Math 441

Floating Point Numbers

Numbers in Computers

• binary
• \(N\) bits
• Integers:

  • Arithmetics modulo \(2^N\)

  • Unsigned integers: \(0\) to \(2^N - 1\)

  • Signed integers: \(-2^{N-1}\) to \(2^{N-1} - 1\).

  • Exercise:

    Write down all three-bit integers, first as binary, then each of them as an unsigned integer, then each as signed integer.

• Floating point numbers

Floating Point Numbers

• \(k\): number of bits for exponent
• \(t\): number of bits for mantissa
• \(1 + k + t = N\)

---

\[x = {\color{#44ee44}(-1)^s} \cdot {\color{red}2^e} \cdot {\color{blue}(1.f)_2} = {\color{#44ee44}(-1)^s} \cdot {\color{red}2^{(c)_2 - b}} \cdot {\color{blue}(1.f)_2}\]

---

Exponent \(e = (c)_2 - b\) where \(b\) is called a bias. Typically \(b = 2^{k-1} - 1\).

8-bit Example:

• \(k = 3\): number of bits for exponent
• \(t = 4\): number of bits for mantissa
• Bias: \(b = 2^{3 - 1} - 1 = 3\)

Largest 8-bit Number:

• \(k = 3\): number of bits for exponent
• \(t = 4\): number of bits for mantissa
• Bias: \(b = 2^{3 - 1} - 1 = 3\)

“Smallest” Positive 8-bit Number:

• \(k = 3\): number of bits for exponent
• \(t = 4\): number of bits for mantissa
• Bias: \(b = 2^{3 - 1} - 1 = 3\)

Exercise:

Consider 6-bit floating point numbers with 3-bit exponent and 2-bit mantissa:

• How many positive floating point numbers are there?
• What is the smallest one?
• What is the largest one?
• Find all of them, convert them to decimal representation, and plot them on a number line. Note how they are distributed!

Some Questions

• How do you express 0?
• Why didn’t we use all 0’s and all 1’s as exponents?
  • These are reserved for special cases.

\(\pm 0\), Subnormal Numbers

\(0\):

\(-0\):

\(\pm\infty\), NaN

\(\pm\infty\):

NaN:

IEEE 64-bit Floats

• \(k = 11\)
• \(t = 52\)