You might already be familiar with binary for example. This is

1 0 1 is equivalent to 5 in decimal that's because this is the ones place. This is the twos place

fours place eights place

16s

32s

64s and 128s

and we have a 1 in the fourth place and a 1 in the ones place 4 plus 1 is is 5

So this is fine, but how might we represent a negative number, so let's say we wanted to [represent] negative 5

Well, there's a couple ways we can do that one way is to take this

128s place and instead of using that as the 128s place use that as a sign so change this to a 1 here?

To indicate [that] this is negative, and then the rest of it is the same

1 0 1 and so this would be 5 here and then instead of this representing

128 it represents that the number is negative, so [negative] 5 now of course. It's important to know how many bits

You're working with right because if we're only using 4 bits then a 5 would be 0 1 0 1 that's equal to 5

[but] then we're going to use this this top bit here in this case

We're only using 4 bits to the top that is this fourth bit and so negative 5 might be a 1 1 0 1

And now instead [of] this being the eights place

This is actually representing a sign

So this would be negative 5 and this would be

5 so it's important to know how [many] bits you're working with so [that] you know which bit is the first bit and therefore which

Bit indicates what the sign of the number is so to keep going with this example. We can look at just

regular counting and binary here of course if we have all 0 0 0 0 0 that's equal to 0

And then if we have a 1 in the ones place that's equal to 1 we have a 1 in the twos place

That's equal to a 2 and a 1 that's equal to 3 and then of course a for is for

a 4 and a 1 is 5 a 4 and a 2 is 6

For a 2 plus a 1 is is 7 so that's simple enough, and then if this this first

Bit here is is indicating our negative sign?

then we can go backwards to so if we have a negative 1 that's negative [one] a negative to a

Negative you know [2] and a 1 is 3 and then a negative negative 3 negative

4 negative

You know 4 and a 1 is 5 with a negative is negative 5

4 2 2 is 6 so this is negative 6 and then a 4 plus 2 plus 1 is 7 and?

That's a negative 7 and so this is our sign bit

And so this first bit is our sign bit and then these other bits are just our 1 2 & 4 place

simple enough

Couple weird things about this though one is that you'll notice there's a negative 0 right because you can have 0 0 0 and then

It can either be 0 in the sign bit place or a 1 in the sign bit place

So you can have you know there's a difference between 0 and negative 0 so that's that's kind of weird

[the] other thing that is maybe a little bit inconvenient

We'll look at some some other approaches that that don't have this problem is

If you try to add these things together things get kind of weird

so let's say we want to add a 5 and a negative 5 so normally 5 plus negative 5 you would expect to get 0

simple enough, but here if we look at 5 0 1 0 1 and

negative 5 is 1 1 0 1

If we add these together [1] plus 1 is 2 which

be a 0 and then carry the 1 1 plus 0 plus 0 is 1

1 plus 1 again is to the but will

that is a 0 and carry the 1 and then here 1 plus 1 again is 2 so 0 and carry the 1

So in this case what we're seeing is 5 plus a negative 5 is not 0 it's 0 0 1 0 which

Well, and we have a carry so we have a carry coming out of this this this one

Bit that we don't you know if we're working with 4 bits?

we're going to ignore this this carry bit and so we have 0 0 1 0

which is

which is 2

Well, that's kind of weird

We're adding 5 and negative 5 we wouldn't expect to get [2] and you can try adding some other things here

It doesn't it doesn't work

So let's take a look at another scenario here. This is called one's complement. What this is

Is again everything from zero to seven is the same same same as we saw before and this first bit here is all zeros

So we know that these are all positive

But what we do for the negative numbers is we actually just flip all of the bits

We take the compliment of all of the bits, so you know [2] here is 0 1 0?

Well 0 0 1 0 negative 2 is 1 1 [0] 1 so we're just flipping each of those bits

so we flipped the 0

401 We Flipped the 1 4 0 with the 0 [4] a 1 and, so on if you look at each of these numbers

So 5 is 0 1 0 1 negative 5 is 1 0 1 0?

So what happens with with this so we still have this kind of strange thing where we have negative 0 and we'll come back to

That in a minute

[but] what we can do is we can try to you know add some numbers, so if we add 5 So 0 1

0 1 that's 5 and

negative 5 1 0 [1] 0

We add those

adding negative 5 here

[what] we end up getting is 1 plus 0 is 1?

0 plus 1 is 1 1 plus 0 is 1 0 plus 1 is 1 we get 1 1 1 1 which is negative zero

So that's closer, right?

You know 5 minus 5 you expect to get 0 negative 0 is I guess the same thing. [so] that's pretty good

And and that actually works for any of these you know we can do 3 minus 3 or 3 so 3 is?

0 0 1 [1] negative 3 is 1 1 0

0 so this is 3 plus negative 3 if we add these together [so] [1] plus 0 is [1]

1 plus 0 is 1 0 plus 1 is 1 0 plus 1 is 1 we get 1 1 1 1 again [and]

[so] again 3 minus 3 is negative 0 so this is definitely doing doing a lot better than

then here where we were saying a 5 minus 5 was was 2

That's definitely not right

So negative 0 definitely a lot closer. Let's try some other some other things kind of see if we can see a pattern here

so if we did let's Say 5

Plus negative three so five minus three let's see what we get there

five is 0 1 0 1

and negative 3 is 1 1 0 0

so 1 plus 0 is 1 0 plus 0 is 0

1 plus 1 is [2] so we'll put a 0 and carry a 1 and

then 1 Plus 1 again is 2 so we'll put a 0 and carry a 1

And so if we ignore this this first bit here because we're working with 4 bits we get 1

So 5 minus 3 is [1] that's close. It should be 2 but you know let's let's see. What happens if we do

Say 6 minus 2 or 6 plus negative 2

So 6 is

0 1 1 0

negative 2 is 1 1 0 1 1 1 0 1

[0] plus 1 is 1 1 [plus] 0 is 1 1 plus 1 is 2 0 and carry the 1 1 plus 1 is 2?

put a 0 and carry the 1

again ignore that first [bix] [we're] working with four bits so 0 0 1 1 is 3 so

[we're] saying [sex] minus 2 is 3

close

But but we're definitely we're definitely seeing an interesting Pattern here in fact if we look

You know these places where we're getting this negative 0 if we just go down 1 we see zero

Which is what we want here 5 minus 3 we'd expect to see two, but we're seeing one

so if we just go down 1 we'd actually see a 2

6 minus 2 we'd expect to see 4 but we're actually getting 3 but if we go down 1 we get the right answer

So we could just always add [1]

To each [of] these of course it's kind of weird to say add 1 2 to negative 0 to get to 0 but

If you kind of bear with me a minute

You'll see that if we add 1 to the binary value in it and it flips over

You do get to the right thing and here you get to the right thing

So the ones compliment. You know just flipping all of these bits is

Close it's very close. It's off by 1 on all on all of our arithmetic

And it also still has that nice property that we saw with the you know with [the] sign bit

Which is you know you can look at this first bit and you can tell these are all negative numbers

And if the first bit is [zero], then they're positive numbers

so it seems like the ones complement has a lot of the

You know a lot of nice properties. [you] know that you'd have in the sign, bit the one thing. That's you know well

I guess it's not too bad is

It to get from to inverter to negate a number

You know if you're going from five to negative five with a sign bit all you'd have to do is flip one bit

You just flip that first bit to a one and you get to negative five

Well with the ones complement you have to invert each bit so it's a little bit more work

But of course inverting bits is is pretty easy to do in hardware if we're building hardware to do this for building a computer

I'm going to be doing some math, so

[once] [compliment] seems pretty good, but it is off by by one

So what we can do is look at another scenario

Which is two's complement and if we compare ones complement to two's complement?

It's basically the same except what it does is it gets rid of this negative zero?

[so] instead of having negative zero we just go [right] to negative one

Negative two three four five six seven and the the kind of weird thing here is we have a negative eight

And we don't have an eight, but the nice the nice property about this is that all [of] our math works out very nicely

so if we have [five] minus five

5 here is zero one zero one and

So that's 5 and negative 5 is 1 0 1 1 1 0 1 1

and if we add these things if we add negative 5

1 plus 1 is 2 so we'll put a 0 and carry the 1 1 plus 1 is to

put a 0 and carry the 1 1 plus 1 is 2

so we'll put a 0 and carry the 1

And 1 plus 1 is to put a 0 and we have [a] one that we're carrying and so you'll notice that each

when we're when we're looking at the the

Negated version of something when we add each of these

Place values we get a 2 each time, [and] that's why it's called two's complement

But in any event you see that we end up with 0 which is just what we would want when we add a number

Or to its its inverse and in fact you can try it with any of these numbers

and you'll find out [that] you you get all [zeroes], so

[Let's] try you know 6 minus 2 so 6 is 0 1 1 0

that's 6 and

Minus 2 is 1 1 1 0 1 1 1 0 and?

if we add these together we add our negative 2 oops Negative 2

adding negative 2 there 0 plus 0 is 0

1 plus 1 is

2 so we'll

Put a 0 and carry the 1 1 plus 1 plus 1 is 3 so [that's] a 1 and a carry a 1

And the 1 plus 1 is

2 so 0 and carry a 1 and again we ignore that fifth bit there and so 0 1 0 0 is

4 and sure enough 6 minus 2 should be equal 4 so this two's complement works works pretty well

[and] it also has this nice property again that that first bit is a sign bit right because all of these negative numbers

Have a 1 there

And what's even more interesting about two's complement is if you look at the place values. It actually makes sense

This is our ones place. This is our twos place. This is our forest place and this place here

It's you know. It looks like a sign bit, but really it's a negative 8 place

and

You can see when [we] have a negative 8 and all zeros

We have a negative 8 but if you have a negative 8 and a 1 you get negative 7 right?

if you have negative 8 and a 4 you get negative 4 if you get negative 8

Plus 4 Plus 2 plus 1 you end up at Negative 1

So this this this bit value actually has has a mathematical meaning. Which is why our math works out, so it's pretty cool

The one thing that's a little bit harder with two's complement

Is is negating a number? So if we want to go from a 5 to a negative 5?

it's kind of a two-step process right so 5 is 0 1 0 [1]

The first thing you have to do is take the ones compliment 1 0 1 0 so flip all the bits

So you're just inverting everything that's that's easy enough, but then you have to add 1

So you invert?

and

Then add [1] so 2 add 1 2 0 [1] 0 1 is just 1 0 1 1 and then this

Invert so this is 5 then we invert it and then we add 1 and that gets us [to] Negative 5

you see negative 5 1 0 1 1

[1] 0 1 1 so it's a little bit more work to go from a positive number to [2] its inverse

Because you have to invert it and then add 1

But what you'll see you know in the next video when we when we build a circuit that can add and subtract

You'll see that it's it's actually not too [hard] to build to build this ain't hardware