Floating Points as Binary

Related: Binary

Wtf is the bias?

It’s a number to add to exponent according to a standard so it’s easier to work with exponents. Why? Avoids using +/- exponents. Simplifies ordering of FP.

Also also:

Terms:

Mantissa: The (exclusively) binary number after the decimal point
Bias: See above lol
Exponent: You already know this, just pay attention to the sign

1. First, convert the FPN from base 10 to base 2.
   • $0.75\equiv 0.11$
   • $0.11=0.5+0.25$
     1. Normalise it:
   • $1.1\times 2^{-1}$
   • Why normalise?
     • Simplifies Machine Representation (you don’t need a fraction separator)
     • Simplifies Comparisons
     • Compact for very large/small numbers
   • Meaning of exponent:
     • How much over to shift the decimal point.
   • Mantissa (significand) is the (only binary) number after the point. Eg: 1
2. Take the exponent and add the bias:
   • -1 +127 (Bias) = 126.
   • Convert exponent to binary: 01111110.
3. Putting it all together:
   • 0 01111110 10000000000000000000000
   • {Sign} {Exponent} {Mantissa padded to fit 32 bits}

Values for for exponent:

Mantissa