Final answer:
The largest positive number is found by setting the sign bit to 0, the exponent to the maximum value just below the special case of all 1s, and the significand to its maximum value, resulting in a value very close to 2 to the power of 1023.
Step-by-step explanation:
The largest positive number that can be represented in a 40-bit floating point system where the exponent is 11 bits and the significand is 28 bits is calculated by setting the sign bit (S) to 0 (for a positive number), the exponent (E) to its maximum value (without considering special values), and the significand (f) to its maximum value. Since we're not considering special values for the exponent like all 1s or all 0s, the maximum exponent value is when all bits are set to 1 except the highest bit that would indicate an infinity or a special value, that is 210 - 1. The significand is all ones for the maximum value.
To get the largest value, we have:
- Sign bit (S) = 0 (positive number)
- Exponent (E) = 211-1 - 1 = 210 - 1 = 1023 (11 bits, the bias is 1023)
- Significand (f) maximum value with all 28 bits set to 1
Using the given formula V = (-1)S × 1.f × 2E-1023, the largest value is given by:
V = (-1)0 × (1 + (1-2-28)) × 21023-1023 = 1.111… (28 times) in binary × 2210 - 1 - 1023 = 1.111… (28 times) in binary × 21023
When exponential notation is used, leading zeros are not significant, meaning only the ones in the significand are used to calculate the value.