3. Data representation

A computer only knows two states: electricity flows or it does not. Everything else, from a photo to a song or the letter “ñ”, is a pattern of zeros and ones. If you understand these patterns you can read a memory dump, fix a file that shows strange characters and know why your new disk is smaller than the box says.

How to use this page

1. Read sections 1 to 6 (about 50 minutes) and try the examples with paper and pencil.
2. Practise with the Bit Lab: the tabs Number bases and Conversions train sections 2 and 3, and the tabs Units and KB vs KiB train section 6. The lab explains the methods in detail, so this page only gives you the essentials.
3. Answer the questions at the end and do Activity 5 and 6.

1. Bits, bytes and words

• A bit (binary digit) is the smallest unit of information: 0 or 1.
• A nibble is 4 bits.
• A byte is 8 bits. With 8 bits you can make 2⁸ = 256 different patterns (from 0 to 255). Memories and files are counted in bytes.
• A word is the group of bits that a processor handles at once (for example 32 or 64 bits).

With n bits you can represent 2ⁿ different values. That is why one bit gives 2 values, 4 bits give 16 and 8 bits give 256.

Number or character?

Do not mix two ideas. A number base (binary, decimal, hexadecimal) is a way to write a number. A character encoding (ASCII, Unicode) is a table that says which number stands for which character. Then that number is stored as one or more bytes.

2. Binary (base 2)

Binary uses only two digits, 0 and 1. Each position has a weight that is a power of 2. In one byte the weights are, from left to right:

Position	7	6	5	4	3	2	1	0
Weight	128	64	32	16	8	4	2	1

Binary to decimal: add the weights of the positions that contain a 1.

0100 0001  =  64 + 1                  =  65
1011 0110  =  128 + 32 + 16 + 4 + 2   =  182

Decimal to binary (method of successive divisions): divide by 2 again and again and write the remainders. Read them from the last one to the first one.

13 ÷ 2 = 6  remainder 1
 6 ÷ 2 = 3  remainder 0
 3 ÷ 2 = 1  remainder 1
 1 ÷ 2 = 0  remainder 1      →  13 = 1101 (read upwards)

You can also subtract weights: for 150, take 128 (150 − 128 = 22), then 16 (22 − 16 = 6), then 4 (6 − 4 = 2), then 2. The bits are 1001 0110.

Quick check

A binary number ending in 0 is even, and ending in 1 is odd. Adding a zero at the end doubles the value (101 is 5 and 1010 is 10).

3. Hexadecimal (base 16)

Binary numbers are long and hard to read. Hexadecimal uses 16 digits: 0–9 and then A–F. The big advantage is that one hex digit is exactly one nibble (4 bits), so two hex digits are one byte.

Decimal	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Binary	0000	0001	0010	0011	0100	0101	0110	0111	1000	1001	1010	1011	1100	1101	1110	1111
Hex	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F

Binary to hexadecimal: group the bits in fours, starting from the right, and replace each group by its hex digit. Hexadecimal to binary: do the opposite.

1010 1111  →  A F    →  0xAF  = 10×16 + 15 = 175
0x4F       →  0100 1111

A prefix such as 0x (0x4F) or a suffix such as h shows that a number is hexadecimal. Hex appears in many places: memory addresses, MAC addresses (3C:52:82:1A:F0:9B), colours on the web (#FF8800 means red 255, green 136, blue 0) and in file dumps.

4. Representing characters

4.1 ASCII

ASCII assigns a number from 0 to 127 (7 bits) to 128 characters: control codes, digits, English letters and punctuation. It is stored in one byte, and the top bit is unused.

Character	Decimal	Hex	Binary
0 (digit)	48	0x30	0011 0000
space	32	0x20	0010 0000
A	65	0x41	0100 0001
Z	90	0x5A	0101 1010
a	97	0x61	0110 0001
line feed (LF)	10	0x0A	0000 1010
carriage return (CR)	13	0x0D	0000 1101

Notice that A and a differ in one bit only: the lowercase code is the uppercase code plus 32 (0x20). The text Hi is stored as the two bytes 48 69 (hex).

ASCII has no letters like ñ, é or ç. Many “extended ASCII” tables appeared that use the 8th bit (codes 128 to 255) for those symbols, for example ISO-8859-1 (Latin-1) and Windows-1252. The problem is that each table gives a different meaning to the same code above 127.

4.2 Unicode and UTF-8

Unicode is a universal catalogue. It gives a unique number, called code point, to each character of almost all the writing systems of the world, plus symbols and emoji (well over 100,000 characters). A code point is written U+ followed by hexadecimal digits:

Character	Code point
A	U+0041
ñ	U+00F1
€	U+20AC
漢	U+6F22
😀	U+1F600

Unicode defines the code points, but not how to store them as bytes. That is the job of the UTF encodings (Unicode Transformation Format):

Encoding	Bytes per character	Notes
UTF-8	1 to 4 (variable)	The first 128 characters are identical to ASCII, so old ASCII files are valid UTF-8. It is the standard on the web and the default in Linux and macOS.
UTF-16	2 or 4	Used internally by Windows and by Java, among others.
UTF-32	4 (fixed)	Simple but wastes space.

Examples in UTF-8 (hexadecimal bytes):

Character	Code point	UTF-8 bytes
A	U+0041	41 (1 byte)
ñ	U+00F1	C3 B1 (2 bytes)
€	U+20AC	E2 82 AC (3 bytes)
😀	U+1F600	F0 9F 98 80 (4 bytes)

So the word Mañana has 6 characters but takes 7 bytes in UTF-8: 4D 61 C3 B1 61 6E 61. On Linux or macOS you can see the bytes of any text with the xxd tool:

printf 'Mañana' | xxd
# 00000000: 4d61 c3b1 616e 61                        Ma..ana

4.3 Mojibake: when the encoding is wrong

If a program reads a file with a different encoding from the one used to write it, you get mojibake: garbage characters. The classic case is a UTF-8 file opened as ISO-8859-1. The bytes C3 B1 of ñ are read as two separate characters, Ã (C3) and ± (B1):

Saved as UTF-8:       España   →   bytes 45 73 70 61 C3 B1 61
Read as ISO-8859-1:   España

The opposite also happens: an ISO-8859-1 file where ñ is the single byte F1 is invalid as UTF-8, so the program shows a replacement symbol (�). The solution is to open the file with the correct encoding, or to convert it (with a text editor or the iconv tool).

5. How the operating system is involved

The file system stores bytes. It does not know if they are text, a photo or a program. The OS and the applications must agree on how to interpret them:

• File names and text files. Modern systems use Unicode for file names, so you can name a file canción.txt. Inside the file, the encoding is a choice of the program that saved it. The encoding is not given by the extension.
• Terminals and fonts. The terminal must use the same encoding as the program that prints in it, and the font must contain a drawing (glyph) for each character. Missing glyphs appear as empty squares.
• Locale. The locale of the system sets the language, the date and number formats, the sorting rules and the default encoding for some tools.
• Line endings. Linux and macOS end a line with LF (0x0A). Windows uses two bytes, CR LF (0x0D 0x0A). A script written in Windows can fail in Linux because of the extra CR.
• BOM. Some UTF files start with a byte order mark. It is useless in UTF-8 and can break scripts; prefer UTF-8 without BOM.

6. Units of information

Digital information is measured in bits and bytes (b is bit, B is byte). Two families of prefixes exist:

Prefix (SI, decimal)	Value	Prefix (IEC, binary)	Value
kilo (kB)	10³ = 1,000 B	kibi (KiB)	2¹⁰ = 1,024 B
mega (MB)	10⁶ = 1,000,000 B	mebi (MiB)	2²⁰ = 1,048,576 B
giga (GB)	10⁹ B	gibi (GiB)	2³⁰ = 1,073,741,824 B
tera (TB)	10¹² B	tebi (TiB)	2⁴⁰ B

The difference grows with the size: about 2.4 % for kilo, 4.9 % for mega, 7.4 % for giga and 10 % for tera.

• Disk manufacturers and network speeds use SI units (decimal).
• Memory is naturally counted in powers of two, so RAM sizes are binary (an 8 GB RAM module is really 8 GiB).
• Each operating system or tool decides how to show sizes. Windows File Explorer divides by 1,024 but writes “GB”. macOS uses decimal units. Linux depends on the tool: many command-line tools (ls -lh, df -h) use powers of 1,024, while df -H uses powers of 1,000. Always check which unit is used.

Why does my 1 TB disk show less? The box says 1 TB = 10¹² bytes. Windows divides by 2³⁰ and writes GB:

10¹² ÷ 2³⁰ = 931.3   →   Windows shows about 931 GB (really 931 GiB)

The disk does not lose space. The two sides use different units.

Network speeds are measured in bits per second. To get bytes per second, divide by 8: a 100 Mb/s connection is at best 12.5 MB/s, so a 1 GB download takes at least 80 seconds.

Check what you have learned

1. Convert 1100 1010 to decimal and to hexadecimal.

   Answer

   128 + 64 + 8 + 2 = 202. In groups of four: 1100 is C and 1010 is A, so the hex value is 0xCA.

2. Convert decimal 99 to binary (8 bits) and to hexadecimal.

   Answer

   99 = 64 + 32 + 2 + 1, so the binary number is 0110 0011. The groups are 6 and 3, so the hex value is 0x63.

3. Which ASCII character has the code 0x4F? And what is the code of b?

   Answer

   0x4F = 79, which is the letter O (A is 65, so O is 65 + 14). The code of b is 98 (a is 97), or 0x62.

4. A web page shows “canción” instead of “canción”. What went wrong, and how do you fix it?

   Answer

   The text was saved in UTF-8, where ó takes two bytes (C3 B3), but the page is read as ISO-8859-1, which shows each byte as one character (Ã and ³). Fix it by choosing UTF-8 as the encoding when reading the file.

5. A shop sells a 2 TB external disk. How large will an operating system that uses binary units show it, approximately?

   Answer

   2 × 10¹² bytes ÷ 2³⁰ ≈ 1,863 GiB, or about 1.82 TiB. The disk is not defective: the box uses decimal units.

Glossary

English	Castellano
bit / byte / nibble	bit / byte (octeto) / cuarteto
word	palabra
binary / decimal / hexadecimal	binario / decimal / hexadecimal
number base	base de numeración
weight (of a position)	peso (de una posición)
character encoding	codificación de caracteres
ASCII	ASCII
Unicode	Unicode
code point	punto de código
UTF-8 / UTF-16 / UTF-32	UTF-8 / UTF-16 / UTF-32
mojibake	texto ilegible por codificación errónea
line ending	fin de línea
byte order mark (BOM)	marca de orden de bytes
prefix	prefijo
kilobyte (kB) / kibibyte (KiB)	kilobyte (kB) / kibibyte (KiB)

Credits

This page adapts the SOM course notes by Moisés Mas Rocamora, Daniel Álvarez Santamaría and Jana Taboada León, licensed under CC BY-NC-SA 4.0. This page is shared under the same licence.