Hello and welcome to the 6th video in the binary series. So
you more or less now understand how you can convert a binary
fraction into a decimal fractions. But how can we do it
the other way around? So let's say I have got a fraction, say 4
3/4. How can I convert that
into? A binary fraction. Again, let's call up on the place value
table. So the rate explained
is here. 1248 and on this side,
half or quarter.
And an 8 now what happens in here when it comes to the whole
part, the four. I know that I can build up from using one for
an nothing else, so I will need the place value holders.
And then the radix point now 3/4 by the obvious thinking would be
always just to put the three in here. Don't forget that we are
in the binary, so we can only use ones and zeros, so we can't
use a 3, but there are quarters. So what happens if I take out
one of those quarters? So if I've got 3/4 and I take away 1/4
from it, I am going to left with
two quarters. But hold on, two quarters is exactly the same as
a half. Well, that solves my problem, because then I can just
put a digit here. So you might have spotted this by knowing the
half and a quarter makes up 3/4. But this means that four and
three quarters in decimal is the same as 100 radix .11.
In binary. Let's look at another example.
3 and 5
eights. Place values again.
1. 24 Radix point half a
quarter. And an 8.
Now three is 2 + 1, so I'm going to use G of this place values
Radix point here, but when it comes to 58 again, I'm going to
be in a little bit of a trouble to find out what combinations
going on in here, and therefore you need to have quite a good
understanding of manipulating fractions, adding and
subtracting them. So if you're not sure, please do recap days,
so don't forget that there are loads of very good online
resource is that you can recap
on. Had to add and subtract fractions, so let's see what
happens with the five 8 now.
I can use up one of the AIDS, but I can use up only one of
them. So what happens if I take an 8 away from the 458? So I've
got 48 left now that Tendai, because remember any function
that has got twice as much as the bottom is at the top, always
cancers down to half, so the remainder here is just half.
That means I need a place for the zero here, so three and five
eights in decimal is the same
as. 11 radix .101.
In binary.
Let's look at a slightly different decimal equivalent, so
6.25 in decimal. What does it look like in binary? Now for
this particular format, it's good to know the fraction and
decimal fraction equivalent, so again, the whole part of the
table will be 1248, etc. Radix
point. And then we would have half a quarter and an 8 here.
Now.
This part of the table is the same, but beyond the radix point
because I have got the decimal fractions. Now I would need to
look at the decimal fractions here, so half is 0.5, a quarter
is 0.25. Well, that tells you something that and then eight is
0.125. So Luckily this is a rather simple function to
convert into binary fraction because six will be the sum of.
Four and two. So I need a placeholder zero for
the one radix point 1/2. I'm not using because all
I'm using is .25 so 6.25 in decimal is the same as
110 radix .01 in binary.
Let's look at a somewhat more complicated
example, so 5.3125.
In decimal, what is it equal to in binary? So again, right down
the place values. So on the left hand side of the radix point
everything is the same as it always has been.
And on the right hand side now I have got
0.5 zero, .25, zero, .125 and then I have got
0.0625.
So let's see which one of these I need to use to make up my
#5 is 4 and one and on the other side of the radix point.
Now 0.5 is too big because the 0.3125 is less than 0.5, so I'm
not going to use this. What happens if I take 0.25 out from
my decimal fraction part? So if I've got the 0.3125?
And I need to take away the 0.25 from it. What do I left with? So
first of all to carry out this obstruction I need to fill in
the zeros 5 -- 0 is five 2 -- 0 is two 1 -- 5. I cannot do so I
need to borrow.
11 -- 5 is Six 2 -- 2 is 0.
.0 -- 0 is zero. Well that seems rather lucky because the
0.3125 is just simply the sum of 0.25 and 0.0 sixty five. So
that's another place value. Hold up there for.
5.3125 in decimal is equivalent to 101 radix
point. 0101 in binary.
Now, I hope that seeing these examples will give you a good
idea about how to go around converting binary fractions to
decimal fractions or decimal fractions into binary fractions.
Also, you probably have a little bit of an idea about how
difficult or hard inconvenient converting decimal fractions
into binary functions can be not to further add to this
difficulties, what if I would need it to convert, such as
simple fraction? S3 and 4 /. 5 now there are no fifths in the
binary place value table because these place values are just
halves of Hobbs, etc etc. So there are no 15 that so you can
sit around there quite a bit of time to sort of think about what
I can do. How can I be that forfeits out of.
Hop hop squatters, 8 entrance, etc. Another rather simple
example that could be also problematic. Let's say if I
wanted to convert 5.4 again, .4. What combinations of 0.5, zero,
.25, zero point 25, etc would make up the .4 bit. This sort
of draws your attention to the limitations of the binary
fractions. While not every single decimal fractions.
Can be expressed as a finite binary fraction. We can always
extend the system to use more an more digits to express the
fractions, but unfortunately this is an inherent property of
the binary system.
In the next video I will show you how we can try to come
about these problems and see a different method of converting
decimal fractions into binary, which can make things slightly
easier. But for now I'm going to leave you with some
practice questions and you will find the answers to these
after the questions.
So these are the practice questions.
And here are the answers.