Denary → Binary & Hexadecimal

Syllabus & Goals 3 min

Cambridge 1.1 · Number systems Paper 1 · Computer Systems

By the end of this lesson you can:

1. Convert a denary number to binary using two methods.
2. Convert between binary and hexadecimal in groups of four bits.
3. State why hexadecimal is used — MAC addresses, IP addresses, HTML colour codes and error codes.

Recap / Warm-Up 5 min

Last lesson you converted binary to denary by adding place values. Now we go the other way, and meet a third system.

Quick starter

Using the headings 128 64 32 16 8 4 2 1, which columns add up to 20?

Key Concept — converting & hexadecimal 14 min

Denary → binary: two methods

Method 1 — subtraction. Take away the largest power of 2 you can, again and again, until you reach 0. Put a 1 in each column you used.

Method 2 — division by 2. Divide the number by 2 repeatedly, writing the remainder each time. Read the remainders from bottom to top.

The hexadecimal (base 16) system

Hexadecimal — hex for short — is a base 16 system, so it needs 16 digits: 0–9 then A B C D E F.

Why bother with hex? Four real uses

A computer only works in binary, but long binary numbers are hard for people to read. Hex is shorter — one hex digit replaces four bits — so engineers use it for:

• Error codes — shown in hex, pointing to a memory location.
• MAC addresses — 48 bits as six pairs of hex digits.
• IP addresses — IPv6 is a 128-bit number written in hex.
• HTML colour codes — #RRGGBB, two hex digits each for red, green and blue.

Worked Example 12 min

(a) Denary 142 → binary

Method 1 (subtraction): 142 = 128 + 8 + 4 + 2.

So 142 = 10001110.

(b) Binary 101111100001 → hex

Split into groups of four from the right, then read each group from the table:

1011 1110 0001 → B E 1 → hex BE1.

(c) Hex 45A → denary

Multiply each digit by its column value (256, 16, 1), remembering A = 10:

(4 × 256) + (5 × 16) + (10 × 1) = 1024 + 80 + 10 = 1114.

The same idea as an algorithm

Cambridge pseudocode

// Convert a positive denary number to 8-bit binary (division-by-2 method)
DECLARE Number, Index : INTEGER
DECLARE Bits : ARRAY[1:8] OF INTEGER
Number ← 142
FOR Index ← 8 TO 1 STEP -1
    Bits[Index] ← Number MOD 2
    Number ← Number DIV 2
NEXT Index
OUTPUT "The 8 bits, most significant first:"
FOR Index ← 1 TO 8
    OUTPUT Bits[Index]
NEXT Index

The same algorithm in Python (IDLE)

# Convert a positive denary number to binary (division-by-2 method)
number = 142
bits = []
while number > 0:
    bits.insert(0, number % 2)
    number = number // 2
print("Binary is", bits)

Output

Binary is [1, 0, 0, 0, 1, 1, 1, 0]

Try It Yourself 12 min

📝 Exam Practice 10 min

Recap & Key Terms 3 min

Convert denary to binary by subtraction or by dividing by 2. Hexadecimal is base 16 (0–9, A–F); four bits make one hex digit, so binary ↔ hex is just grouping in fours. Hex is used for MAC/IP addresses, HTML colours and error codes.

Hexadecimal number system

A base 16 system using the digits 0–9 and letters A–F.

MAC address

A 48-bit number (six pairs of hex digits) that uniquely identifies a device's network interface card.

HTML colour code

A six-digit hex value #RRGGBB giving the red, green and blue parts of a colour.

Nibble

A group of 4 bits — exactly one hexadecimal digit.

Homework 1 min

Task (≤ 15 min): (a) Convert 200 to binary. (b) Convert that binary number to hex. (c) Convert hex 7D4 to denary.