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In this video I'm going to show
you a different way of using

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this complement notation pending
on which method makes more

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sense. Which method is easier
for you? You've got the choice.

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Doesn't matter which method you
use, so just to recap why we're

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doing the twos complement
notation that this is one way to

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represent negative numbers in
the binary number system. And

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again to recap, if the number
starts with a zero in the twos

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complement notation that we know
that this is a positive number.

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And if a number starts with N1,
we know that this is a negative

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number, so the 1st digit in the
twos complement notation is used

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for the sign bit. Therefore
there's a limitation. So

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depending on how many bits you
using for this compliment

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notation, we can only use one
less bit for building the actual

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numbers. So again, just to
recap, in case you haven't seen

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the previous video, every single

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number. In the two complement
notation have to be used at a

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fixed length, so you can't talk
about, just generally the twos

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complement notation. You always
have to say what length of bits

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you talking about, so you can
talk about the four bits twos

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compliment. You can talk about
the 7 bits this compliment, or

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you can talk about the 22 bits
2's complement, but you always

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have to give whoever reads
numbers the length, the fixed

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length of the numbers. Otherwise
the deuce complement notation.

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It doesn't necessarily work, so
this second method is slightly

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different from the previous one,
but the answers will be the

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same. So in the previous method,
we converted and then we kind of

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inversion process. In this
method we also using something

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similar, so step one.

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Again, we will be to convert.

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Into binary.

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But then Step 2.

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Would be just a simple
inversion.

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So we don't have to do
anything like copying bits

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or whatever. We just invert
all of the digits, but then

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comes on next tab.

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And that is odd.

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1.

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And I'm going to show you
why this is working and how

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we can do this, but let's
look at an exact example.

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So what would be negative four
in four Bits 2's complement? OK,

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so step one.

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Convert so again, we're using 4

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bits.

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1248
and four is just

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100. And don't
forget we need the

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zero first here.

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OK.

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Then Step 2 is the invert.

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So what happens here then
everywhere I have got a zero, I

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turn it into your one and
everywhere when I have a one, I

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turn it into a 0 so zero become
one and one becomes zero. And

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then comes the extra step.

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Is the adding one so one?

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1 + 1 makes 0.

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Carry the one 1 + 1 makes
0 carry the one 1 + 0

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makes one and one plus
nothing keeps one, so minus

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four is the same as 1100 in
4 bits 2's complement.

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So let's look at minus six and
again converted using the four

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bits to complement, so step one

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convert. So 4 bits on using
1, two for an 8 and I know

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that the six I will the
negative six. I will be able

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to show because it's less
than 8, so six is 4 + 2.

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And again I need the rest
of the zeros and I need to

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0 here as well because I'm
talking about four bits 2's

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complement, then Step 2.

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Invert.

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So zeros becomes ones
and ones become zero.

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And then the last step.

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Is plus one so one comes here
and I just need to add them

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together. 1 + 1 makes 0 karida
one 1 + 0 makes 1 zero plus

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nothing is 01 plus nothing is 1,
so minus six is the same as 1010

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in 4 bits. This compliment now
don't forget that this is not

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the same as 1010.

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Just as an ordinary binary
number. So if you just look at

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this from context, it needs to
be cleared. If it's a compliment

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notation, or if it's just
ordinary binary notation,

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because depending on the
difference between complements

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and ordinary numbers when you
trying to convert it back into

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to find out the decimal
equivalent, you would do

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different process is.

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Let's look at another example.
So what is 7 in five Bits 2's

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complement notation? So 75 bits
2's complement. Now this is a

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positive number, so when it's a
positive number I don't need to

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do the inversion process. So
again, just like in the

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previous video, positive
numbers are the easy examples

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because all we need to do is
just step one the conversion.

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So the place values again, we're
using that 5 bits, so make sure

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that you using order 5 bits 124,
eight, 16 that's all 12345 bits.

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Then, which one of these do we
need to be at? 7 again? 7 is 1

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less than 8, so I will need to
use every single digits that are

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less than 8. So 1 + 2 + 4 makes
7 altogether, but because it's 5

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minutes to complement notation,
I need to put the zeros in front

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of it. So positive 7 is the
same as 00111 in five bits,

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this complement notation.

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Now how can we do it the other
way around? So let's say that

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you are given a number 10011 and
you are told that this is a 5

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bits 2's complement notation

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number. So how can we find
out what number is this one

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now? Remember that we went
from when we went from

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decimal to binary, viewed
under conversion. Then we

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done the inversion and then
we done the odd one. So when

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you've got the binary
equivalent and you want to go

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back to the decimal, then you
have to do that step by step

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going backwards. So the Step
1 first here be.

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Take away one. Let's do that
now. So 10011 I need to take

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away one. One takeaway one is

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0. One takeaway, nothing is 1
zero takeaway. Nothing is 00

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takeaway, nothing is zero and
one takeaway zero is 1 so

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that is the first stage. Then
remember that the middle step

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was.

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The inversion, So what I need to
do now I need to invert this

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number so zero becomes 11,
becomes 00, becomes 10, becomes

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one and one becomes zero, and
this is the ordinary binary

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equivalent of this 5 bits 2's
complement an. Now what I can do

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is do the last step.

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Which in this case will be
the conversion.

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So just put the place values on
top of the number.

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8 + 4 + 1.

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That makes 13, so this number is
the equivalent is minus 13 OK

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and we have done this conversion
in the previous video, so you

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can double check there and
convince yourself that this is a

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correct way of doing it.

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So look at one more example
01011. This is again A5 bits.

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Two's complement notation.

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Now again, this is a positive
number because it starts with

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zero, so therefore this number
has not gone through the

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inversion, and adding one
process. So when I see a number

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like this, I know it's going to
be an easy calculation because

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all I need to do is just one
step and that is the conversion.

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So how do I convert this
number back to ordinary

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binary? Let's just copy
this one down here again.

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And the easiest and quickest
way, at least for me, is to put

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the place values on top.

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And then just add those numbers
together. So 8 + 2 + 1 makes 11,

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so this number is positive 11.

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So let's look at a couple of
eight bits to complement

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examples because that is quite
widely used. So how could we

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represent minus? 94 in eight
Bits 2's complement.

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So remember step one good. It's
a negative number. I need to go

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through all three steps. Step
one will be conversion.

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So 8 bits 124
eight 1630 two 64128,

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so 12345678 different bits
to represent my number.

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So I need to
build up 94 so.

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64 I'm going to use and the
difference between 94 and 64.

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It's a handy 30, so the next
step is how can I build up 30

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from the rest of the digits? Now
32 is too big because I'm only

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need 30, so the next biggest
place where you can use is 16,

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so it takes 16 away from this
and the remainder is 14.

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I can take eight from the 14.

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The remainder is 6 and six is 4
+ 2 even number, so I'm

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expecting the last bit to be 0
and it is because I'm not using

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the one and again, don't forget
that you always need to make

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sure that if there are empty
spaces before your number, you

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need to fill that in with zeros
to make sure that your number

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is 8 bits long. OK, so that was
first step, the Step 2 then was

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the inversion.

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So what we need to do now,
invert every single bits to the

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opposite, so zero become one,
one becomes 0.

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So zero becomes 11, becomes
zero and zero become one.

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This is the inverted number.
But what happens now?

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Is I need to add 1. OK,
so let's just copy the

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number down here again.

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An other one to it.

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1 + 1 is zero karida one 1 +
0 is 1.

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And from here on I don't
need to add anything, so

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I can just copy them.

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So the edmister is binary
equivalent of minus 94. Is this

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number now? You might ask
yourself a question that you

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know when I'm adding binary
numbers together, sometimes I do

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have to do loads of carrying.
And what happens if I carried

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this number over? So what
happens if I needed to add 1 to

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it and then I need to carry it?

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Well, that. Extra bit here would
be called the overflow.

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And what the computer does is
just simply ignores it. So if

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after the addition you end up
with a 9 bits long number, all

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you need to do is ignore the
first bit and your answer will

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be just that 8 bit.

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I hope that you now understand
how the Two's complement

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notation works, and I encourage
you to try out the practice

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questions that will follow and
you can check your work by

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looking at the answers for this
video. We will have two sets of

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questions. Here are the
first set.

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This is the answer
to the first set.

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Here is the 2nd set.

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And here are the answers to
the second set.