Computer • Chapter 03

🔢 Number System

Number System
Binary • Octal • Decimal • Hexadecimal • Conversions • Fractions
Source coverage: uploaded Computer eBook — Chapter 3 (PDF pages 11–14). Content kept source-grounded; obvious print/typesetting issues are flagged rather than silently changed.
Base 2
Binary • 0, 1
Base 8
Octal • 0–7
Base 10
Decimal • 0–9
Base 16
Hex • 0–9, A–F
🧠 Introduction
  • A Number System is a method of representing numbers using a set of symbols (digits).
  • A number system is a method of representing numbers using a set of symbols called digits.
  • Computers use binary (0 and 1) because electronic circuits work through ON (1) and OFF (0) states.
  • Computers use binary (0 and 1) because their operations are based on the ON (1) and OFF (0) states of electronic circuits.
NUMBER SYSTEMNumber System BINARYBase 2Digits: 0, 1 OCTALBase 8Digits: 0–7 DECIMALBase 10Digits: 0–9 HEXADECIMALBase 160–9, A–F
📋 Types of Number Systems
Number SystemBaseDigits UsedSource Example
Binary20, 1(1011)2
Octal80 to 7(725)8
Decimal100 to 9(593)10
Hexadecimal160–9, A–F(2A)16
2
Binary
Digits: 0, 1
Simple for computer circuits
8
Octal
Digits: 0–7
3 binary bits = 1 octal digit
10
Decimal
Digits: 0–9
Human-friendly representation
16
Hexadecimal
Digits: 0–9, A–F
4 binary bits = 1 hex digit
0️⃣1️⃣ Binary Number System — Base 2
  • Digits: 0, 1
  • Place values: 20, 21, 22, 23 ...
  • Advantages: simple for computer circuits; less chance of error in digital processing.
(1011)2 = 1×23 + 0×22 + 1×21 + 1×20 = 1110
8️⃣ Octal Number System — Base 8
  • Digits: 0 to 7
  • Place values: 80, 81, 82 ...
  • Octal is used as a short form of binary.
  • 3 binary bits = 1 octal digit
(725)8 = 7×82 + 2×81 + 5×80 = 448 + 16 + 5 = 46910
🔟 Decimal Number System — Base 10
  • Digits: 0 to 9
  • Place values: 100, 101, 102 ...
  • Use: Human-friendly representation of numbers.
(593)10 = 5×102 + 9×101 + 3×100
PDF PRINT NOTE
Correct arithmetic: 500 + 90 + 3 = 593. The units digit contributes 3 × 10⁰ = 3.
🅰️ Hexadecimal Number System — Base 16
  • Digits: 0–9 and A–F
  • A = 10, B = 11 ... F = 15
  • Place values: 160, 161, 162 ...
  • Used as a short form of binary.
  • 4 binary bits = 1 hexadecimal digit
(2A)16 = 2×161 + 10×160 = 32 + 10 = 4210
🔄 Relationship Between Number Systems
3 Binary Bits ⇄ 1 Octal Digit
Binary ↔ Octal grouping
4 Binary Bits ⇄ 1 Hex Digit
Binary ↔ Hexadecimal grouping
  • Decimal → another base: divide the integer part repeatedly; multiply the fractional part repeatedly by the target base. Reverse conversion uses positional weights.
  • Binary/Octal/Hexadecimal → Decimal: positional values are used.
Decimal → Binary
  • Repeatedly divide by 2.
  • Write remainders.
  • Read the remainders in reverse order.
13÷2 = 6 R1 → 6÷2 = 3 R0 → 3÷2 = 1 R1 → 1÷2 = 0 R1
Answer = 11012
Binary → Decimal

Multiply each bit by 2n, where n is the position from the right starting at 0.

(1011)2 = 1×23 + 0×22 + 1×21 + 1×20 = 1110
Binary ⇄ Octal
  • Binary → Octal: group integer bits in 3 from the right; group fractional bits in 3 from the binary point towards the right. Pad with zeros at the outer ends if needed.
  • Octal → Binary: convert each octal digit to 3 binary bits.
(101101)2 = (101 101) → 558
(725)8 → (111010101)2
Binary ⇄ Hexadecimal
  • Binary → Hex: group integer bits in 4 from the right; group fractional bits in 4 from the binary point towards the right. Pad with zeros at the outer ends if needed.
  • Hex → Binary: each hex digit becomes 4 binary bits.
(11010110)2 → D616
(2F)16 → (00101111)2
8️⃣ / 1️⃣6️⃣ Decimal → Octal / Hexadecimal
  • Use repeated division by 8 for octal and by 16 for hexadecimal.
  • Source example: 12510 → divide by 8 → remainder method → 1758.
➗ Fractions in Number System
  • For fractional part conversion, Decimal → Binary uses repeated multiplication by 2.
  • At each step, take the integer part and repeat the process with the fractional part.
0.625 × 2 = 1.25 (take 1) → 0.25 × 2 = 0.5 (take 0) → 0.5 × 2 = 1.0 (take 1). Therefore (0.625)₁₀ = (0.101)₂.
PDF TYPESETTING NOTE
The first product is 1.25. Read the extracted integer bits in the order they are produced: 1, 0, 1.
(0.101)2 = 1×2−1 + 0×2−2 + 1×2−3 = 0.5 + 0 + 0.125 = 0.625
🎯 Advantages & Uses
SystemUse / Advantage from PDF
BinaryProcessing in computers; simple for circuits; less chance of error in digital processing.
OctalShort representation of binary in older systems.
DecimalHuman readable / human-friendly representation.
HexadecimalMemory addresses, color codes, machine instructions.
⚡ Master Summary Table
BaseNameDigits UsedBits Relation
2Binary0, 11 bit
8Octal0–73 bits
10Decimal0–9—
16Hexadecimal0–9, A–F4 bits
⚡ Quick Recall — Bases
Binary = 2 • Octal = 8 • Decimal = 10 • Hexadecimal = 16
⚡ Quick Recall — Grouping
3 binary bits = 1 octal digit
4 binary bits = 1 hexadecimal digit
⚠ Exam Trap
Octal digits are only 0–7; digits 8 and 9 are not part of octal.
⚠ Exam Trap
Hexadecimal uses A–F for values 10–15.
🎯 Conversion Memory Rule
Decimal → another base: repeated division (integer part).
Decimal fraction → binary: repeated multiplication by 2.
🎯 Use Memory Rule
Binary → computer processing • Octal → older binary shorthand • Decimal → human readable • Hex → memory/color/machine instructions.