Data representation (3)
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1.
Convert the following hexadecimal number to its decimal equivalent. Show your working.
Hexadecimal: 3A16
To convert 3A16 to decimal, we multiply each digit by the corresponding power of 16 and sum the results.
3A16 = (3 × 161) + (A × 160)
Since A represents 10 in decimal, we have:
3A16 = (3 × 16) + (10 × 1)
3A16 = 48 + 10
3A16 = 5810
Therefore, the decimal equivalent of 3A16 is 58.
2.
Question 1: Explain why computers use binary to represent data. Include in your answer a discussion of the advantages of using a binary system over a decimal system.
Computers use binary (base-2) to represent data because the electronic circuits within a computer can easily represent two distinct states: 'on' (represented by 1) and 'off' (represented by 0). This is directly achievable with electrical signals. Using a binary system is advantageous for several reasons:
- Simplicity of Implementation: It's much easier and more reliable to create electronic circuits that can reliably distinguish between two voltage levels (representing 0 and 1) than to create circuits that can represent ten distinct voltage levels (needed for decimal).
- Reliability: Binary signals are less susceptible to noise and interference compared to analog signals that might be used to represent decimal values.
- Digital Logic: Binary is the foundation of digital logic, which is how computers perform calculations and store information. Logic gates (AND, OR, NOT, etc.) operate directly on binary inputs and outputs.
- Efficient Storage: Binary allows for efficient storage of data. Each bit can represent a distinct piece of information.
A decimal system would require significantly more complex and less reliable circuitry to represent the ten digits (0-9).
3.
Explain, in terms of binary numbers, why an overflow occurs during addition. Your answer should include a clear definition of overflow and illustrate your explanation with an example using 4-bit numbers.
Overflow occurs when the result of an arithmetic operation exceeds the maximum value that can be represented by the available number of bits. In binary addition, overflow happens when the sum of two numbers is larger than the largest possible value that can be stored in the given number of bits.
Consider a 4-bit binary number system. This system can represent numbers from 0 to 15 (24 - 1). Let's say we add 11 (binary 1011) and 5 (binary 0101):
- 11 (1011) + 5 (0101) = 16 (10000)
The result, 16, is larger than the maximum value representable by 4 bits (15). The extra bit (the most significant bit) '1' is discarded, resulting in an overflow. The 4-bit system can only represent numbers from 0 to 15, and any result exceeding this range will overflow.
This is because each position in a binary number represents a power of 2. When adding, the sum of the digits in a position can exceed the value that position can hold, leading to a carry-over that isn't properly handled within the limited number of bits.