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This BCA(5th Semester) assignment has seven questions of total 80 marks. Answer all the questions. You may use illustrations and diagrams to enhance explanations. Illustrations/ examples, where-ever required, should be different from those given in the course material.
Q. Explain each of the following concepts, along with at least one suitable example for each:(For 6 Marks)
(i) Fixed-point number representation
(ii) round-off error
(iii) representation of zero as floating point number
(iv) significant digits in a decimal number representation
(v) normalized representation of a floating point number
(vi) overflow
---
Below is the explanation for the given concepts along with suitable examples. Each part is worth **6 marks**.
---
### (i) **Fixed-point Number Representation**
Fixed-point representation is a way to store real numbers in a computer by allocating a fixed number of bits for the integer part and the fractional part. Unlike floating-point representation, the decimal point is fixed and does not move.
#### Example:
- Assume a fixed-point representation with 8 bits:
- 4 bits for the integer part.
- 4 bits for the fractional part.
- The binary number `1101.0101`:
- Integer part: `1101` = \( 13 \).
- Fractional part: `0.0101` = \( 0.3125 \) (calculated as \( 1/4 + 1/16 \)).
- Result: \( 13.3125 \).
Fixed-point numbers are used in systems like microcontrollers where hardware constraints favor simpler arithmetic.
---
### (ii) **Round-off Error**
Round-off error occurs when a number is approximated due to the limitations in precision of a computer's representation. Computers cannot represent all real numbers exactly due to finite memory, leading to small errors.
#### Example:
- Representing \( \frac{1}{3} \) in a computer with 5 decimal places:
- Actual value: \( 0.33333\ldots \).
- Stored value: \( 0.33333 \).
- Round-off error: \( 0.00000\ldots + \ldots \).
This error can accumulate in iterative calculations, leading to significant inaccuracies in results.
---
### (iii) **Representation of Zero as a Floating-Point Number**
Floating-point numbers are represented using the **sign**, **exponent**, and **mantissa**. Zero is a special case and is represented by:
- **Sign**: 0 (positive zero) or 1 (negative zero).
- **Exponent**: All bits set to 0.
- **Mantissa**: All bits set to 0.
#### Example:
In IEEE 754 single-precision (32-bit):
- Positive zero: `0 00000000 00000000000000000000000`.
- Negative zero: `1 00000000 00000000000000000000000`.
While positive and negative zeros are mathematically equivalent, they may behave differently in certain operations, such as division.
---
### (iv) **Significant Digits in a Decimal Number Representation**
Significant digits are the meaningful digits in a number, representing its precision. This includes all non-zero digits and any zeros between them or trailing zeros in a decimal fraction.
#### Example:
- Number: \( 0.004560 \):
- Significant digits: \( 4560 \) (4 digits: 4, 5, 6, 0).
- Leading zeros are not counted.
- In scientific computations, significant digits determine the precision of a measurement or a result.
---
### (v) **Normalized Representation of a Floating-Point Number**
In normalized floating-point representation, the number is adjusted such that the leading digit (most significant bit) of the mantissa is non-zero (for binary, it is always 1). This ensures maximum precision for the given number of bits.
#### Example:
- Given: \( 0.00125 \) (in binary: `0.00101`).
- Normalize: Shift decimal point to the right by 3 places.
- Result: \( 1.01 \times 2^{-3} \).
Normalized representation is standard in floating-point formats like IEEE 754, where it optimizes storage and precision.
---
### (vi) **Overflow**
Overflow occurs when a number exceeds the maximum value that can be represented by the chosen data type. This typically happens in fixed-size representations during arithmetic operations.
#### Example:
- In an 8-bit unsigned integer:
- Maximum value: \( 2^8 - 1 = 255 \).
- Adding \( 255 + 1 \):
- Expected result: \( 256 \).
- Actual result: \( 0 \) (due to wrap-around).
Overflow can lead to unexpected behavior and is a critical concern in systems like financial or safety-critical applications.