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.