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Consider the Following IEEE 32 bit representation:
0 01111110 10100000000000000000000
Its decimal equivalent is ___________ (Upto 4 decimal places)
- a)0.8121
- b)0.8129
- c)0.8125
- d)0.8127
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Consider the Following IEEE 32 bit representation:0 01111110 101000000...
The IEEE format for 32 bit floating point arithmetic is
Sign Bit= 0 (positive sign)
Biased Exponent= 01111110(126)
It is less than bias then it is negative
Actual Exponent= 126- 127= -1
So it can be represented using normalised form as:
1.101 * 2-1
è 0.1101
è 0.8125
Most Upvoted Answer
Consider the Following IEEE 32 bit representation:0 01111110 101000000...
Given IEEE 32-bit representation: 0 01111110 10100000000000000000000
Step 1: Identifying the Sign, Exponent, and Mantissa
The given representation can be divided into three parts: sign, exponent, and mantissa.
Sign: The leftmost bit is the sign bit. Here, the sign bit is 0, which represents a positive number.
Exponent: The next 8 bits represent the exponent. In this case, it is 01111110.
Mantissa: The remaining 23 bits represent the mantissa. In this case, it is 10100000000000000000000.
Step 2: Converting the Exponent from Binary to Decimal
To convert the exponent from binary to decimal, we need to subtract the bias value. The bias value for a 32-bit representation is 127.
Exponent: 01111110 (binary)
Converting to decimal: 126 (decimal)
Step 3: Converting the Mantissa from Binary to Decimal
To convert the mantissa from binary to decimal, we need to calculate the fractional part by multiplying each bit with the corresponding power of 2 and then summing them up.
Mantissa: 10100000000000000000000 (binary)
Converting to decimal: 0.5 + 0.25 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 = 0.75
Step 4: Calculating the Decimal Equivalent
The decimal equivalent of the given IEEE 32-bit representation can be calculated using the formula:
Decimal Equivalent = (-1)^sign * (1 + mantissa) * 2^(exponent - bias)
In this case, the sign is positive (0), the mantissa is 1.75 (0.75 + 1), the exponent is 126, and the bias is 127.
Decimal Equivalent = (-1)^0 * (1 + 0.75) * 2^(126 - 127)
Decimal Equivalent = 1 * 1.75 * 2^(-1)
Decimal Equivalent = 0.875
Step 5: Rounding to Four Decimal Places
Rounding the decimal equivalent to four decimal places, we get 0.8125.
Therefore, the correct answer is option C) 0.8125.
Step 1: Identifying the Sign, Exponent, and Mantissa
The given representation can be divided into three parts: sign, exponent, and mantissa.
Sign: The leftmost bit is the sign bit. Here, the sign bit is 0, which represents a positive number.
Exponent: The next 8 bits represent the exponent. In this case, it is 01111110.
Mantissa: The remaining 23 bits represent the mantissa. In this case, it is 10100000000000000000000.
Step 2: Converting the Exponent from Binary to Decimal
To convert the exponent from binary to decimal, we need to subtract the bias value. The bias value for a 32-bit representation is 127.
Exponent: 01111110 (binary)
Converting to decimal: 126 (decimal)
Step 3: Converting the Mantissa from Binary to Decimal
To convert the mantissa from binary to decimal, we need to calculate the fractional part by multiplying each bit with the corresponding power of 2 and then summing them up.
Mantissa: 10100000000000000000000 (binary)
Converting to decimal: 0.5 + 0.25 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 = 0.75
Step 4: Calculating the Decimal Equivalent
The decimal equivalent of the given IEEE 32-bit representation can be calculated using the formula:
Decimal Equivalent = (-1)^sign * (1 + mantissa) * 2^(exponent - bias)
In this case, the sign is positive (0), the mantissa is 1.75 (0.75 + 1), the exponent is 126, and the bias is 127.
Decimal Equivalent = (-1)^0 * (1 + 0.75) * 2^(126 - 127)
Decimal Equivalent = 1 * 1.75 * 2^(-1)
Decimal Equivalent = 0.875
Step 5: Rounding to Four Decimal Places
Rounding the decimal equivalent to four decimal places, we get 0.8125.
Therefore, the correct answer is option C) 0.8125.
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Consider the Following IEEE 32 bit representation:0 01111110 10100000000000000000000Its decimal equivalent is ___________ (Upto 4 decimal places)a)0.8121b)0.8129c)0.8125d)0.8127Correct answer is option 'C'. Can you explain this answer?
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