## 3.3 Data Organization

In pure mathematics a value may take an arbitrary number of bits. Computers, on the other hand, generally work with some specific number of bits. Common collections are single bits, groups of four bits (called nibbles), groups of eight bits (bytes), groups of 16 bits (words), groups of 32 bits (double words or dwords), groups of 64-bits (quad words or qwords), and more. The sizes are not arbitrary. There is a good reason for these particular values. This section will describe the bit groups commonly used on the Intel 80x86 chips.

### 3.3.1 Bits

The smallest "unit" of data on a binary computer is a single bit. Since a single bit is capable of representing only two different values (typically zero or one) you may get the impression that there are a very small number of items you can represent with a single bit. Not true! There are an infinite number of items you can represent with a single bit.

With a single bit, you can represent any two distinct items. Examples include zero or one, true or false, on or off, male or female, and right or wrong. However, you are not limited to representing binary data types (that is, those objects which have only two distinct values). You could use a single bit to represent the numbers 723 and 1,245. Or perhaps 6,254 and 5. You could also use a single bit to represent the colors red and blue. You could even represent two unrelated objects with a single bit. For example, you could represent the color red and the number 3,256 with a single bit. You can represent any two different values with a single bit. However, you can represent only two different values with a single bit.

To confuse things even more, different bits can represent different things. For example, one bit might be used to represent the values zero and one, while an adjacent bit might be used to represent the values true and false. How can you tell by looking at the bits? The answer, of course, is that you can't. But this illustrates the whole idea behind computer data structures: data is what you define it to be. If you use a bit to represent a boolean (true/false) value then that bit (by your definition) represents true or false. For the bit to have any real meaning, you must be consistent. That is, if you're using a bit to represent true or false at one point in your program, you shouldn't use the true/false value stored in that bit to represent red or blue later.

Since most items you'll be trying to model require more than two different values, single bit values aren't the most popular data type you'll use. However, since everything else consists of groups of bits, bits will play an important role in your programs. Of course, there are several data types that require two distinct values, so it would seem that bits are important by themselves. However, you will soon see that individual bits are difficult to manipulate, so we'll often use other data types to represent boolean values.

### 3.3.2 Nibbles

A nibble is a collection of four bits. It wouldn't be a particularly interesting data structure except for two items: BCD (binary coded decimal) numbers1 and hexadecimal numbers. It takes four bits to represent a single BCD or hexadecimal digit. With a nibble, we can represent up to 16 distinct values since there are 16 unique combinations of a string of four bits:

```0000

0001

0010

0011

0100

0101

0110

0111

1000

1001

1010

1011

1100

1101

1110

1111

```

In the case of hexadecimal numbers, the values 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F are represented with four bits (see "The Hexadecimal Numbering System" on page 60). BCD uses ten different digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and requires four bits (since you can only represent eight different values with three bits). In fact, any sixteen distinct values can be represented with a nibble, but hexadecimal and BCD digits are the primary items we can represent with a single nibble.

### 3.3.3 Bytes

Without question, the most important data structure used by the 80x86 microprocessor is the byte. A byte consists of eight bits and is the smallest addressable datum (data item) on the 80x86 microprocessor. Main memory and I/O addresses on the 80x86 are all byte addresses. This means that the smallest item that can be individually accessed by an 80x86 program is an eight-bit value. To access anything smaller requires that you read the byte containing the data and mask out the unwanted bits. The bits in a byte are normally numbered from zero to seven as shown in Figure 3.1.

Figure 3.1 Bit Numbering

Bit 0 is the low order bit or least significant bit, bit 7 is the high order bit or most significant bit of the byte. We'll refer to all other bits by their number.

Note that a byte also contains exactly two nibbles (see Figure 3.2).

Figure 3.2 The Two Nibbles in a Byte

Bits 0..3 comprise the low order nibble, bits 4..7 form the high order nibble. Since a byte contains exactly two nibbles, byte values require two hexadecimal digits.

Since a byte contains eight bits, it can represent 28, or 256, different values. Generally, we'll use a byte to represent numeric values in the range 0..255, signed numbers in the range -128..+127 (see "Signed and Unsigned Numbers" on page 69), ASCII/IBM character codes, and other special data types requiring no more than 256 different values. Many data types have fewer than 256 items so eight bits is usually sufficient.

Since the 80x86 is a byte addressable machine (see the next volume), it turns out to be more efficient to manipulate a whole byte than an individual bit or nibble. For this reason, most programmers use a whole byte to represent data types that require no more than 256 items, even if fewer than eight bits would suffice. For example, we'll often represent the boolean values true and false by 000000012 and 000000002 (respectively).

Probably the most important use for a byte is holding a character code. Characters typed at the keyboard, displayed on the screen, and printed on the printer all have numeric values. To allow it to communicate with the rest of the world, PCs use a variant of the ASCII character set (see "The ASCII Character Encoding" on page 97). There are 128 defined codes in the ASCII character set. PCs typically use the remaining 128 possible values for extended character codes including European characters, graphic symbols, Greek letters, and math symbols.

Because bytes are the smallest unit of storage in the 80x86 memory space, bytes also happen to be the smallest variable you can create in an HLA program. As you saw in the last chapter, you can declare an eight-bit signed integer variable using the int8 data type. Since int8 objects are signed, you can represent values in the range -128..+127 using an int8 variable (see "Signed and Unsigned Numbers" on page 69 for a discussion of signed number formats). You should only store signed values into int8 variables; if you want to create an arbitrary byte variable, you should use the byte data type, as follows:

```static

byteVar: byte;

```

The byte data type is a partially untyped data type. The only type information associated with byte objects is their size (one byte). You may store any one-byte object (small signed integers, small unsigned integers, characters, etc.) into a byte variable. It is up to you to keep track of the type of object you've put into a byte variable.

### 3.3.4 Words

A word is a group of 16 bits. We'll number the bits in a word starting from zero on up to fifteen. The bit numbering appears in Figure 3.3.

Figure 3.3 Bit Numbers in a Word

Like the byte, bit 0 is the low order bit. For words, bit 15 is the high order bit. When referencing the other bits in a word, use their bit position number.

Notice that a word contains exactly two bytes. Bits 0 through 7 form the low order byte, bits 8 through 15 form the high order byte (see Figure 3.4).

Figure 3.4 The Two Bytes in a Word

Naturally, a word may be further broken down into four nibbles as shown in Figure 3.5.

Figure 3.5 Nibbles in a Word

Nibble zero is the low order nibble in the word and nibble three is the high order nibble of the word. We'll simply refer to the other two nibbles as "nibble one" or "nibble two. "

With 16 bits, you can represent 216 (65,536) different values. These could be the values in the range 0..65,535 or, as is usually the case, -32,768..+32,767, or any other data type with no more than 65,536 values. The three major uses for words are signed integer values, unsigned integer values, and UNICODE characters.

Words can represent integer values in the range 0..65,535 or -32,768..32,767. Unsigned numeric values are represented by the binary value corresponding to the bits in the word. Signed numeric values use the two's complement form for numeric values (see "Signed and Unsigned Numbers" on page 69). As UNICODE characters, words can represent up to 65,536 different characters, allowing the use of non-Roman character sets in a computer program. UNICODE is an international standard, like ASCII, that allows commputers to process non-Roman characters like Asian, Greek, and Russian characters.

Like bytes, you can also create word variables in an HLA program. Of course, in the last chapter you saw how to create sixteen-bit signed integer variables using the int16 data type. To create an arbitrary word variable, just use the word data type, as follows:

```static

w: word;

```

### 3.3.5 Double Words

A double word is exactly what its name implies, a pair of words. Therefore, a double word quantity is 32 bits long as shown in Figure 3.6.

Figure 3.6 Bit Numbers in a Double Word

Naturally, this double word can be divided into a high order word and a low order word, four different bytes, or eight different nibbles (see Figure 3.7).

Figure 3.7 Nibbles, Bytes, and Words in a Double Word

Double words can represent all kinds of different things. A common item you will represent with a double word is a 32-bit integer value (which allows unsigned numbers in the range 0..4,294,967,295 or signed numbers in the range -2,147,483,648..2,147,483,647). 32-bit floating point values also fit into a double word. Another common use for dword objects is to store pointer variables.

In the previous chapter, you saw how to create 32-bit (dword) signed integer variables using the int32 data type. You can also create an arbitrary double word variable using the dword data type as the following example demonstrates:

```static

d: dword;

```

1Binary coded decimal is a numeric scheme used to represent decimal numbers using four bits for each decimal digit.
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://www.plantation-productions.com/Webster/www.artofasm.com/Linux/HTML/DataRepresentation2.html", "fetch_time": 1713195797000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2024-18/segments/1712296817002.2/warc/CC-MAIN-20240415142720-20240415172720-00878.warc.gz", "warc_record_offset": 892115768, "warc_record_length": 6421, "token_count": 2512, "char_count": 10640, "score": 4.09375, "int_score": 4, "crawl": "CC-MAIN-2024-18", "snapshot_type": "latest", "language": "en", "language_score": 0.9360381960868835, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00051-of-00064.parquet"}