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Given the following binary number in 32 bit (single precision) IEEE-754 format:
00111110011011010000000000000000
Q. The decimal value closest to this floating-point number is:
- a)1.45 X 101
- b)1.45 X 10-1
- c)2.27 X 10-1
- d)2.27 X 101
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Given the following binary number in 32 bit (single precision) IEEE-75...
In 32-bit IEEE-754 format
1st bit represent sign
2-9th bit represent exponent and 10-32 represent Mantissa (Fraction part)
2-9th bit represent exponent and 10-32 represent Mantissa (Fraction part)
Sign = 0, so positive
2-9 bits — 01111100 when subtracted by 01111111 i.e., 126 decimal value gives -> 0000 0011
Which is -3.(negative as the value is less than 126)
As number is less than 126 it is subtracted otherwise 126 would have been subtracted from it in 32 bit representation.
Mantissa is normal ,hence, 1.M can be used .Which is 1.1101101.
Thus,
Data + 1.1101101 * 2^-3 (±M * B^(±e))
Mantissa shift right 3 times ->
+0.0011101101
= 0.228
= 2.28 * 10^-1
Thus, option c is correct.
2-9 bits — 01111100 when subtracted by 01111111 i.e., 126 decimal value gives -> 0000 0011
Which is -3.(negative as the value is less than 126)
As number is less than 126 it is subtracted otherwise 126 would have been subtracted from it in 32 bit representation.
Mantissa is normal ,hence, 1.M can be used .Which is 1.1101101.
Thus,
Data + 1.1101101 * 2^-3 (±M * B^(±e))
Mantissa shift right 3 times ->
+0.0011101101
= 0.228
= 2.28 * 10^-1
Thus, option c is correct.
Most Upvoted Answer
Given the following binary number in 32 bit (single precision) IEEE-75...
Given information:
The given binary number is in 32-bit (single precision) IEEE-754 format: 00111110011011010000000000000000Q.
Explanation:
To convert the given binary number into decimal, we need to understand the IEEE-754 format of representing floating-point numbers.
IEEE-754 format for single precision:
The single precision or 32-bit format in IEEE-754 consists of 3 parts:
1. Sign bit (1 bit) - Determines the sign of the number (0 for positive and 1 for negative).
2. Exponent bits (8 bits) - Represent the exponent in biased form. The exponent is biased by a value of 127.
3. Mantissa bits (23 bits) - Represent the fractional part of the number.
Binary representation:
The given binary number is 00111110011011010000000000000000Q.
1. The most significant bit (leftmost bit) is 0, which implies the number is positive.
2. The next 8 bits, 01111100, represent the biased exponent.
- Converting 01111100 to decimal gives 124.
- Subtracting the bias (127) from the exponent gives -3.
- Therefore, the exponent is -3.
3. The remaining 23 bits, 11011010000000000000000, represent the mantissa.
- Since the leading bit is always 1 in IEEE-754 format, we can assume it and drop it.
- After dropping the leading bit, the mantissa becomes 1.11011010000000000000000.
- Converting the mantissa to decimal gives 1.828125.
Final calculation:
To calculate the decimal value of the floating-point number, we can use the formula:
Value = (-1)^sign * (1 + mantissa) * 2^exponent
Plugging in the values, we get:
Value = (-1)^0 * (1 + 1.828125) * 2^(-3)
= 1 * 2.828125 * 0.125
= 0.353515625
The closest decimal value to 0.353515625 is 0.227, which can be written in scientific notation as 2.27 × 10^(-1).
Conclusion:
Therefore, the decimal value closest to the given binary number is 2.27 × 10^(-1), which matches option C.
The given binary number is in 32-bit (single precision) IEEE-754 format: 00111110011011010000000000000000Q.
Explanation:
To convert the given binary number into decimal, we need to understand the IEEE-754 format of representing floating-point numbers.
IEEE-754 format for single precision:
The single precision or 32-bit format in IEEE-754 consists of 3 parts:
1. Sign bit (1 bit) - Determines the sign of the number (0 for positive and 1 for negative).
2. Exponent bits (8 bits) - Represent the exponent in biased form. The exponent is biased by a value of 127.
3. Mantissa bits (23 bits) - Represent the fractional part of the number.
Binary representation:
The given binary number is 00111110011011010000000000000000Q.
1. The most significant bit (leftmost bit) is 0, which implies the number is positive.
2. The next 8 bits, 01111100, represent the biased exponent.
- Converting 01111100 to decimal gives 124.
- Subtracting the bias (127) from the exponent gives -3.
- Therefore, the exponent is -3.
3. The remaining 23 bits, 11011010000000000000000, represent the mantissa.
- Since the leading bit is always 1 in IEEE-754 format, we can assume it and drop it.
- After dropping the leading bit, the mantissa becomes 1.11011010000000000000000.
- Converting the mantissa to decimal gives 1.828125.
Final calculation:
To calculate the decimal value of the floating-point number, we can use the formula:
Value = (-1)^sign * (1 + mantissa) * 2^exponent
Plugging in the values, we get:
Value = (-1)^0 * (1 + 1.828125) * 2^(-3)
= 1 * 2.828125 * 0.125
= 0.353515625
The closest decimal value to 0.353515625 is 0.227, which can be written in scientific notation as 2.27 × 10^(-1).
Conclusion:
Therefore, the decimal value closest to the given binary number is 2.27 × 10^(-1), which matches option C.
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Given the following binary number in 32 bit (single precision) IEEE-754 format:00111110011011010000000000000000Q. The decimal value closest to this floating-point number is:a)1.45 X 101b)1.45 X 10-1c)2.27 X 10-1d)2.27 X 101Correct answer is option 'C'. Can you explain this answer?
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