https:/.../emt051080p.mp4

Welcome to the 50 there in the
binary number series. In this

video I'm going to show you how
we can deal with fractions in

the binary system to be able to
do that, first we're going to

talk a little bit more about the
power definitions, and then

we're going to extend the binary
place value system. After that,

I'm going to show you a couple
of examples how you can convert

binary fractions into decimal
fractions. So let's extend or

recap our knowledge about the

powers. So we all remember that
any number to the zero power, by

definition, is always one, and
if we talk about the negative

powers, that is the same as one
over the positive equivalent

over the same power. So what
does it mean for us? Let's have

a look at some examples. So if
we talk about 3:00 to the minus

four, that is just simply one

over. Three to four, and if we
talk about two to the minus

three, that is just simply 1 /
2 to three last example.

Done to the minus two, that's
the same as 1 / 10 to the power

of 2. Now Lisa, you need to
know the negative powers, but

remember when we were talking
about the decimal place value

system, we talked about the
smallest place where you being

tend to zero and then then to

the one. Dented it too tentative
310 to the four etc. Now when

you look at the powers the
powers grow by one as we go from

right to left. But what happens
when we go from left to right?

The powers decreased by one and
then if you introduce the

decimal point you can further
extend this place value table so

we can talk about 10 to the
minus one because zero last one

is minus one then.

Into the minus 210 to the
minus three etc now.

If we look at it with the number
equivalents, this would be one.

This would be 1000.

1000 and 10,000. This way this
would be 1 / 10 this would be

1 / 100 because 10 to the power
of three is 100 and it will be

1 / 1000 now.

It might be a bit more familiar
if we look at this way. 1 / 10

is 0.11, / 100 is 0.01 and 1 /
1000 is 0.001, so you might

depending on how you would talk
about it, you either and this

format or this format now. Why
is it good for us? But basically

when we start to talk about
decimal numbers and we can talk

about 3.4. Two, but that's
basically means that I have.

Three times. 10 to the
0 + 4 * 10 to the

minus 1 + 2 times.

10 To the minus two, so
the place values are

built into the system.

Let's look at how it differs
when we talk about binary

numbers. So the binary place
value system we started with two

to the power of 0, two to the
power of 1 two three power of 2

to the power of three, and so
on. Now we can also introduce

something new called a Radix
point and then we can extend our

knowledge of place values to the
binary system because again, if

you look at the powers grow from
right to left and then buy one

and then they decrease by one
from left to right.

So we can extend this to two to
the minus one.

Due to the minus 2, two to the
minus three, and so on depending

on how many digits you want. So
what are these numbers here?

This is 1.

This is to this is for this is
it this one? You are quite

familiar with by now. Now the
radix point comes here. What is

2 to the minus one? Well that's
1 / 2 one is 2 to the minus two

that's 1 / 4 what is 2 to the
minus three? That's 1 / 8

remember two to two is 4 and two
to three is 8. Now again what we

learned about the binary place
value system is that if I'm go

from. Right to left. The place.
Why use double?

So you can see you know double
at 1 S, 2 W, 3 S, 4 W 4 is 8,

etc. Now if we go from left to
right, the place values gets

halved, so half of eight is 4.
Half of four is 2, half of two

is 1. Now that's the true on the
right hand side from the radix

point because half of 1 is half
half of half is 1/4 and half of

quarter is an 8. If you can't
really remember this or you have

forgotten about it, try to draw
them as functions as.

Slices of pizza to convince
yourself that this is how it

works, and again, the double
oven 8 is 1/4 and a double of

1/4 is 1/2. Because remember two
quarters Constanta your half and

two 8 sconces down to 1/4.

Now again. In
some cases you will need the

fraction numbers, but it's
always a good thing to know what

they look like as decimals. So
half would always be a .5

quarter would be 0.25 and eight
would be 0.125. Again, if you

half in .5 you get .25 and if
you half in .25 you got.

.125 If you're not sure, double
check it with the Calculator.

So why is it good for us?
Let's look at binary fraction,

so let's say I've got 11 radix
.1 in binary. Now let's see what

kind of decimal number is here

so. Anything that's on the left
hand side from the Radix point

works like just ordinary binary
numbers, so the place values

here. One and two, and on the
right hand side from the radix

point is the new system that we
just introduced. So this place

value is equivalent to 1/2. So
what I've got here.

Is I bought 1 * 2
+ 1 * 1.

Plus 1 * 1/2 This is 2 +
1, three and a half, which I can

also write S 3.5, so that's the
decimal equivalent of 1, one

radix .1 and listen when you're
reading out the binary

equivalent, you don't save, just
simply point you, say radix

point to make the difference
distinction between binary and

decimal fractions. Let's look at
a different example and see how

we can convert that binary
fraction into its decimal

equivalent. So our number this
case will be one radix, .101.

I can again build up the
binary place value table, so

here's the radix point. I will
have two to zero.

2 to the one, two to two, and so
on. On the left hand side and I

would have two to the minus 1,
two to the minus, 2 two to the

minus, three on the right hand

side. The equivalent.

1. 24 keep the
point half a quarter.

And an 8 now when it comes

to our. Binary number.

It's very important that you
place them correctly so the

points have to line up so the
digits I'm using is 1 radix

.101. What it means to me? I'm
using one of the ones.

I'm using one of the halves.

And I'm using Zita over
the quarters and I'm

using one of the AIDS.

Again, 0 times anything. Just
going to give me zero and what

I've got here is 1 + 1/2 plus
and eight, so how my adding

together half aninat? Well,
whenever I'm adding fractions

together I always need to make
them the common denominator. So

what's the common denominator of
two and eight wow 8 so I know

that half is 48 and the 18.

Is just 118, so I've got one.

Plus four 8 + 1 eight
altogether makes 5 eight, so

one radix .101 in binary is
equivalent to one and 5 eights

in decimal. Now if you want to
convert the five 8 into

decimal.

And look just keep it as a
fraction what you need to do.

You need to do the division.

So 5 / 8 is the same as 5
/ 8, so let's do the traditional

division eight into five dozen
guy bring into O and appoint an

borrow 0. 8 into 50 Go
6 * 6 * 8 is 48,

so we got a two remainder borrow
0, again 18 to 20 goes twice

2 * 8016, Four remainder,
bringing another zero an 18 to

40 goes five times. That means
that 5 / 8 is equal to

0.625, so that means that one
and 5 eights.

Is also the same as 1.625?
Really depends on if you

happy having the equivalent
instructions or you want

them as decimal fractions.

Let's look at another
example. So if I

have got 111 radix

.1001. The binary place
values again start with the

point. I have one.

2 four and I won't need any
higher place values because I've

only got 3 digits and then I

have 1/2. 1/4 on

8. And the 16. And
again because I only have got 4

digits in here, I only need four
place values in here. Now let's

put the corresponding digits
under the place values. So 111

radix .1001. So what it means to
us is that we have 1 * 4

+ 1 * 2 + 1 * 1
+ 1 * 1/2.

Plus 0 * 1/4
+ 0 times on

8 + 1 *

16 Again. Or the place
values which carry zeros will

not be taken into account. So we
have got 4 + 2 + 1 +

1/2 + 16. Now the whole part of
the number B 4 + 2 + 1

seventh and the fraction part of

the number. Is half and a
16 again to add them together? I

need to make them into fractions
with common denominator 16 will

be a common denominator, so 1 /
16 plus half breaking up into

sixteens is just 8 / 16. You
might remember that a half can

always be represented with the
numbers where the ratio top to

bottom is one to two. So anytime

when. Bottom part of the
fraction is double of the top

part of the fractions. That's
always 1/2, so one 816 + 1 six

teens make 916's, so it's 7 and
9 / 16. Now I'm going to spare

you the long division this case
and I'm just going to pick up a

Calculator and calculate what
this is equivalent to do. So

this will be 7 point.

5625 Remember when you want to
calculate the equivalent of it

as a decimal fraction, you just
need to divide 9 by 16.

So 1 one
1.1001 in binary

is equivalent to
7.5625 in decimal.

I hope you now have an idea of
how to convert binary

fractions to decimal fractions
and had the extended place

value system works in the
following pages you will have

some practice questions and
the answers will follow.

So these are the practice
questions.

And here are the answers.