In this video, we're going to look at the binary fractions
again, but from a slightly different angle. I will show you
an alternative method to convert decimal fractions into binary
fractions, which will work in most cases. I will show you some
examples, and again I will draw your attention to the
limitations of this method.
So let's look at an example. So what happens if you would need
to convert 13.3125 into binary? From here on, I'm going to split
the number I'm going to convert the whole number and the
fraction part separately. The whole number 413. We're going to
use the normal way of converting decimal numbers into binary's,
so use the place where you
table. 124 eight 16 So what combination of these makes up 13
while 8 + 4 makes 12 + 1 makes 13, so the
whole part of this decimal fraction is 1101.
How about the decimal part? What I'm going to do now? I'm going
to just separately right down the decimal part.
And the trick is here to keep doubling the number. So what's
the double of three 125?
Double of five is 10, carried A1 double of two is 4 + 1
makes 5 double of two is 2 and double of three is 6. Now what
happened in here is that I do not have any overflow into the
whole number part of this part of the structure and therefore
I'm going to record 0 after the radix point.
Then I'm going to keep doubling. Obviously double of 00, so I
don't even need to think about that. 2 * 5 is 10, carried A one
2 * 2 is 4 + 1 makes it five. 2 * 6 is 12. So record the two
and the one is an overflow. So this digit. Now I'm going to
pick up and record as the 2nd digit of the.
Binary fraction.
Imagine like if you were picking this one up from here recorded
here. So from now it disappears. The next step that I'm going to
do, I'm going to double again, but without the whole part. So
I'm just going to 55.
Which may extend double of two is 4 + 1 is 5 and there again I
have got no overflow into the whole parts, so I'm going to
record a 0 here.
Double again 2 * 5 is 10.
There is an overflow. This is my last digit here because from
here on I've got no more fractional parts, so you stop
when you end up with a zero in the fractional part. Now pulling
the two together.
13.3125, in decimal is the same
as 1101 Radix .0101 in binary.
The second example is 9.1875. Again separates the
number into whole and fractional part. The whole
part is 9.
Which is 8 + 1, so eight no four, no two and one.
The decimal part now.
0.1875 Let's keep doubling it. 2 * 5 is 10 carry one. 2 *
7 is 14 + 1 makes it 15 carried A one 2 * 8016 + 1
is 17, carried A one. 2 * 1 is 2 + 1 is 3 again. I
did not have any overflow into the whole number part, so the
1st digit behind the radix point that I'm going to record.
Is a 0.
Double again.
2 * 5 is 10, carried a warm 2 * 7 is 14 + 1 makes it 15.
Carry the one 2 * 3 is 6 + 1, seven again no overflow into
the whole part. So I'm going to record 0 here.
Double again 2 * 5 is 10. Carried a wamp. 2 * 7 is 14
+ 1 makes it 15. Now I have gotten overflowing here, so I'm
going to record this.
As the next digit after the radix point and then double
again, don't forget that this one is not here anymore because
I picked up and recorded it in
here. So 2 * 5 is 10.
So the next digit is 1 again. So for the two things together.
9.1875 in decimal is the same
as 1001 radix .0011 in binary.
The next number is 0.6875. Luckily, this number doesn't
have any whole parts, so we just need to concentrate on the
decimal fraction part so I can just simply keep doubling this
number. 2 * 5 is 10. Carry one 2 * 7 is 14 +
1 is 15, carried A one 2 * 8016 + 1 is 17.
Either one 2 * 6 is.
12 + 1 is 13, so I've got an
overflow. So the 1st digit I'm going to record behind
the radix point is 1.
Double 2 * 5 is 10, carried A one 2 * 7 is 14 + 1 makes it
15. Carry the one 2 * 3 is 6 + 1. Seven this case I did not
have an overflow, so the next day did after the radix point is
0 double again.
2 * 510 carried A one 2 * 7 is 14 + 1 makes it 15
overflow, so the next digit is 1. Remember that's gone now and
O makes no difference there. 2 * 5 is 10, so it's 1.0. So we
have got one more digit here, which is a one so 0.6875 in
decimal. As the same is O radix
.1011. In binary.
Now let's look at a nice and easy decimal number. The one
that we didn't quite know how to deal with at the end of the last
video. So let's look at 3.4. So let's separate the number again
in two whole and fractional parts, so three is 2 + 1, which
is 1 one and the fractional part. Let's just double 2 * 4 is
0.8, so after the radix point, the 1st digit will be 0.
2 * 8 is 16.
So 1.6 the next digit is 1.
2 * 6 is 12, so 2.
.1 again carry the one.
Double of two is 4 no carry, 004 is 8 again, no
overflow. Double F8 is 16 so I've got one here now.
WF6 is 12. I've got another one here. Now double F2 is 4.
Put down a zero and hold on. I'm repeating myself look.
Point 4.8. 6248624862
so this. Simple decimal fraction 3.4 is
an infinitely recurring binary fraction, so that's again shows
you some difficulties when it comes to converting that simple
fractions to binary fractions. So this would be.
3.4 in decimal would be 1
one radix .01100110.
212345678 places.
As I mentioned it in the last video, this is something that's
fundamentally inherent property of the binary number system. We
can't really do anything about it, but by using more binary
digits to represent the decimal numbers, we can minimize this
problem. Let's look at another simple
example, 4.715. Separate it again to whole and
fractional. Part 4 is just 100. Remember this is 1, two and four
and the fractional part will be 0.715. Now let's keep doubling
it. 2 * 5 is then carried A1 three times, one is 2 + 1 is
three 2 * 7 is 14, so I've got one of the overflow, so the 1st
digit in after the radix point will be one.
Now those two unnecessary anymore. So double again 2 * 3
is Six 2 * 4 is 8, so there is no overflow. This digit will
be at 0.
Double it again. 2 * 6 is 12, carried A one 2 * 8016 + 1 makes
17. I've got an overflow here now, so that's number one.
Double it again. 2 * 2 is four 2 * 7 is 14.
So next digit is 1.
Double against that digits gone 2 * 4 is eight 2 * 4 is 8
no overflow, so this digit phobia, 0 double 2 * 8 is 16,
carried A one 2 * 8 is 16 + 1 is 17. So there is one as an
overflow that's gone. Now double again 2 * 6 is 12, carried A one
2 * 7 is 14 + 1 makes it 15, so I've got one as an overflow.
Double again 2 * 2 is four 2 * 5 is 10.
Overflow, so that's another one
in there. Times 2 is 8000 is the next digit? Well, I don't know
about you, but I'm getting exhausted in here and look there
is not even a sign anywhere for a repetition, so this function
looks even worse than the previous one. And again just
look at it how simple this is in decimal. So yes, the binary
number system indeed have got quite a few limitations which
can get quite a bit annoying
well. What kind of things have been discovered about the binary
number system? Well, basically we know that not all decimal
fractions can be expressed as a finite binary fraction.
Unfortunately, this cannot be avoided, but can be minimized by
using more bids. Also, if you look at the examples through the
video again, you can see that the radix point is different for
different numbers, so the position of the radix point is
changing from number to number.
That can get quite confusing for the computer, but Luckily for
this problem we do have a solution and that is the
floating point notation which we will talk about in more details
in one of the following videos. For now I've prepared some
examples for you, so please look at them. Try them and you will
find the answers later, so these are the practice questions.
And here are the answers.