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In this video, we're going to
look at the binary fractions

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again, but from a slightly
different angle. I will show you

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an alternative method to convert
decimal fractions into binary

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fractions, which will work in
most cases. I will show you some

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examples, and again I will draw
your attention to the

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limitations of this method.

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So let's look at an example. So
what happens if you would need

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to convert 13.3125 into binary?
From here on, I'm going to split

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the number I'm going to convert
the whole number and the

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fraction part separately. The
whole number 413. We're going to

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use the normal way of converting
decimal numbers into binary's,

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so use the place where you

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table. 124 eight 16 So what
combination of these makes up 13

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while 8 + 4 makes 12
+ 1 makes 13, so the

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whole part of this decimal
fraction is 1101.

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How about the decimal part? What
I'm going to do now? I'm going

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to just separately right down
the decimal part.

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And the trick is here to keep
doubling the number. So what's

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the double of three 125?

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Double of five is 10, carried
A1 double of two is 4 + 1

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makes 5 double of two is 2 and
double of three is 6. Now what

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happened in here is that I do
not have any overflow into the

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whole number part of this part
of the structure and therefore

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I'm going to record 0 after
the radix point.

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Then I'm going to keep doubling.
Obviously double of 00, so I

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don't even need to think about
that. 2 * 5 is 10, carried A one

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2 * 2 is 4 + 1 makes it five.
2 * 6 is 12. So record the two

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and the one is an overflow. So
this digit. Now I'm going to

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pick up and record as the 2nd
digit of the.

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Binary fraction.

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Imagine like if you were picking
this one up from here recorded

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here. So from now it disappears.
The next step that I'm going to

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do, I'm going to double again,
but without the whole part. So

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I'm just going to 55.

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Which may extend double of two
is 4 + 1 is 5 and there again I

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have got no overflow into the
whole parts, so I'm going to

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record a 0 here.

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Double again 2 * 5 is 10.

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There is an overflow. This is my
last digit here because from

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here on I've got no more
fractional parts, so you stop

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when you end up with a zero in
the fractional part. Now pulling

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the two together.

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13.3125, in decimal
is the same

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as 1101 Radix
.0101 in binary.

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The second example is
9.1875. Again separates the

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number into whole and
fractional part. The whole

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part is 9.

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Which is 8 + 1, so eight
no four, no two and one.

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The decimal part now.

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0.1875 Let's keep doubling it. 2
* 5 is 10 carry one. 2 *

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7 is 14 + 1 makes it 15
carried A one 2 * 8016 + 1

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is 17, carried A one. 2 * 1
is 2 + 1 is 3 again. I

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did not have any overflow into
the whole number part, so the

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1st digit behind the radix point
that I'm going to record.

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Is a 0.

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Double again.

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2 * 5 is 10, carried a warm 2
* 7 is 14 + 1 makes it 15.

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Carry the one 2 * 3 is 6 +
1, seven again no overflow into

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the whole part. So I'm going to
record 0 here.

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Double again 2 * 5 is 10.
Carried a wamp. 2 * 7 is 14

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+ 1 makes it 15. Now I have
gotten overflowing here, so I'm

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going to record this.

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As the next digit after the
radix point and then double

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again, don't forget that this
one is not here anymore because

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I picked up and recorded it in

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here. So 2 * 5 is 10.

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So the next digit is 1 again. So
for the two things together.

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9.1875 in decimal
is the same

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as 1001 radix
.0011 in binary.

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The next number is 0.6875.
Luckily, this number doesn't

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have any whole parts, so we just
need to concentrate on the

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decimal fraction part so I can
just simply keep doubling this

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number. 2 * 5 is 10. Carry
one 2 * 7 is 14 +

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1 is 15, carried A one 2
* 8016 + 1 is 17.

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Either one 2 * 6 is.

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12 + 1 is 13, so I've got an

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overflow. So the 1st digit
I'm going to record behind

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the radix point is 1.

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Double 2 * 5 is 10, carried A
one 2 * 7 is 14 + 1 makes it

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15. Carry the one 2 * 3 is 6 +
1. Seven this case I did not

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have an overflow, so the next
day did after the radix point is

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0 double again.

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2 * 510 carried A one 2 *
7 is 14 + 1 makes it 15

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overflow, so the next digit is
1. Remember that's gone now and

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O makes no difference there. 2 *
5 is 10, so it's 1.0. So we

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have got one more digit here,
which is a one so 0.6875 in

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decimal. As the same is O radix

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.1011. In binary.

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Now let's look at a nice and
easy decimal number. The one

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that we didn't quite know how to
deal with at the end of the last

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video. So let's look at 3.4. So
let's separate the number again

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in two whole and fractional
parts, so three is 2 + 1, which

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is 1 one and the fractional
part. Let's just double 2 * 4 is

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0.8, so after the radix point,
the 1st digit will be 0.

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2 * 8 is 16.

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So 1.6 the next digit is 1.

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2 * 6 is 12, so 2.

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.1 again carry the one.

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Double of two is 4 no
carry, 004 is 8 again, no

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overflow. Double F8 is 16 so
I've got one here now.

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WF6 is 12. I've got another one
here. Now double F2 is 4.

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Put down a zero and hold on.
I'm repeating myself look.

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Point 4.8. 6248624862

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so this.
Simple decimal fraction 3.4 is

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an infinitely recurring binary
fraction, so that's again shows

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you some difficulties when it
comes to converting that simple

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fractions to binary fractions.
So this would be.

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3.4 in decimal
would be 1

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one radix .01100110.

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212345678 places.

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As I mentioned it in the last
video, this is something that's

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fundamentally inherent property
of the binary number system. We

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can't really do anything about
it, but by using more binary

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digits to represent the decimal
numbers, we can minimize this

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problem. Let's look
at another simple

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example, 4.715. Separate
it again to whole and

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fractional. Part 4 is just 100.
Remember this is 1, two and four

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and the fractional part will be
0.715. Now let's keep doubling

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it. 2 * 5 is then carried A1
three times, one is 2 + 1 is

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three 2 * 7 is 14, so I've got
one of the overflow, so the 1st

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digit in after the radix point
will be one.

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Now those two unnecessary
anymore. So double again 2 * 3

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is Six 2 * 4 is 8, so there
is no overflow. This digit will

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be at 0.

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Double it again. 2 * 6 is 12,
carried A one 2 * 8016 + 1 makes

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17. I've got an overflow here
now, so that's number one.

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Double it again. 2 * 2 is
four 2 * 7 is 14.

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So next digit is 1.

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Double against that digits gone
2 * 4 is eight 2 * 4 is 8

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no overflow, so this digit
phobia, 0 double 2 * 8 is 16,

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carried A one 2 * 8 is 16 + 1
is 17. So there is one as an

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overflow that's gone. Now double
again 2 * 6 is 12, carried A one

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2 * 7 is 14 + 1 makes it 15,
so I've got one as an overflow.

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Double again 2 * 2 is
four 2 * 5 is 10.

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Overflow, so that's another one

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in there. Times 2 is 8000 is the
next digit? Well, I don't know

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about you, but I'm getting
exhausted in here and look there

138
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is not even a sign anywhere for
a repetition, so this function

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looks even worse than the
previous one. And again just

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look at it how simple this is in
decimal. So yes, the binary

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number system indeed have got
quite a few limitations which

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can get quite a bit annoying

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well. What kind of things have
been discovered about the binary

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number system? Well, basically
we know that not all decimal

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fractions can be expressed as a
finite binary fraction.

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Unfortunately, this cannot be
avoided, but can be minimized by

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using more bids. Also, if you
look at the examples through the

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video again, you can see that
the radix point is different for

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different numbers, so the
position of the radix point is

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changing from number to number.

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That can get quite confusing for
the computer, but Luckily for

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this problem we do have a
solution and that is the

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floating point notation which we
will talk about in more details

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in one of the following videos.
For now I've prepared some

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examples for you, so please look
at them. Try them and you will

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find the answers later, so these
are the practice questions.

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And here are the answers.