Number Systems & Binary

Syllabus & Goals 3 min

Cambridge 1.1 · Number systems Paper 1 · Computer Systems

By the end of this lesson you can:

1. Explain why computers represent all data in binary.
2. Describe the denary (base 10) and binary (base 2) number systems.
3. Convert an 8-bit binary number to denary using place values.

Recap / Warm-Up 5 min

This is your first lesson, so let's start from something you already know. You count every day in denary — the base 10 system with the digits 0–9. Computers do not. They use only two digits.

Quick starter

A light switch has just two states: off and on. If off means 0 and on means 1, how many different patterns can three switches make?

Key Concept — binary represents data 14 min

Inside a computer are millions of tiny switches (transistors). Each switch is either on or off — nothing in between. We write on as 1 and off as0. A system with only two digits is called binary.

Denary — the base 10 system you already use

Denary uses ten digits (0–9) and column headings that are powers of 10 — units, tens, hundreds, thousands:

Binary — the base 2 system computers use

Binary works the same way, but each heading is a power of 2. For an 8-bit number the headings are:

Worked Example — binary → denary 12 min

Problem: convert the binary number 11101110 to denary.

Method: write the number under the place-value headings. Wherever there is a 1, add that heading to a running total.

Add the headings that sit above a 1:

128 + 64 + 32 + 8 + 4 + 2 = 238

The same idea as an algorithm

Cambridge pseudocode

// Convert an 8-bit binary number to denary
DECLARE Bits : ARRAY[0:7] OF INTEGER
DECLARE PlaceValue, Total, Index : INTEGER
Bits ← [1, 1, 1, 0, 1, 1, 1, 0]
Total ← 0
PlaceValue ← 128
FOR Index ← 0 TO 7
    IF Bits[Index] = 1
        THEN
            Total ← Total + PlaceValue
    ENDIF
    PlaceValue ← PlaceValue DIV 2
NEXT Index
OUTPUT "Denary value is ", Total

The same algorithm in Python (IDLE)

# Convert an 8-bit binary number to denary
bits = [1, 1, 1, 0, 1, 1, 1, 0]
place_value = 128
total = 0
for bit in bits:
    if bit == 1:
        total = total + place_value
    place_value = place_value // 2
print("Denary value is", total)

Output

Denary value is 238

Try It Yourself 12 min

📝 Exam Practice 10 min

Answer the way the examiner expects — the command word and the marks tell you how much to write.

Recap & Key Terms 3 min

Computers store everything in binary because their switches have two states. Binary is base 2 with column headings 128 64 32 16 8 4 2 1. To convert binary to denary, add the headings above every 1.

Binary number system

A base 2 number system that uses only the digits 0 and 1.

Denary number system

The everyday base 10 system, using the digits 0–9.

Bit

A single binary digit (0 or 1).

Byte

A group of 8 bits.

Place value

The value of a column — in binary each column is double the one to its right.

Homework 1 min

Task (≤ 15 min): Convert these binary numbers to denary: 00111100, 10000001 and 11111111.