1.1.4 | NUMBER SYSTEMS

Topics from the Cambridge IGCSE (9-1) Computer Science 0984 syllabus 2023 - 2025.

OBJECTIVES
​1.1.4 (a) Add two positive 8-bit binary integers
(b) Understand the concept of overflow and why it occurs in binary addition

ALSO IN THIS TOPIC
1.1.1 NUMBER SYSTEMS
1.1.2 NUMBER SYSTEMS
1.1.3 NUMBER SYSTEMS 
YOU ARE HERE | 1.1.4 NUMBER SYSTEMS
1.1.5 NUMBER SYSTEMS
1.1.6 NUMBER SYSTEMS
1.2.1 TEXT, SOUND AND IMAGES
1.2.2 TEXT, SOUND AND IMAGES
1.2.3 TEXT, SOUND AND IMAGES
1.3.1 STORAGE AND COMPRESSION
​1.3.2 STORAGE AND COMPRESSION
​1.3.3 STORAGE AND COMPRESSION
​1.3.4 STORAGE AND COMPRESSION
TOPIC 1 KEY TERMINOLOGY
TOPIC 1 ANSWERS
TOPIC 1 TEACHER RESOURCES (CIE)

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DATA REPRESENTATION

BINARY ADDITION

It is simple to just convert a Binary representation to Denary, add the Denary values then convert back to Binary, however Binary addition is also easy if you follow the rules below:

• Zero plus Zero = 0
• Zero plus One = 1
• One plus One = 0 and carry a 1
• Three Ones = 1 plus carry a 1​

Number 1 = 11
Number 2 = 6
Answer = 18​

Starting from the right(least significant bit) of the binary

• 1 + 1 = 0 and carry 1 (shown in red in the 2s column)
• 1 + 1 + 1 = 1 and carry 1 (shown in red in the 4s column)
• 1 + 1 =  0 and carry 1 (shown in red in the 8s column)
• 1 + 1 =  0 and carry 1 (shown in red in the 16s column)
• 1 + 0 = 1
• Answer 1 0 0 1 0 which is 18

Use a pen and paper and for each of the questions below work out the correct answer and show your working out.
1: What is the result of adding the binary numbers 11010110 and 00101111?
A) 100100100
B) 11100101
C) 11111111
D) 01000000

2: What is the result of adding the binary numbers 10011101 and 01100110?
A) 11000011
B) 00001111
C) 11111111
D) 01011011

3: What is the result of adding the binary numbers 10101010 and 01010101?
A) 11111111
B) 00000000
C) 11111110
D) 11111101

4: What is the result of adding the binary numbers 11001100 and 10101010?
A) 01100110
B) 11111111
C) 11010110
D) 10010100

5: What is the result of adding the binary numbers 1111000011100001 and 0110101001101011?
A) 10101011010011000
B) 11011110101110110
C) 10001101010001010
D) 10100100111011011

OVERFLOW

Overflow in binary addition occurs when the result of adding two binary numbers exceeds the maximum value that can be represented by the number of bits allocated for the calculation. When this happens, the result cannot be represented accurately within the available bits, and the most significant bit (MSB) of the result is lost or "overflows" beyond the maximum value that can be represented.

To understand this concept, consider adding two 8-bit binary numbers, 11111111 and 00000001. The sum of these two numbers is 1,00000000, but since only 8 bits are allocated for the result, the most significant bit of the sum is lost, resulting in a final answer of 00000000. In this case, overflow has occurred because the sum of the two numbers exceeds the maximum value that can be represented by 8 bits.

Overflow can have serious consequences in computer systems and can lead to incorrect results or unexpected behavior. To avoid overflow, it is important to ensure that the number of bits allocated for binary calculations is sufficient to represent the expected range of values accurately. If overflow does occur, appropriate error handling mechanisms should be in place to alert the user or take corrective action.

1: What is overflow in binary addition?
A) A type of error that occurs when two binary numbers are subtracted
B) A condition where the result of adding two binary numbers exceeds the maximum value that can be represented by the number of bits allocated for the calculation
C) A method of compressing digital images
D) A type of sorting algorithm

2: Which of the following can cause overflow in binary addition?
A) Adding two negative binary numbers
B) Adding two positive binary numbers
C) Subtracting two binary numbers
D) Multiplying two binary numbers

3: What happens when overflow occurs in binary addition?
A) The result is rounded up to the nearest binary number
B) The result is rounded down to the nearest binary number
C) The most significant bit of the result is lost or "overflows" beyond the maximum value that can be represented
D) The result is calculated using a different method

4: How can overflow in binary addition be mitigated?
A) By using a larger number of bits for the calculation
B) By only adding binary numbers with the same sign
C) By using a different binary arithmetic operation, such as subtraction or multiplication
D) By rounding the result up to the nearest binary number

5: Which of the following is a consequence of overflow in binary addition?
A) The result is always accurate
B) The result is incorrect and may cause unexpected behavior in a computer system
C) The result is always rounded up to the nearest binary number
D) The result is always rounded down to the nearest binary number

ALSO IN THIS TOPIC

1.1.1 NUMBER SYSTEMS | Why Computers use Binary
1.1.2 NUMBER SYSTEMS | Binary, Hex and converting between number systems 
1.1.3 NUMBER SYSTEMS | Benefits of Hex
1.1.4 NUMBER SYSTEMS | Binary addition and Overflow 
1.1.5 NUMBER SYSTEMS | Binary Shifts
1.1.6 NUMBER SYSTEMS | Negative Number representation, Two's Complement
1.2.1 TEXT, SOUND AND IMAGES | How Computer represent Text
1.2.2 TEXT, SOUND AND IMAGES | How Computers represent Sound
1.2.3 TEXT, SOUND AND IMAGES | How Computers represent Images
1.3.1 STORAGE AND COMPRESSION | How data storage is measured
​1.3.2 STORAGE AND COMPRESSION | Calculating image and sound file size
​1.3.3 STORAGE AND COMPRESSION | Purpose of Data Compression
​1.3.4 STORAGE AND COMPRESSION | Lossy and Lossless, how files are compressed
TOPIC 1 KEY TERMINOLOGY
TOPIC 1 ANSWERS
TOPIC 1 TEACHER RESOURCES (CIE)