🔢 Number Systems & Data Representation

Why Number Systems Matter

Humans think in decimal (base-10, using digits 0-9) because we have 10 fingers. But computers think in binary (base-2, using only 0 and 1) because electronic circuits have only two states — ON (1) and OFF (0).

Understanding number systems helps you grasp how computers store and process all data — from a simple number to a high-definition video.

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The Four Number Systems

System	Base	Digits Used	Used By
Binary	2	0, 1	Computers internally
Octal	8	0, 1, 2, 3, 4, 5, 6, 7	Older computer systems
Decimal	10	0, 1, 2, 3, 4, 5, 6, 7, 8, 9	Humans (everyday use)
Hexadecimal	16	0-9, A(10), B(11), C(12), D(13), E(14), F(15)	Memory addresses, colors, programming

Unary is the simplest counting idea, where one symbol is repeated to show quantity, but it is not a practical computer number system like binary, octal, decimal, or hexadecimal. It is mainly useful as a contrast point when understanding why place-value systems are more powerful.

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Binary Number System (Base-2)

The binary system is the language of computers. Every piece of data — text, images, audio, video — is ultimately stored as sequences of 0s and 1s.

Each binary digit is called a bit. Eight bits make one byte.

Place Values in Binary

Each position represents a power of 2 (from right to left):

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

Example: Binary 1101 = 1×8 + 1×4 + 0×2 + 1×1 = 13 in decimal.

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Decimal to Binary Conversion

Method: Divide by 2 repeatedly, note the remainders from bottom to top.

Example: Convert 25 to binary

Division	Quotient	Remainder
25 ÷ 2	12	1
12 ÷ 2	6	0
6 ÷ 2	3	0
3 ÷ 2	1	1
1 ÷ 2	0	1

Read remainders bottom to top: 25 = 11001 in binary

Binary to Decimal Conversion

Method: Multiply each bit by its place value and add.

Example: Convert 11010 to decimal = 1×16 + 1×8 + 0×4 + 1×2 + 0×1 = 16 + 8 + 0 + 2 + 0 = 26

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Octal Number System (Base-8)

Uses digits 0-7. Each octal digit represents exactly 3 binary bits.

Octal	Binary
0	000
1	001
3	011
5	101
7	111

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Hexadecimal Number System (Base-16)

Uses 0-9 and A-F (where A=10, B=11, C=12, D=13, E=14, F=15). Each hex digit represents exactly 4 binary bits (1 nibble).

Hex	Decimal	Binary
0	0	0000
5	5	0101
9	9	1001
A	10	1010
F	15	1111

Common use: Memory addresses, HTML colors (#FF5733 = Red:FF, Green:57, Blue:33), MAC addresses.

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Data Representation — How Computers Store Text

Since computers only understand 0s and 1s, every character (letter, digit, symbol) must be assigned a unique binary number. This is done using coding systems:

ASCII (American Standard Code for Information Interchange)

ASCII uses 7 bits to represent characters, giving 128 possible characters (0-127).

Character	ASCII Code	Binary
A	65	1000001
B	66	1000010
Z	90	1011010
a	97	1100001
0	48	0110000
Space	32	0100000

Extended ASCII uses 8 bits = 256 characters (includes accented letters, symbols).

Unicode

Unicode is a modern standard that supports over 1,00,000 characters from virtually every language in the world — including Hindi (Devanagari), Arabic, Chinese, Japanese, emojis, and mathematical symbols.

• UTF-8 — Most common encoding on the internet (uses 1-4 bytes per character)
• UTF-16 — Used by Windows and Java

ASCII can represent only English; Unicode can represent ALL languages including Hindi (हिंदी).

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Types of Data in Computers

Data Type	What it Represents	How it's Stored
Text	Letters, numbers, symbols	ASCII or Unicode codes
Numbers	Integers, decimals	Binary (integer) or Floating point
Images	Photos, graphics	Pixels with color values (BMP, JPEG, PNG)
Audio	Music, voice	Sampled waveforms (MP3, WAV)
Video	Moving images + audio	Frames + audio track (MP4, AVI)

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Summary Points

Concept	Key Details
Binary	Base-2 (0,1) — computer's native language
Decimal	Base-10 (0-9) — human system
Octal	Base-8 (0-7) — 1 octal digit = 3 bits
Hexadecimal	Base-16 (0-F) — 1 hex digit = 4 bits (1 nibble)
Decimal→Binary	Divide by 2, remainders bottom to top
Binary→Decimal	Multiply bits by powers of 2, add up
Bit	Smallest unit — 0 or 1
Byte	8 bits
Nibble	4 bits = 1 hex digit
ASCII	7-bit, 128 characters, English only
Extended ASCII	8-bit, 256 characters
Unicode	1,00,000+ characters, all languages including Hindi
UTF-8	Most common Unicode encoding on internet
A in ASCII	65
a in ASCII	97
0 in ASCII	48