Convert hexadecimal to binary scientific notation using IEEE 754 single-precision floating point

Question

I am trying to convert these numbers to binary scientific notation, but I cannot figure out the process. Could someone please the process of going about solving this?

For IEEE 754 single precision floating point, what is the number, as written in binary scientific notation, whose hexadecimal representation is the following?

0061 0000

I can get this converted from hex to unsigned binary:

0000 0000 0110 0001 0000 0000 0000 0000

but I can't figure out how to properly represent this using binary scientific notation. Thanks in advance!

Answer 1

You can find many explanations of the IEEE-754 format on the Web. In short, each 32-bit single-precision floating-point number consists of three parts (we assume that the bits numbering begins from zero):

• sign, bit 31
• exponent, bits 30-23 (eight bits in total)
• significand (or "mantissa"), bits 22-0 (twenty three bits in total)

However, in your case the exponent is zero, which is a special case, well explained here as:

Underflow: If the exponent has minimum value (all zero), special rules for denormalized values are followed. The exponent value is set to -126 while the "invisible" leading bit for the mantissa is no longer used. The range of the mantissa is now [0:1).

So, the additional leading bit is not used, and your answer is:

0.1100001E-126

More detailed discussion of all special cases can be found here.