Question

Represent -0.75 in IEEE 754 single and double precision floating point representation.

Answer 1

The number -0.75 is represented in IEEE 754 single precision as 1 01111110 10000000000000000000000 and in double precision as 1 01111111110 1000000000000000000000000000000000000000000000000000, with steps explaining how to find the sign bit, exponent, and mantissa.

Step-by-step explanation:

The student has asked to represent the number -0.75 in IEEE 754 single and double precision floating point representation. To represent any number in IEEE 754 format, we convert it to binary, normalize it, and then encode it into the specified format.

In binary, -0.75 is -11 x 2-1 or -0.11 in binary notation. To encode this in single precision (32-bit), the steps would be:

1. Find the sign bit. Since the number is negative, the sign bit is 1.
2. Normalize the binary number, which is already normalized as -1.1.
3. Extract the exponent value and add the bias (127 for single precision). The exponent is -1, so we add the bias: -1 + 127 = 126. This is 01111110 in binary.
4. Find the mantissa by writing the normalized value without the leading 1, which gives us 10000000000000000000000 as we need 23 bits for the mantissa in single precision.

Thus, the IEEE 754 single precision representation is:

1 01111110 10000000000000000000000

For double precision (64-bit), the process is similar but with a bias of 1023 and 52 bits for the mantissa. The result is:

1 01111111110 1000000000000000000000000000000000000000000000000000