Activity 2.3.1 — Hexadecimal and Octal Number Systems¶

Learning Objectives¶

By the end of this lesson, students will be able to:

Explain why hexadecimal and octal number systems are used in digital electronics.
Count in decimal, binary, octal, and hexadecimal.
Convert between decimal ↔ octal using Successive Division and Weighted Multiplication.
Convert between decimal ↔ hexadecimal using Successive Division and Weighted Multiplication.
Convert directly between binary, octal, and hexadecimal using the grouping shortcut.

Vocabulary¶

Vocabulary (click to expand)

Term	Definition
Base / Radix	The number of unique symbols a number system uses (e.g., Base 10 = decimal)
Octal	Base-8 number system; digits 0–7
Hexadecimal	Base-16 number system; digits 0–9 and A–F
LSD	Least Significant Digit — the rightmost digit (lowest place value)
MSD	Most Significant Digit — the leftmost digit (highest place value)
Data Bus	A group of parallel wires that transfer data within a computer
Successive Division	Method for converting decimal → another base by repeated division
Weighted Multiplication	Method for converting any base → decimal by multiplying by place values
Bit-Weighting Factor	The positional value of a digit (e.g., 2⁰=1, 2¹=2, 8⁰=1, 16¹=16)

Part 1: Why Do We Need More Number Systems?¶

The Problem with Binary¶

Humans think in decimal (base 10). Computers operate in binary (base 2). The mismatch gets worse as data buses get wider.

A value on a 32-bit data bus looks like this in binary:

0110 1001 0111 0001 0011 0100 1100 1010

That's 32 characters — hard to read, easy to mistype.

The Solution: Compact Notation¶

System	Base	Digits	Compact?
Binary	2	0, 1	❌ Very long
Octal	8	0–7	✅ Shorter
Hexadecimal	16	0–9, A–F	✅✅ Most compact

The same 32-bit value above in hex is just: 697134CA — 8 characters!

Why powers of 2? Binary, octal (2³=8), and hexadecimal (2⁴=16) are all powers of 2, which means they have a clean, direct relationship to binary. This is why they are used in computing.

Part 2: Counting in All Four Systems¶

The table below shows the numbers 0–20 in all four number systems. Study the patterns — especially where the hexadecimal system uses letters instead of multi-digit numbers.

Decimal	Binary	Octal	Hexadecimal
0	00000	0	0
1	00001	1	1
2	00010	2	2
3	00011	3	3
4	00100	4	4
5	00101	5	5
6	00110	6	6
7	00111	7	7
8	01000	10	8
9	01001	11	9
10	01010	12	A
11	01011	13	B
12	01100	14	C
13	01101	15	D
14	01110	16	E
15	01111	17	F
16	10000	20	10
17	10001	21	11
18	10010	22	12
19	10011	23	13
20	10100	24	14

Key insight: In hexadecimal, A represents 10, B = 11, C = 12, D = 13, E = 14, and F = 15. A number system needs unique symbols — "10" is already two symbols, so letters A–F fill the gap.

Subscript notation: Always subscript the base to avoid confusion.
Examples: 94₁₀ = decimal 94, 134₈ = octal 134, 5E₁₆ = hex 5E
An "H" suffix is also acceptable: 5EH

Part 3: The Two Conversion Methods (Universal)¶

These two processes work for any base conversion, not just binary.

Method 1: Successive Division (Decimal → Any Base)¶

Used to convert FROM decimal TO any other base.

Steps:

Divide the decimal number by the target base.
Record the remainder — this becomes the LSD (rightmost digit).
If the quotient is 0, stop. Otherwise, use the quotient as the new number and repeat from Step 1.
Read remainders from bottom to top → this is your converted number.

Method 2: Weighted Multiplication (Any Base → Decimal)¶

Used to convert FROM any base TO decimal.

Steps:

Assign each digit its bit-weighting factor based on its position:
Position 0 (rightmost) = Base⁰
Position 1 = Base¹
Position 2 = Base²
And so on...
Multiply each digit by its bit-weighting factor.
Sum all the products to get the decimal result.

Part 4: Decimal ↔ Octal Conversion¶

Decimal → Octal (Successive Division by 8)¶

Worked Example: Convert 94₁₀ to octal.

94 ÷ 8 = 11  remainder  6  ← LSD
11 ÷ 8 =  1  remainder  3
 1 ÷ 8 =  0  remainder  1  ← MSD (stop, quotient = 0)

Read remainders bottom to top:  1, 3, 6

Result:  94₁₀ = 136₈

Remember: Read remainders from BOTTOM (last) to TOP (first) — that's MSD to LSD order.

Practice Problem — Decimal → Octal¶

Convert 189₁₀ into its octal equivalent. Work it out before checking the solution below.

Show Solution

189 ÷ 8 = 23  remainder  5  ← LSD
 23 ÷ 8 =  2  remainder  7
  2 ÷ 8 =  0  remainder  2  ← MSD

Read bottom to top:  2, 7, 5

Result:  189₁₀ = 275₈

Octal → Decimal (Weighted Multiplication by powers of 8)¶

Bit-weighting factors for octal:

Position	3	2	1	0
Factor	8³=512	8²=64	8¹=8	8⁰=1

Worked Example: Convert 136₈ to decimal.

  1 × 8² = 1 × 64 =  64
  3 × 8¹ = 3 × 8  =  24
  6 × 8⁰ = 6 × 1  =   6
                   ------
  Sum =             94

Result:  136₈ = 94₁₀  ✓

Practice Problem — Octal → Decimal¶

Convert 134₈ into its decimal equivalent. Work it out before checking the solution below.

Show Solution

  1 × 8² = 1 × 64 =  64
  3 × 8¹ = 3 × 8  =  24
  4 × 8⁰ = 4 × 1  =   4
                   ------
  Sum =             92

Result:  134₈ = 92₁₀

Part 5: Decimal ↔ Hexadecimal Conversion¶

Decimal → Hexadecimal (Successive Division by 16)¶

Worked Example: Convert 94₁₀ to hexadecimal.

94 ÷ 16 = 5  remainder  14 → E  ← LSD
 5 ÷ 16 = 0  remainder   5  ← MSD (stop, quotient = 0)

Read bottom to top:  5, E

Result:  94₁₀ = 5E₁₆  (or 5EH)

Reminder: Remainders 10–15 become letters A–F!

Remainder	Hex Digit
10	A
11	B
12	C
13	D
14	E
15	F

Practice Problem — Decimal → Hexadecimal¶

Convert 429₁₀ into its hexadecimal equivalent. Work it out before checking the solution.

Show Solution

429 ÷ 16 = 26  remainder  13 → D  ← LSD
 26 ÷ 16 =  1  remainder  10 → A
  1 ÷ 16 =  0  remainder   1  ← MSD

Read bottom to top:  1, A, D

Result:  429₁₀ = 1AD₁₆  (or 1ADH)

Hexadecimal → Decimal (Weighted Multiplication by powers of 16)¶

Bit-weighting factors for hexadecimal:

Position	3	2	1	0
Factor	16³=4096	16²=256	16¹=16	16⁰=1

Worked Example: Convert 5E₁₆ to decimal.

  5 × 16¹ = 5 × 16 = 80
  E × 16⁰ = 14 × 1 = 14
                    -----
  Sum =             94

Result:  5E₁₆ = 94₁₀  ✓

Practice Problem — Hexadecimal → Decimal¶

Convert B2EH into its decimal equivalent. Work it out before checking the solution.

Show Solution

B = 11, 2 = 2, E = 14

  B × 16² = 11 × 256 = 2816
  2 × 16¹ =  2 × 16  =   32
  E × 16⁰ = 14 × 1   =   14
                       -------
  Sum =                 2862

Result:  B2EH = 2862₁₀

Part 6: Cross-Base Conversions (Hex ↔ Octal, Octal ↔ Binary)¶

When converting between non-decimal bases (hex → octal, octal → binary), there is no direct formula. You must use decimal as a stepping stone:

Hex → Decimal → Octal

Octal → Decimal → Binary

Practice Problem — Hexadecimal → Octal¶

Convert 5AH into its octal equivalent.

Show Solution

Step 1: Hex → Decimal

  5 × 16¹ = 5 × 16 = 80
  A × 16⁰ = 10 × 1 = 10
                     ----
  Sum =              90

  5AH = 90₁₀

Step 2: Decimal → Octal

90 ÷ 8 = 11  remainder  2  ← LSD
11 ÷ 8 =  1  remainder  3
 1 ÷ 8 =  0  remainder  1  ← MSD

Result:  90₁₀ = 132₈

Final:  5AH = 132₈

Practice Problem — Octal → Binary¶

Convert 132₈ into its binary equivalent.

Show Solution

Step 1: Octal → Decimal

  1 × 8² = 1 × 64 =  64
  3 × 8¹ = 3 × 8  =  24
  2 × 8⁰ = 2 × 1  =   2
                   ------
  Sum =             90

  132₈ = 90₁₀

Step 2: Decimal → Binary

90 ÷ 2 = 45  remainder  0  ← LSD
45 ÷ 2 = 22  remainder  1
22 ÷ 2 = 11  remainder  0
11 ÷ 2 =  5  remainder  1
 5 ÷ 2 =  2  remainder  1
 2 ÷ 2 =  1  remainder  0
 1 ÷ 2 =  0  remainder  1  ← MSD

Result:  90₁₀ = 1011010₂

Final:  132₈ = 1011010₂

Part 7: The Binary ↔ Octal ↔ Hex Shortcut ⚡¶

Since binary, octal, and hexadecimal are all powers of 2, there is a direct shortcut that bypasses decimal entirely.

Conversion	Group Size	Reason
Binary ↔ Octal	3 binary bits	2³ = 8
Binary ↔ Hexadecimal	4 binary bits	2⁴ = 16

Binary → Octal¶

Start from the right, group binary digits into sets of 3.
Add leading zeros to the leftmost group if needed.
Convert each group of 3 bits to its octal digit (0–7).

Binary → Hexadecimal¶

Start from the right, group binary digits into sets of 4.
Add leading zeros to the leftmost group if needed.
Convert each group of 4 bits to its hex digit (0–F).

Worked Example — Shortcut¶

Binary:  1 0 1 1 0 1 0₂

Octal grouping (sets of 3, right to left):
  001 | 011 | 010
   1     3     2
Result: 132₈ ✓

Hex grouping (sets of 4, right to left):
  0101 | 1010
    5      A
Result: 5AH ✓

This shortcut only works between number systems whose bases are powers of 2. It does NOT work for decimal conversions.

Practice Problem — Shortcut Technique¶

Use the shortcut to convert A6₁₆ into binary and octal. Check with a calculator.

Show Solution

Step 1: Hex → Binary (expand each hex digit to 4 binary bits)

A = 1010
6 = 0110

A6₁₆ = 1010 0110₂

Step 2: Binary → Octal (group binary into sets of 3)

Rearrange:  001 | 010 | 011 | 0  → need 8 bits: 10100110
Group right-to-left: 010 | 100 | 110 → Wait, let's be precise:

10100110₂ → split right to left into groups of 3:
  010 | 100 | 110
   2     4     6

Result: A6₁₆ = 10100110₂ = 246₈

Calculator check: A6₁₆ = 166₁₀ = 246₈ ✓

Summary: Conversion Road Map¶

                ┌─────────────────────┐
                │      DECIMAL (10)   │
                └──────┬──────────────┘
          Succ. Div ↑↓ Weighted Mult
       ┌───────────────┼───────────────┐
       ▼               ▼               ▼
  BINARY (2)      OCTAL (8)       HEX (16)
       │               │               │
       └───────────────┴───────────────┘
              ↕ Direct shortcut (grouping)
              (only works between powers of 2)

Conversion Direction	Method
Decimal → Any Base	Successive Division (divide by base, collect remainders bottom→top)
Any Base → Decimal	Weighted Multiplication (multiply each digit × base^position, sum all)
Binary ↔ Octal	Group binary into 3s
Binary ↔ Hex	Group binary into 4s
Hex ↔ Octal	Go through decimal (Hex→Dec→Oct or Oct→Dec→Hex)

Key Reminders for the Activity¶

Always subscript your bases to avoid confusion (e.g., 94₁₀, 136₈, 5EH).
When using successive division, read remainders bottom to top (LSD → MSD).
When converting hex → decimal, replace letters with their numeric values first (A=10, B=11, C=12, D=13, E=14, F=15).
The binary ↔ octal/hex shortcut requires adding leading zeros to fill out groups on the left side.
Cross-base conversions (like hex → octal) require a decimal step in the middle.

📝 Quick Reference: Hex Digit Values¶

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