how can we act together?

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Fractions need to make them to
be the same denominators, so it

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will be 1 + 2 eight +18,
which is altogether one and

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three eights. And remember,
we had one to start with,

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so 10011011 in eight bits
floating point notation is

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the same as minus one and
three eights in decimal.

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Let's look at the
next example, which

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is 11011100 again.

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The first step is to identify
the sign. Now the sign is 1

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again. So we have got a
negative number.

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Then Step

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2. Find
the exponent.

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Remember the exponent is 101
now. This is in three Bits 2's

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complement, so the 1st digit
tells me that this is a negative

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number. So this is a negative
exponent, so it went through the

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inversion process. So what I
need to do to find out what

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positive equivalent is in here
is to redo that inversion

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process, so copy the number
until you copy the one.

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Then invert everything else.

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So that zero becomes one. This
one becomes a 0, so this will be

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the positive equivalent of this
negative number, and the

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positive equivalent in here
using the place values.

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This is equal to

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R. Three, so this original
number is negative three, so our

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exponent is negative 3.

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And that 3.

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Using the mantissa.

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Now the normalized mantissa is
the last four bits. Remember,

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it's always starts with 0.1100,
but what that Azar's done?

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The exponent was negative three,
so the computer was stored to

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move it 3 places to the negative
direction. So what does that

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mean? One place to place

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three places. Filling the zeros.

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So the original number
was a very small

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number O radix .0001100.

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So what is this number?

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Use the place values again this
is 1 radix point or half or

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quarter on 8R16R32 and the rest
of them are zero, so we don't

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really need to bother about
them. So this number here is

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basically the sum of the 16th
and the 32.

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Again, how can I add
fractions together? I need to

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make them to be the same
denominator, so how can I

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make 30 twos from sixteens? I
just need to double it.

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So it's altogether 3 / 32.
And remember I had a negative

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number because my assignment was
one so one.

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1011100 As an 8 bits floating
point is exactly the same as

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minus 3 / 32.

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Let's do one more example of
this floating point notation and

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let's look at what number is 0.

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101
1011

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Step one, find

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the sign. The
first bit is 0, so this

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is a positive number.

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Then comes Step

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2. Using
the exponent.

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Now our exponent.

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Is 101.

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Which is a negative number.

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So again, going through the
inversion process copied the

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number until your Capital One,
then invert everything else.

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And. This is again.

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Positive three so this is still
negative. 3 The difference here

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now will be that my month is a
is slightly different, so let's

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look at the last step and deal
with the mantissa.

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So I'm normalized mantissa
is 0.1011. The computer

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was told to move
this mantissa or the

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radix point. Three in the
negative direction, so it would

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need to move it 123.

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So 0.00.
So the original

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number was 0.0001011.
The accompanying place

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values are one
radix point 1/2

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or quarter an
8 or 16

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or 3264 and

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128th. So I need to
talk together. Now is 1 / 16 +

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1 / 64 + 1 / 128. Now
again to be able to add them

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together I need to get them all
to be the same denominator. Now

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this is a common denominator
because I can get 228 from

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doubling all of them but at

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different numbers. So from.

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16 to get 228 what I've done. I
doubled 123 times. So that is

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basically 16 * 8 going to give
me the 128. So what this is

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telling me that 8 / 128 is the
same as 1 / 1664 and 120. There

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is only one doubling point in
here, so that is just 2 / 128.

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Plus 1 / 128, so
this is altogether giving me

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8 + 2 is 10
plus one is 11 /

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128 so 01011011 in eight
bits within point.

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Is the same as 11 / 128?
Remember this time it was zero,

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so this is a positive number and
for positive numbers we don't

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write out the positive sign.

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Now we went through the 8 bits
floating point notation and you

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might find it still a little bit
confusing. That's absolutely

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fine, because this is like an
orchestra you pulling together

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everything that you've learned
about binary numbers. So you

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putting together the two bits
complement notation, some normal

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mathematical knowledge about
moving decimal places applied 2

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in binary numbers and so on. So
this is probably one of the most

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difficult questions that you can
get in binary numbers, but once

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you understand them then you
will really know.

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What needed to be known
about this system, and

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you know how the computer
looks at the binary code.

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And now it is your turn to try
these conversions. You will find

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the answers to these questions
shortly after the questions

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appear. So these are the
practice questions.

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And here are the answers.