Select the correct answer that corresponds to the floating-point representation of the decimal number: -8.625 Select one: a. • S=1 • E=10000010 • Fraction=1110101 b. • S=0 • E=10000010 • Fraction=000101 c. • S=0 • E=00000010 • Fraction=110101 d. • S=1 • E=10000010 • Fraction=000101
The Correct Answer and Explanation is:
To represent the decimal number -8.625 in IEEE 754 single-precision floating-point format, we need to go through a series of steps. The IEEE 754 single-precision format consists of:
- 1 sign bit (S)
- 8 exponent bits (E)
- 23 fraction (mantissa) bits
Step 1: Convert the decimal to binary
First, let’s convert the absolute value of the number, 8.625, into binary.
- Integer part (8):
Convert 8 to binary:
8 = 1000₂ - Fractional part (0.625):
Multiply the fractional part by 2 repeatedly until you reach 0 or a repeating pattern:
0.625 × 2 = 1.25 → Take the integer part (1).
0.25 × 2 = 0.5 → Take the integer part (0).
0.5 × 2 = 1.0 → Take the integer part (1).
So, the fractional part is 0.101₂.
Thus, the binary representation of 8.625 is:
1000.101₂.
Step 2: Normalize the binary number
In normalized scientific notation, the binary number should be expressed as a number between 1 and 2. This is done by shifting the binary point.
For 1000.101₂, we shift the binary point three places to the left to get:
1.000101 × 2³.
Step 3: Determine the exponent and bias
In IEEE 754 single-precision, the exponent is stored with a bias of 127. So, the exponent for 2³ is:
Exponent = 3 + 127 = 130.
In binary, 130 is:
10000010₂.
Step 4: Determine the fraction (mantissa)
The mantissa is the part of the number after the binary point in the normalized form, but without the leading 1 (since it is implied). So, for 1.000101, the mantissa is:
000101.
Step 5: Sign bit
Since the original number is negative (-8.625), the sign bit (S) will be 1.
Putting it all together:
- Sign bit (S) = 1 (since the number is negative)
- Exponent (E) = 10000010₂ (which is 130 in decimal)
- Fraction (Mantissa) = 000101 (just the bits after the leading 1)
Thus, the correct IEEE 754 representation is:
S = 1, E = 10000010, Fraction = 000101.
Correct Answer:
The correct answer is d. S=1, E=10000010, Fraction=000101.
