WEBVTT

00:00:16.540 --> 00:00:20.824
Welcome to the 50 there in the
binary number series. In this

00:00:20.824 --> 00:00:25.465
video I'm going to show you how
we can deal with fractions in

00:00:25.465 --> 00:00:30.106
the binary system to be able to
do that, first we're going to

00:00:30.106 --> 00:00:34.033
talk a little bit more about the
power definitions, and then

00:00:34.033 --> 00:00:37.960
we're going to extend the binary
place value system. After that,

00:00:37.960 --> 00:00:42.601
I'm going to show you a couple
of examples how you can convert

00:00:42.601 --> 00:00:45.814
binary fractions into decimal
fractions. So let's extend or

00:00:45.814 --> 00:00:47.599
recap our knowledge about the

00:00:47.599 --> 00:00:53.286
powers. So we all remember that
any number to the zero power, by

00:00:53.286 --> 00:00:58.159
definition, is always one, and
if we talk about the negative

00:00:58.159 --> 00:01:03.032
powers, that is the same as one
over the positive equivalent

00:01:03.032 --> 00:01:08.791
over the same power. So what
does it mean for us? Let's have

00:01:08.791 --> 00:01:14.993
a look at some examples. So if
we talk about 3:00 to the minus

00:01:14.993 --> 00:01:17.651
four, that is just simply one

00:01:17.651 --> 00:01:24.572
over. Three to four, and if we
talk about two to the minus

00:01:24.572 --> 00:01:31.004
three, that is just simply 1 /
2 to three last example.

00:01:32.030 --> 00:01:37.760
Done to the minus two, that's
the same as 1 / 10 to the power

00:01:37.760 --> 00:01:43.740
of 2. Now Lisa, you need to
know the negative powers, but

00:01:43.740 --> 00:01:48.420
remember when we were talking
about the decimal place value

00:01:48.420 --> 00:01:53.100
system, we talked about the
smallest place where you being

00:01:53.100 --> 00:01:56.376
tend to zero and then then to

00:01:56.376 --> 00:02:02.282
the one. Dented it too tentative
310 to the four etc. Now when

00:02:02.282 --> 00:02:07.770
you look at the powers the
powers grow by one as we go from

00:02:07.770 --> 00:02:12.866
right to left. But what happens
when we go from left to right?

00:02:12.866 --> 00:02:17.178
The powers decreased by one and
then if you introduce the

00:02:17.178 --> 00:02:21.490
decimal point you can further
extend this place value table so

00:02:21.490 --> 00:02:26.586
we can talk about 10 to the
minus one because zero last one

00:02:26.586 --> 00:02:28.154
is minus one then.

00:02:28.490 --> 00:02:34.210
Into the minus 210 to the
minus three etc now.

00:02:34.890 --> 00:02:41.221
If we look at it with the number
equivalents, this would be one.

00:02:41.221 --> 00:02:43.169
This would be 1000.

00:02:43.850 --> 00:02:50.710
1000 and 10,000. This way this
would be 1 / 10 this would be

00:02:50.710 --> 00:02:58.550
1 / 100 because 10 to the power
of three is 100 and it will be

00:02:58.550 --> 00:03:00.510
1 / 1000 now.

00:03:01.040 --> 00:03:08.144
It might be a bit more familiar
if we look at this way. 1 / 10

00:03:08.144 --> 00:03:14.804
is 0.11, / 100 is 0.01 and 1 /
1000 is 0.001, so you might

00:03:14.804 --> 00:03:20.132
depending on how you would talk
about it, you either and this

00:03:20.132 --> 00:03:25.904
format or this format now. Why
is it good for us? But basically

00:03:25.904 --> 00:03:31.232
when we start to talk about
decimal numbers and we can talk

00:03:31.232 --> 00:03:36.790
about 3.4. Two, but that's
basically means that I have.

00:03:40.610 --> 00:03:47.910
Three times. 10 to the
0 + 4 * 10 to the

00:03:47.910 --> 00:03:50.415
minus 1 + 2 times.

00:03:51.060 --> 00:03:56.342
10 To the minus two, so
the place values are

00:03:56.342 --> 00:03:58.214
built into the system.

00:03:59.870 --> 00:04:04.160
Let's look at how it differs
when we talk about binary

00:04:04.160 --> 00:04:08.840
numbers. So the binary place
value system we started with two

00:04:08.840 --> 00:04:14.456
to the power of 0, two to the
power of 1 two three power of 2

00:04:14.456 --> 00:04:19.019
to the power of three, and so
on. Now we can also introduce

00:04:19.019 --> 00:04:23.231
something new called a Radix
point and then we can extend our

00:04:23.231 --> 00:04:27.092
knowledge of place values to the
binary system because again, if

00:04:27.092 --> 00:04:32.006
you look at the powers grow from
right to left and then buy one

00:04:32.006 --> 00:04:35.516
and then they decrease by one
from left to right.

00:04:35.520 --> 00:04:39.260
So we can extend this to two to
the minus one.

00:04:39.880 --> 00:04:45.424
Due to the minus 2, two to the
minus three, and so on depending

00:04:45.424 --> 00:04:50.176
on how many digits you want. So
what are these numbers here?

00:04:51.820 --> 00:04:52.849
This is 1.

00:04:53.360 --> 00:04:58.176
This is to this is for this is
it this one? You are quite

00:04:58.176 --> 00:05:02.304
familiar with by now. Now the
radix point comes here. What is

00:05:02.304 --> 00:05:08.152
2 to the minus one? Well that's
1 / 2 one is 2 to the minus two

00:05:08.152 --> 00:05:13.312
that's 1 / 4 what is 2 to the
minus three? That's 1 / 8

00:05:13.312 --> 00:05:18.816
remember two to two is 4 and two
to three is 8. Now again what we

00:05:18.816 --> 00:05:22.944
learned about the binary place
value system is that if I'm go

00:05:22.944 --> 00:05:26.600
from. Right to left. The place.
Why use double?

00:05:27.250 --> 00:05:33.254
So you can see you know double
at 1 S, 2 W, 3 S, 4 W 4 is 8,

00:05:33.254 --> 00:05:37.362
etc. Now if we go from left to
right, the place values gets

00:05:37.362 --> 00:05:42.102
halved, so half of eight is 4.
Half of four is 2, half of two

00:05:42.102 --> 00:05:46.526
is 1. Now that's the true on the
right hand side from the radix

00:05:46.526 --> 00:05:51.266
point because half of 1 is half
half of half is 1/4 and half of

00:05:51.266 --> 00:05:55.374
quarter is an 8. If you can't
really remember this or you have

00:05:55.374 --> 00:05:58.534
forgotten about it, try to draw
them as functions as.

00:05:58.590 --> 00:06:02.847
Slices of pizza to convince
yourself that this is how it

00:06:02.847 --> 00:06:07.878
works, and again, the double
oven 8 is 1/4 and a double of

00:06:07.878 --> 00:06:12.135
1/4 is 1/2. Because remember two
quarters Constanta your half and

00:06:12.135 --> 00:06:14.457
two 8 sconces down to 1/4.

00:06:15.180 --> 00:06:21.171
Now again. In
some cases you will need the

00:06:21.171 --> 00:06:25.384
fraction numbers, but it's
always a good thing to know what

00:06:25.384 --> 00:06:29.980
they look like as decimals. So
half would always be a .5

00:06:29.980 --> 00:06:34.576
quarter would be 0.25 and eight
would be 0.125. Again, if you

00:06:34.576 --> 00:06:39.938
half in .5 you get .25 and if
you half in .25 you got.

00:06:40.440 --> 00:06:45.290
.125 If you're not sure, double
check it with the Calculator.

00:06:46.650 --> 00:06:52.998
So why is it good for us?
Let's look at binary fraction,

00:06:52.998 --> 00:07:00.404
so let's say I've got 11 radix
.1 in binary. Now let's see what

00:07:00.404 --> 00:07:03.578
kind of decimal number is here

00:07:03.578 --> 00:07:09.914
so. Anything that's on the left
hand side from the Radix point

00:07:09.914 --> 00:07:14.454
works like just ordinary binary
numbers, so the place values

00:07:14.454 --> 00:07:19.866
here. One and two, and on the
right hand side from the radix

00:07:19.866 --> 00:07:24.822
point is the new system that we
just introduced. So this place

00:07:24.822 --> 00:07:28.952
value is equivalent to 1/2. So
what I've got here.

00:07:30.040 --> 00:07:36.460
Is I bought 1 * 2
+ 1 * 1.

00:07:36.970 --> 00:07:44.650
Plus 1 * 1/2 This is 2 +
1, three and a half, which I can

00:07:44.650 --> 00:07:50.410
also write S 3.5, so that's the
decimal equivalent of 1, one

00:07:50.410 --> 00:07:55.210
radix .1 and listen when you're
reading out the binary

00:07:55.210 --> 00:08:00.010
equivalent, you don't save, just
simply point you, say radix

00:08:00.010 --> 00:08:04.330
point to make the difference
distinction between binary and

00:08:04.330 --> 00:08:10.770
decimal fractions. Let's look at
a different example and see how

00:08:10.770 --> 00:08:16.170
we can convert that binary
fraction into its decimal

00:08:16.170 --> 00:08:22.770
equivalent. So our number this
case will be one radix, .101.

00:08:22.770 --> 00:08:29.370
I can again build up the
binary place value table, so

00:08:29.370 --> 00:08:35.370
here's the radix point. I will
have two to zero.

00:08:35.390 --> 00:08:40.626
2 to the one, two to two, and so
on. On the left hand side and I

00:08:40.626 --> 00:08:45.246
would have two to the minus 1,
two to the minus, 2 two to the

00:08:45.246 --> 00:08:47.094
minus, three on the right hand

00:08:47.094 --> 00:08:50.190
side. The equivalent.

00:08:51.220 --> 00:08:58.834
1. 24 keep the
point half a quarter.

00:08:59.410 --> 00:09:03.071
And an 8 now when it comes

00:09:03.071 --> 00:09:06.780
to our. Binary number.

00:09:08.200 --> 00:09:12.590
It's very important that you
place them correctly so the

00:09:12.590 --> 00:09:18.297
points have to line up so the
digits I'm using is 1 radix

00:09:18.297 --> 00:09:23.565
.101. What it means to me? I'm
using one of the ones.

00:09:24.950 --> 00:09:28.778
I'm using one of the halves.

00:09:28.930 --> 00:09:32.548
And I'm using Zita over
the quarters and I'm

00:09:32.548 --> 00:09:34.558
using one of the AIDS.

00:09:35.880 --> 00:09:41.472
Again, 0 times anything. Just
going to give me zero and what

00:09:41.472 --> 00:09:47.996
I've got here is 1 + 1/2 plus
and eight, so how my adding

00:09:47.996 --> 00:09:51.724
together half aninat? Well,
whenever I'm adding fractions

00:09:51.724 --> 00:09:56.850
together I always need to make
them the common denominator. So

00:09:56.850 --> 00:10:02.908
what's the common denominator of
two and eight wow 8 so I know

00:10:02.908 --> 00:10:06.170
that half is 48 and the 18.

00:10:06.220 --> 00:10:09.755
Is just 118, so I've got one.

00:10:10.380 --> 00:10:15.814
Plus four 8 + 1 eight
altogether makes 5 eight, so

00:10:15.814 --> 00:10:21.742
one radix .101 in binary is
equivalent to one and 5 eights

00:10:21.742 --> 00:10:27.670
in decimal. Now if you want to
convert the five 8 into

00:10:27.670 --> 00:10:28.164
decimal.

00:10:29.530 --> 00:10:32.364
And look just keep it as a
fraction what you need to do.

00:10:32.364 --> 00:10:33.672
You need to do the division.

00:10:34.200 --> 00:10:41.144
So 5 / 8 is the same as 5
/ 8, so let's do the traditional

00:10:41.144 --> 00:10:46.352
division eight into five dozen
guy bring into O and appoint an

00:10:46.352 --> 00:10:53.569
borrow 0. 8 into 50 Go
6 * 6 * 8 is 48,

00:10:53.569 --> 00:11:00.975
so we got a two remainder borrow
0, again 18 to 20 goes twice

00:11:00.975 --> 00:11:06.794
2 * 8016, Four remainder,
bringing another zero an 18 to

00:11:06.794 --> 00:11:13.671
40 goes five times. That means
that 5 / 8 is equal to

00:11:13.671 --> 00:11:18.432
0.625, so that means that one
and 5 eights.

00:11:18.470 --> 00:11:24.234
Is also the same as 1.625?
Really depends on if you

00:11:24.234 --> 00:11:28.426
happy having the equivalent
instructions or you want

00:11:28.426 --> 00:11:30.522
them as decimal fractions.

00:11:31.590 --> 00:11:38.070
Let's look at another
example. So if I

00:11:38.070 --> 00:11:41.310
have got 111 radix

00:11:41.310 --> 00:11:47.920
.1001. The binary place
values again start with the

00:11:47.920 --> 00:11:50.460
point. I have one.

00:11:51.040 --> 00:11:55.456
2 four and I won't need any
higher place values because I've

00:11:55.456 --> 00:11:58.032
only got 3 digits and then I

00:11:58.032 --> 00:12:01.516
have 1/2. 1/4 on

00:12:01.516 --> 00:12:08.494
8. And the 16. And
again because I only have got 4

00:12:08.494 --> 00:12:14.916
digits in here, I only need four
place values in here. Now let's

00:12:14.916 --> 00:12:19.856
put the corresponding digits
under the place values. So 111

00:12:19.856 --> 00:12:27.266
radix .1001. So what it means to
us is that we have 1 * 4

00:12:27.266 --> 00:12:33.194
+ 1 * 2 + 1 * 1
+ 1 * 1/2.

00:12:33.230 --> 00:12:39.862
Plus 0 * 1/4
+ 0 times on

00:12:39.862 --> 00:12:43.178
8 + 1 *

00:12:43.178 --> 00:12:49.312
16 Again. Or the place
values which carry zeros will

00:12:49.312 --> 00:12:56.422
not be taken into account. So we
have got 4 + 2 + 1 +

00:12:56.422 --> 00:13:04.006
1/2 + 16. Now the whole part of
the number B 4 + 2 + 1

00:13:04.006 --> 00:13:06.850
seventh and the fraction part of

00:13:06.850 --> 00:13:13.616
the number. Is half and a
16 again to add them together? I

00:13:13.616 --> 00:13:18.632
need to make them into fractions
with common denominator 16 will

00:13:18.632 --> 00:13:24.560
be a common denominator, so 1 /
16 plus half breaking up into

00:13:24.560 --> 00:13:30.488
sixteens is just 8 / 16. You
might remember that a half can

00:13:30.488 --> 00:13:35.504
always be represented with the
numbers where the ratio top to

00:13:35.504 --> 00:13:38.696
bottom is one to two. So anytime

00:13:38.696 --> 00:13:43.350
when. Bottom part of the
fraction is double of the top

00:13:43.350 --> 00:13:48.732
part of the fractions. That's
always 1/2, so one 816 + 1 six

00:13:48.732 --> 00:13:54.942
teens make 916's, so it's 7 and
9 / 16. Now I'm going to spare

00:13:54.942 --> 00:14:00.738
you the long division this case
and I'm just going to pick up a

00:14:00.738 --> 00:14:04.878
Calculator and calculate what
this is equivalent to do. So

00:14:04.878 --> 00:14:06.948
this will be 7 point.

00:14:07.610 --> 00:14:12.406
5625 Remember when you want to
calculate the equivalent of it

00:14:12.406 --> 00:14:17.638
as a decimal fraction, you just
need to divide 9 by 16.

00:14:18.180 --> 00:14:24.984
So 1 one
1.1001 in binary

00:14:24.984 --> 00:14:31.788
is equivalent to
7.5625 in decimal.

00:14:32.910 --> 00:14:37.362
I hope you now have an idea of
how to convert binary

00:14:37.362 --> 00:14:40.701
fractions to decimal fractions
and had the extended place

00:14:40.701 --> 00:14:44.411
value system works in the
following pages you will have

00:14:44.411 --> 00:14:47.379
some practice questions and
the answers will follow.

00:14:49.800 --> 00:14:52.008
So these are the practice
questions.

00:14:57.700 --> 00:14:59.400
And here are the answers.