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.