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 and their place values.
3. Convert a binary number of 8, 12 or 16 bits to denary, showing the working.

Textbook: Chapter 1, §1.1.1–1.1.2 (pp. 2–4)

Recap / Warm-Up 5 min

This is the first lesson of the course, so 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 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

1 · Why computers use binary

A computer contains millions of tiny switches (transistors). Each switch is either on or off — nothing in between. On is written as 1 and off as 0. A system with only these two digits is binary.

So every kind of data — numbers, text, sound, images, program instructions — must be converted into binary before the computer can store or process it. The switches are built from logic gates (Unit 10).

2 · Denary — base 10

Denary uses ten digits (0–9). Each column heading is a power of 10: units, tens, hundreds, thousands. Moving one column left multiplies the heading by 10.

3 · Binary — base 2

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

4 · Converting binary to denary

Write the binary number under its headings. Each time a 1 appears, add that column's heading to a running total. Ignore every column holding a 0.

The method works for any length. A 12-bit number adds four more headings (256, 512, 1024, 2048); a 16-bit number goes up to 32 768.

Worked Example — binary → denary 12 min

Worked example 1 · an 8-bit number

Question: convert 10110101 to denary. Show your working. [2]

1. Write the headings 128 64 32 16 8 4 2 1 above the bits.each bit is only worth something once you know its column.
2. Pick out the columns holding a 1: 128 + 32 + 16 + 4 + 1.
3. Add them: 128 + 32 = 160; 160 + 16 = 176; 176 + 4 = 180; 180 + 1 = 181.a running total avoids adding five numbers in your head at once.
4. Answer: 181. Marks: correct place values added (1); correct answer (1).

Worked example 2 · a 12-bit number

Question: convert 010011010110 to denary.

1. Extend the headings four places left: 2048 1024 512 256 then the usual eight.each new column doubles the one to its right.
2. The 1s sit under 1024 + 128 + 64 + 16 + 4 + 2.
3. 1024 + 128 = 1152; + 64 = 1216; + 16 = 1232; + 4 = 1236; + 2 = 1238.
4. Answer: 1238.

Worked example 3 · a 16-bit number

Question: convert 0010110001101011 to denary.

1. Group the bits in fours to keep your place: 0010 1100 0110 1011.16 digits in a row are easy to misalign; groups of four are not.
2. The 1s sit under 8192 + 2048 + 1024 + 64 + 32 + 8 + 2 + 1.
3. 8192 + 2048 = 10 240; + 1024 = 11 264; + 64 = 11 328; + 32 = 11 360; + 8 = 11 368; + 2 = 11 370; + 1 = 11 371.
4. Answer: 11 371.

The same method as an algorithm

// Convert an 8-bit binary string to denary
DECLARE Binary : STRING
DECLARE PlaceValue, Total, Index : INTEGER
Binary ← "10110101"
Total ← 0
PlaceValue ← 128
FOR Index ← 1 TO 8
  IF MID(Binary, Index, 1) = "1"
    THEN
      Total ← Total + PlaceValue
  ENDIF
  PlaceValue ← PlaceValue DIV 2
NEXT Index
OUTPUT "Denary value is ", Total

# Convert an 8-bit binary string to denary
binary = "10110101"
total = 0
place_value = 128
for bit in binary:
    if bit == "1":
        total = total + place_value
    place_value = place_value // 2
print("Denary value is", total)

Denary value is 181

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: each heading doubles, 128 64 32 16 8 4 2 1. To convert binary to denary, add the headings above every 1. The same method works for 12 and 16 bits.

Bit

The basic computing element that is either 0 or 1; short for binary digit.

Binary number system

A number system based on 2 that can only use the values 0 and 1.

Denary number system

A number system based on 10 that uses the digits 0 to 9.

Byte

A group of 8 bits.

Place value (column heading)

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

Most / least significant bit

The left-most bit (largest place value) / the right-most bit (place value 1).

Homework 1 min

Task (≤ 15 min): convert these binary numbers to denary, showing the headings you used: 00111100, 10000001, 11111111 and the 12-bit number 100000000001. [4]