1  Aufgabe 1

- Igor Dimitrov
- Malte Herzog
- Raphael Stumm

\[\begin{align*} [0.5731E5]_8 &= (5 \times 8^{-1} + 7 \times 8^{-2} + 3 \times 8^{-3} + 1 \times 8^{-4}) \times 8^5 \\ &= 20480 + 3584 + 192 + 8 \\ &= 24264 \\ &= [0.24264E5]_{10} \in \mathbb{F}(10, 5, 1) \end{align*}\]

\[\begin{align*} &0.3 \times 2 = 0.6 &&m_1 = 0 \\ &0.6 \times 2 = 1.2 &&m_2 = 1 \\ &0.2 \times 2 = 0.4 &&m_3 = 0 \\ &0.4 \times 2 = 0.8 &&m_4 = 0 \\ &0.8 \times 2 = 1.6 &&m_5 = 1 \\ &0.6 \times 2 = 1.2 &&m_6 = 1 \tag{pattern repeats} \\ &\ldots \end{align*}\]

Somit:

\[\begin{align*} 0.3 &= [0.0100110011001\ldots]_2 \\ &= [0.100110011001\ldots E-1]_2 \\ &= [0.10011001100E-1]_2 \tag{abrunden r = 11} \\ &\in \mathbb{F}(2, 11, 2) \end{align*}\]

\([0.10011001100E-1]_2 = 0.2998046875\ldots\)

Es gilt:

\[\begin{align*} &r=1\quad &&0.\underline{2}998046875 = 0.3 = 0.3\\ &r=2\quad &&0.\underline{29}98046875 = 0.30 = 0.3\\ &r=3\quad &&0.\underline{299}8046875 = 0.300 = 0.3\\ &r=4\quad &&0.\underline{2998}046875 = 0.2998 \neq\ 0.3 \end{align*}\]

D.h der Fehler erst ab r = 4 bemerkbar