.
In this segment, we're going to take an example of floating
point representation.
So, we're
going to take a hypothetical example, because in a regular single-precision
number you'll have to use . . . people use thirty-two bits.|So if you're using thirty-two bits,
and then you have certain bits for . . .
for the sign of the number, then you'll have certain bits for the exponent of the number, and certain bits for
the mantissa of the number, but since we are talking about thirty-two bits, that might be too many bits to write
down on the board.|So that's why I'm going to take a hypothetical word for your floating point representation,
but you should understand how things are really stored in real
life, in single-precision you'll be using thirty-two bits to have a single-precision number.
So let's go ahead and look at this, so what I want to do is I'm going to take an eight-bit word.
So what we are assuming is that we have a floating point representation
which is going to use eight bits.|And I'm going to distribute
now these bits, so I have one, two, three, four,
five, six, seven, and eight.|And this, for example,
is, let's suppose, used for the sign of the number,
and this one is used for sign of
exponent.|Generally, there's
no bit used for the sign of exponent in real life, because they use a biased exponent, we'll take that
example in another segment.|But to keep things simple, to be able to understand what
floating point representation is all about, we are using one bit for the sign of the exponent.
And then I'm going to use three bits . . . I'm going to use three bits,
this is one, two, and three bits for the exponent itself,
and the rest of the three, I'm going to use for the . . . for
the mantissa.|So I have an eight-bit word,
the first bit is going to be used for the sign of the number, the second bit is going to be used for the sign of the exponent,
the next three are going to be used for the exponent itself, and the
next three are going to be used for the mantissa, it makes a total of eight.|So if somebody tells me, hey, go ahead and
put -13.9 to the base-10 into the
floating point.|Can you
represent this number, -13.9, into our floating point representation,
when you are given that your eight-bit word, first bit is used for the sign of the
number, second bit is used for the sign of the exponent, next three for the exponent itself, and
next three for the mantissa.|Take this number, -13.9 in the base-10,
and put it in the floating point representation.|So let's
go ahead and see how we can go about doing that.|So we have -13.9
to the base-10, that's our number which we want to be able to converting to our floating point representation,
right?|So, the first thing which we have to do is that we have to convert 13
into base-2, and 13 in base-2 is
1101, and I'm going to leave this as homework, because we
already talked about binary representations.|So go ahead and see that how 13 is
same as 1101 base-2.|The same thing,
0.9, in the base-10, will be 0,
radix point, 11100
base-2, and this will be an approximation, because, again, as we learned in
binary representation of 0.9 cannot be represented exactly, but this is the . . . these are the
first five digits which you have of 0s and 1s after the
radix point for the representation of 0.9.|So, this again, I'm going to
leave this as your homework.|So once you have done these two pieces of
homework, what I can write down is that -13.9 to the
base-10 is nothing but 1101, then the radix
point, 11100
base-2, and this is an approximation, of course.|And then the minus sign right here
because I have a minus sign right there.|Now, in order to be able to write down the floating
point representation, what I have to do is I have to say
-1.10111100
base-2.|So what I've done here is that I have to move this
radix point all the way up to here, because I have to have the first . . .
the number before the radix point has to be a 1.|So in order to be able to do
that, it can be only 1, nothing else, it cannot be 0, it cannot be more than 1,
so it is just 1, so that means I have to move the radix point all the way three places
to the left here, so that's why I get 1, radix point, 101, from here, and
then the three 1s which are right here, and so on and so forth, multiplied by 2 to the power 3,
the reason why it's 2 to the power 3 is because I have moved the radix point three places
to the left here.|So if that is the case, I'm going to, again, approximate it
by -1.101 base-2
times 2 to the power 3.|And the reason why I'm doing that is because I can only . . .
I only have three places for the mantissa, so I'm going to, rather than writing down the whole number,
or the longer version of the number, I'm going to just say it's approximately -1,
radix point, 101, and forget about these 1s, because I cannot represent
them anyway.|In the mantissa, I only have places for three bits,
so it'll be the bit after the radix point, 101.|This is automatically taken into consideration
because every floating point representation will have a 1 before the radix point, so I don't need to
store that, so 101 will be stored in the mantissa.|Now
I already know that the sign bit of the number is negative, so I’ll have a 1 in the
sign bit, I'll have 101 in the mantissa, but I've still got to figure out
what is the sign of the exponent, and what is the value of the exponent.|Now
I already know the sign 2 raised power 3, the sign of the exponent is positive, so I'll have a 0 in the sign of the
exponent.|So I've got to figure out what to do with this 3 now.|The 3, again, I know that
is 11 base-2.|And since I have three
bits in the exponent I'm going to put as 011 base-2.
Again, I'm going to leave this as homework, for you to show me that
why 3 is 11 base-2.|Now why you've got to put a 0 here
is because I have three places for my exponent, so since it is 11 base-2,
in order to be able to fill up the third space, I'll put the 0 in the front, which does not make any difference for
the number equivalent, which is still 3, so I'll have 011 there.
So let me go ahead and write down the whole number now in the base-2
format, and that will make it very clear that what numbers go where.
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