[MUSIC]. Let's take a look at how we represent information using these zero's and one's these low voltages and high voltages. We're going to group our binary digits, bits, into groups of eight, that we're going to call bytes. And, if we do that, you'll notice that we can have a range of values that go from 000 to 11111. And that gives a range for the kinds of numbers we can represent in a, in a single byte. That's going to correspond to zero decimal to 255. And how did we get that 255? Well, if we think about numbers being represented by a ones digit. And then of course in decimal we would have a 10's digit here, but in binary that's a 2's digit and then a 4's digit and an 8's and 16's and 32's, 64's and 128's. So, if we add all of those values we would get 255. As a little example. Let's take a look at this number here. 00101011. That corresponds to a 1 in the 32's column, a 1 in the 8's column, a 1 in the 2's column, and lastly, a 1 in the 1's column. When we add those values, we get the decimal number 43. In the other direction, if we started with the number 26 in decimal, that's a 16 plus an 8 plus a 2, all powers of 2. And that yields the following binary number, 00101010. So the conversion is pretty simple. It pretty much involves just the powers of two. Now, one of things that we do because it gets tiring to say zero zero one zero one zero one zero, We'll take a byte and represent it as. Two hexadecimal digits. Hexadecimal digits are base 16 digits. So now instead of a ones column or a twos column. We have a ones column and a 16 columns. So in this case we would represent a byte by the digits 0, 0. Each one of these is a group of four binary digits. And we can range all the way up to FF. The, case of all one binary digits, like we had above. The table on the right, shows you the correspondence of the numbers from 0 to 15, in both binary and hexadecimal. So, you can see that we use the numbers 0 through 9 because we're already familiar with those. We use those in our, in our decimal system, but those other six values we need from 10 to 15 in bay 16, we'll use the letters A through F. And the correspondence to the binary digits is right here. So, we can see we can go, we go from the initials all 0's to a final all 1's. 'Kay. In the C language, we can write these expressions these values, very easily by using the notation 0x to indicate a hex number. And a number in base 16. Here, we see an eight digit hexidecimal number. Since the digits of 4 bits each, that's a 32 bit long number that will be represented in for bytes. And in C, it doesn't matter whether we use upper case or lower case letters. Either one will work. Okay, so now that we have an idea of how the numb-, numbers are represented we'll be going a lot more into that soon. But we have our basic integers here. How do we find these in memory? So the first thing we should understand is that memory has been organized also in bytes. Basically it's just a big long list of bytes, and each one of these bytes has and address or an index that we can use to refer to it. the if we start at the index that begins with all zeros and we end at the index of all one's or all f's in hexidecimal notation. All right, so basically we have a byte. At the start followed by another byte another byte another byte all the way to the end. How many bytes do we have? Well, that depends on how many bits we have in the address. that's possible and we're going to see that we have 32-bit machines that use 32-bit addresses. And the more modern 64-bit machines that use 64-bits of address. All right. Where do those numbers go in these little bytes in memory, these locations, is dependent on the compiler and the run-time system of our machine. It will decide where the various numbers are stored, and allocates them by finding a place for them in that memory. Alright so let's talk a little bit about machine words we've been I've been mentioned that maybe we'll have machines with 32-bit addresses or 64-bit addresses. that, that amount corresponds to the word size of the machine. its all the word sizes also the nominal size of all the integer values we use so when a 32-bit machine will have integers that can go up to 32 binary digits. And this includes address represented in words. So what, as I mentioned before, until recently, most machines used 32-bits, or 4 byte words. In the old days, we also had machines with 16-bit words or even at the very beginning, 8-bit words. This limits addresses to 4 gigabytes. That's how many different numbers we can have in 32-bits. Therefore how many different addresses that we can have in the memory. This has become now too small for memory, memory for many memory intensive applications. Many of our applications use much more space than that. So computers have changed to use 64 bit words 8 bytes long and now we can have 2 to the 64 different addresses. Which leads to 18 exabytes of data to be addressed. That's a huge amount, and we haven't run into that wall yet, but we probably will one day. So for backward compatibility though, many CPUs still support different word sizes. we'll be looking primarily at 64 and 32-bits. But there are some instructions that you'll encounter that might manipulate 16-bit quantities or 8-bit, one byte quantities. Alright, so how do we refer to then these larger things in memory? We have mentioned bytes so far. Well, that's what we see here on the right, in the slide. We have an address corresponding to each byte of the memory, each individual little cell, where we can put an 8 bit quantity. But I just said that we use 32-bit words to represent numbers in many of our machines. So 4 bytes will have to be grouped together to represent a single word. Well, what address are we going to give that word? The next one after it will also require an address what will we use for that. In a 64-bit machine of course, will have to group the first 8 bytes into a word. And then we have to decide on the address for that. To have a uniform policy, what we do is we say, the address of the word is the address of the first byte in the word. So in this case we would have the first 32-bit word would have the address 000 because it starts at that byte. The next word's address though is 0004. And it doesn't make any sense to talk about the words between those at addresses 1, 2, and 3, because we're going to be grouping things. In this 4 byte structure. So we want to move on to the next four byte structure. So the next address will be four, the next address will be eight, the next address after that will be 12, and so on. And these are in base 10.