IEEE 754 Single-Precision Binary Representation of 63.25

The binary representation of 63.25 in 32-bit IEEE 754 single-precision format is:

0 10000010 01111110000000000000000

Here's how it's derived:

1. Sign Bit: Since 63.25 is positive, the sign bit is 0.

2. Exponent:

   • The decimal number 63.25 is first converted to binary: 1000000'11' (63).
   • The exponent field needs to represent a bias of 127, meaning we add 127 to the actual exponent. Since the exponent of 63 is 6, the biased exponent is 133 (6 + 127) which in binary is 10000101.
   • We need to adjust the exponent to accommodate the fractional part (0.25). Since 0.25 is 2^-2, the exponent decreases by 2 (133 - 2 = 131), and the biased exponent becomes 10000010 in binary.

3. Fraction:

   • The fractional part (0.25) in binary is 0.01.
   • We combine the fraction from the integer part (11) and the fractional part (01) to get 11111110.
   • Since the exponent is adjusted, we shift the binary point of the fraction three places to the right: 111.1110.
   • The fractional part of the IEEE 754 representation is 01111110000000000000000.

Therefore, the complete binary representation of 63.25 in IEEE 754 single-precision format is:

0 10000010 01111110000000000000000