Answer :
In the IEEE single format, the significand has 23 bits, which allows for a precision of approximately 7 decimal places.
The IEEE single format is a standard for representing floating-point numbers in computers. To determine the IEEE single format of 101.2, we need to convert it into binary and represent it in the format:
1 bit for the sign (0 for positive, 1 for negative)
8 bits for the exponent
23 bits for the significand (also known as the mantissa or fraction)
First, let's convert 101.2 into binary:
101.2 = 1100101.00110011001100110011... (repeating)
To represent this number in the IEEE single format, we need to normalize it. Normalization means shifting the binary point to the left until there is only one non-zero digit to the left of the binary point. In this case, the normalized form is:
1.1001010011001100110011... x 2⁶
Now, we have the binary significand and the exponent. The exponent is obtained by adding the bias value (127) to the power of 2 that corresponds to the number of places the binary point was shifted. In this case, the exponent is
6 + 127 = 133, which in binary is:
10000101
To fit this exponent into 8 bits, we need to pad it with leading zeros:
0010000101
Finally, we put all the components together in the IEEE single format:
0 10000101 10010100110011001100110
This represents the number 101.2 in the IEEE single format.
If the number 101.2 is not a machine number, we need to round it to the nearest machine number. To determine the nearest machine number, we compare the significand to the halfway point between two consecutive machine numbers.
In the IEEE single format, the significand has 23 bits, which allows for a precision of approximately 7 decimal places. Therefore, the nearest machine number to 101.2 would be the machine number with the closest representation within 7 decimal places.
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