3.2 · Number Systems

Goal: convert integers fluently between denary (decimal), binary, and hexadecimal in both directions, with full working.

The three number systems on the syllabus

System	Base	Digits used
Denary (decimal)	10	0 1 2 3 4 5 6 7 8 9
Binary	2	0 1
Hexadecimal (hex)	16	0 1 2 3 4 5 6 7 8 9 A B C D E F

You will be asked to convert between any pair, in both directions.

Place value, the universal idea

Any number system has place values that are powers of the base.

Denary (base 10)

Place	1000	100	10	1
Power	10³	10²	10¹	10⁰
Digit	2	3	7	4

2 × 1000 + 3 × 100 + 7 × 10 + 4 × 1 = 2374

Binary (base 2)

Place	128	64	32	16	8	4	2	1
Power	2⁷	2⁶	2⁵	2⁴	2³	2²	2¹	2⁰
Digit	1	0	1	1	0	1	0	1

128 + 32 + 16 + 4 + 1 = 181₁₀

Hexadecimal (base 16)

Place	4096	256	16	1
Power	16³	16²	16¹	16⁰
Digit	0	2	A	7

2 × 256 + 10 × 16 + 7 × 1 = 679₁₀

Quick reference table (0–15)

You must memorise this table cold.

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

Denary → Binary

Method 1 · Successive division by 2

Divide repeatedly by 2 and read remainders bottom to top.

Convert 45₁₀:

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   ← top

Reading bottom to top → 101101₂.

Method 2 · Subtract the largest power of 2

Power of 2	Used?	Remaining
32	✓	45 − 32 = 13
16	✗	13
8	✓	13 − 8 = 5
4	✓	5 − 4 = 1
2	✗	1
1	✓	1 − 1 = 0

Result: 101101₂.

Binary → Denary

Multiply each bit by its place value and sum.

11010110₂ → 128 + 64 + 16 + 4 + 2 = 214₁₀.

Tip: write the place values above the bits before adding.

Denary → Hex

Method 1 · Successive division by 16

Convert 2750₁₀:

2750 ÷ 16 = 171 remainder 14 (E)
 171 ÷ 16 = 10  remainder 11 (B)
  10 ÷ 16 = 0   remainder 10 (A)   ← top

Reading bottom to top → ABE₁₆.

Method 2 · Via binary

Convert to binary first, then group in 4s (right to left).

2750₁₀ = 1010 1011 1110₂ → A B E → ABE₁₆.

Hex → Denary

Multiply each digit by its place value and sum.

3F₁₆ = 3 × 16 + 15 × 1 = 63₁₀

Binary ↔ Hex (the shortcut)

This conversion is the fastest because 1 hex digit = 4 binary bits.

Binary  : 1101 1010 0011 1111
            D    A    3    F
Hex     :        D A 3 F

Both directions take seconds once you have memorised the 0–15 table.

Always group binary in 4s before converting to hex

• Group from the right: 11010100110₂ → 0110 1010 0110 (pad with leading zeros).
• Then read: 6 A 6 → 6A6₁₆.

Common conversion practice

Denary	Binary	Hex
13	1101	D
25	11001	19
100	1100100	64
255	11111111	FF
256	100000000	100
1024	10000000000	400
65535	1111111111111111	FFFF

Memorising 255 = 0xFF and 65535 = 0xFFFF saves time in exam.

How many bits do I need?

For an unsigned integer N, the number of bits needed is ⌈log₂(N+1)⌉.

Range	Bits needed
0–1	1
0–3	2
0–7	3
0–15	4
0–255	8
0–1023	10
0–65535	16

n bits can represent 2ⁿ distinct values. You should be able to derive this for any small n.

Why hexadecimal at all?

Computers are binary. Why bother with hex?

1. Compactness — 1 hex digit replaces 4 binary digits.
2. Readability — RGB colour #FF8800 is much clearer than 11111111 10001000 00000000.
3. Easy bit-mapping — each hex digit corresponds neatly to 4 bits.
4. Standard in addresses — MAC addresses, IPv6, memory dumps all use hex.

Common student mistakes

• Reading division remainders top to bottom (it's bottom to top).
• Forgetting that hex A–F map to 10–15 (writing A = 11 is a frequent slip).
• Missing leading zeros when grouping binary into 4s — pad first.
• Confusing the base with the number of digits.

Practice activity

Convert without a calculator:

1. 87₁₀ → binary, hex
2. 0xC8 → denary, binary
3. 10110101₂ → denary, hex
4. 0xBEEF → binary
5. 255₁₀ + 1 → binary, hex

Answers

1. 1010111₂, 57₁₆
2. 200₁₀, 11001000₂
3. 181₁₀, B5₁₆
4. 1011 1110 1110 1111₂
5. 100000000₂, 100₁₆

Exam-style question

Q (4 marks): (a) Convert 2024₁₀ to binary, showing all working. (b) Convert your answer in (a) to hexadecimal.

Sample answer:

(a) Successive division by 2:

2024 ÷ 2 = 1012 r 0
1012 ÷ 2 =  506 r 0
 506 ÷ 2 =  253 r 0
 253 ÷ 2 =  126 r 1
 126 ÷ 2 =   63 r 0
  63 ÷ 2 =   31 r 1
  31 ÷ 2 =   15 r 1
  15 ÷ 2 =    7 r 1
   7 ÷ 2 =    3 r 1
   3 ÷ 2 =    1 r 1
   1 ÷ 2 =    0 r 1

2024₁₀ = 11111101000₂

(b) Group in 4s from the right (pad with leading 0s):

0111 1110 1000 → 7 E 8 → 2024₁₀ = 7E8₁₆

Key takeaways

• Three bases: 10, 2, 16.
• Master two methods for denary → binary (successive division + subtract powers).
• 1 hex digit = 4 binary bits is the fastest conversion shortcut.
• Memorise 0–15 in all three bases.

➡️ Next: 3.3 Binary Arithmetic & Overflow