Can you give an example?

Sure! Let's consider an example of a 32-bit single-precision floating-point number represented in binary.

Example:

Let's take the decimal number -5.75 and see how it is represented in IEEE 754 format.

1. Convert to Binary:

   • The integer part (5) in binary is 101.
   • The fractional part (0.75) in binary is 0.11 (since 0.75 = 0.5 + 0.25).
   • Therefore, -5.75 in binary is -101.11.

2. Normalize the Binary:

   • Normalize it to scientific notation: -1.0111 × 2^2.

3. Determine the Sign Bit:

   • Since the number is negative, the sign bit is 1.

4. Calculate the Exponent:

   • The exponent is 2. In IEEE 754, we use a bias of 127 for single precision.
   • So, the biased exponent is 2 + 127 = 129, which in binary is 10000001.

5. Determine the Mantissa:

   • The mantissa is the fractional part after the binary point in the normalized form. For 1.0111, we take 0111 and fill the remaining bits with zeros to make it 23 bits: 01110000000000000000000.

Final Representation:

Putting it all together, the 32-bit representation of -5.75 in IEEE 754 format is:

• Sign Bit: 1
• Exponent: 10000001
• Mantissa: 01110000000000000000000

So, the complete binary representation is:

1 10000001 01110000000000000000000

In hexadecimal, this would be represented as 0xA1C00000.