3b. Fractions

Hexadecimal - 4 bit representations Octal = 3-bit representations

Base 2 Decimal

first bit: $2^{-1}$ second bit: $2^{-2}$ and so on

Converting from decimal fractions to base 2 representations:

• multiply decimal by 2 until 1.0 is reached. Then subtract 1 for the next bit

$$\begin{array}{r}\text{x2}\\ 0.36_{10}\\ 0.72\\ 1.44\\ 0.88\\ ...\\ 0.01011100_2\end{array}$$

Fixed point (Q15.16) 32-bit system

Bit assignment: 1 - sign, 15 - integer, 16 - fractional

Floating point (IEEE-754)

Bit assignment: 1 - sign, 8 - exponent, 23 - fractional $\left(\begin{array}{c}\text{mantissa}\\ \text{significand}\end{array}\right)$ or $-1^{\text{sign}}\cdot 2^{\text{exp-127}}\cdot 1.\text{fractional}$ Example:

$$-6\frac{5}{8}⟹\text{conversion}⟹110.101_2⟹\text{normalize}⟹1.10101\times 2^2$$

sign - 1 exponent = 129 (since 129 - 127 = 2) fractional = 101010…0