Two's Complement
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Video Script
In the previous video, we looked at binary numbers that are natural numbers, which are whole numbers greater than zero. There are a couple other binary data types that we’re going to look at in this module. The first one would be a signed integer, and assigned integer allows us to have negative values by changing the sign at the front of the number. So just like with decimal numbers, where we put a minus sign in the front to differentiate between positive numbers and negative numbers, we can do something similar with binary numbers by changing a sign bit at the front of the number to determine if it’s positive or negative. But there are a couple of caveats to that. And let’s take a look at that in this video.
So to understand negative numbers in binary, let’s start with our positive number again. On the last slide, we saw that the number 101010 in binary is the decimal number 42. In a signed binary number, instead of using this first digit as 128, we’ll use it as our assignments. And we will say by convention that if this bit is zero, it’s a positive number. And if this bit is one, it’s a negative number. So of course, we want an easy way to switch between our positive and our negative numbers.
So one of the first ways you can think about negative numbers in binary is what’s called the ones compliments. And so to calculate the ones compliment, we simply invert all of the bits. And so we would say that this number is negative 42, in ones compliments. So as we saw in that example, ones compliment is a very easy way that we can calculate the negative number of a binary number, we just invert all of the bits and declare the first bit to be the sign bit where zero is positive and one is negative.
However, there is a problem with the ones complement method of finding negative numbers in binary, which is why we don’t actually use it in practice. Let’s take a look at what that problem is and see if we can spot it. One of the most important things about binary numbers and any negative numbers, in fact is we should be able to add a positive value and a negative value together and get a logical result.
So let’s try some ones complement addition and see what happens. So we’ve already seen the positive value of 42. Before. And previously In this video, we calculated the negative value of 42 using one’s complement. So with these two values, if we add them together, we should get a value equal to zero. And remember, in binary, a value equal to zero is all zeros. So let’s try this addition and see what happens. Since a binary number is just like any other number, addition should work just like we expect. So here we would do zero plus one, and we get one, we do one plus zero, we get one. And very quickly, we’ll realize that since we inverted these values, each place is going to have a different value. So when we do positive 42 plus negative 42. As one’s complement, we get this number that is all ones. So when we add the positive value of 42, and the negative value of 42 using one’s complement, we end up with this number that is all ones.
So what is this number? Well, we know that this number is a negative number, because the first bit the sign bit is one. And so to find its positive value, we would invert all the bits. So we would invert all of these ones to zeros. And so we know that this number is zero in decimal. So if this is zero, then this number must be negative zero.
But wait a minute, is there such a thing as negative zero? That really doesn’t make any sense? And that is where the problem with one’s compliment lies. If we take the negative value and its positive value, and we add them together using one complement, we end up with this weird number called negative zero, which doesn’t exist. So one’s compliment while it seems to make sense on the surface, mathematically, it really doesn’t work out. And so here’s the result of this example. As we saw, we end up with a number that is negative zero.
So we need to come up with a better way for us to calculate a negative number in binary. The answer to that is a process we call two’s complement. Two’s complement is very similar to one’s complement with one extra step. In two’s complement, we will invert all of the bits, and then we will add one to the value of the number. So let’s try that on our example number of 42.
To calculate the two’s complement of 42, the first thing we will do is we will invert all of the bits and then we will add the value one to that resulting number. So we have one plus one in binary is a zero and then we will carry the one, then we’ll have one plus zero here, we will get one, and then the rest of this will just carry down. So the value negative 42 in two’s complement looks like this 11010110. There is a shortcut way to do two’s complement, where you start at the end of the number and you find the first one. Everything after that stays the same, and everything in front of that gets invert. So, again we see one zero stays the same, and and everything in front of that one gets inverted. That’s a quick shortcut to do two’s complement. So remember, to perform two’s complement, we first invert all of the bits, and then we add one and we will get this value for negative 42.
So now that we’ve calculated the value of negative 42 using two’s complement. Let’s do that same example before we add these two values together and check that our result makes sense. So once again, we want to do our addition example and make sure that this works. So now if we start over at this column, we will have zero plus zero is zero, we will have one plus one will become zero, we will carry a one, then we will have one plus one is zero, carry the one, zero, carry the one, zero, you can kind of see a pattern here where we have zero, carry the one zero, carry the one, zero, and then we will have this one that overflows. And right now, we’re not going to worry about that most math processors just ignore this one. But there is technically a one that does overflow when we do this, but now you’ll notice that we get a number that is all zeros, which in binary is clearly the value zero. So two complement addition works just like we expect it to.
If we take 42 plus negative 42, we will get the value positive zero. And once again, here’s the example on this slide completely worked out. So it works. That’s pretty cool. There are, of course, a few other binary values that are important to understand. We know that binary values can be both signed and unsigned. And so what’s really interesting is if we start counting up from zero, the sign values and the unsigned values will be the same all the way up to 127. Then when the first bit becomes one, the sine value and the unsigned value will diverge. At that point, the unsigned value will be 128, but the sine value will be negative 128. And so as we continue to count up in binary, the unsigned value will continue to get larger, whereas the sine value will start to get smaller all the way back down to negative one. This is why we get some really interesting things in the range of these values.
And that particular problem is known as integer overflow. If you’re working with a signed integer, but you’re always counting up, eventually you’ll reach a point where it will overflow and go from the highest possible value to the lowest possible value. And this is really hilariously explained in this XKCD comic, where if you count sheep and you get to 32,767, you will overflow and get to negative 32,768. And you’ll have to start all over again with your sheep going the wrong way over the fence. Because of this eight bit binary numbers have an interesting range and unsigned eight bit binary number can go from the value zero all the way to the value of two to the eighth minus one which is 255. Assigned value, however, can go from negative two to the seventh, which is negative 128 to positive to the seventh minus one, which is positive 127. So So in general, a binary number within bits, if it’s unsigned can go all the way to two to the n minus one, where if it’s signed, it’s negative two to the n minus one to positive two to the n minus one minus one.