| Nächste Überarbeitung | Vorhergehende Überarbeitung |
| introduction_to_digital_systems:calc_binary_example [2021/09/15 03:50] – angelegt tfischer | introduction_to_digital_systems:calc_binary_example [2021/09/17 06:21] (aktuell) – tfischer |
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| Idea: The numeral $2658.47$ is only the representation with the digits $[0..9]$, but what is the value behind it? | Idea: The numeral $10011101$ is only the representation with the digits $[0,1]$, but what is the value behind it? |
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| <---- | <---- |
| \begin{smallmatrix} | \begin{smallmatrix} |
| \color{black}{\text{numeral}:} & \color{black}{} & \color{black}{1} & \color{black}{0} & \color{black}{0} & \color{black}{1 } & \color{black}{1} & \color{black}{1} & \color{black}{0} & \color{black}{1} \\ | \color{black}{\text{numeral}:} & \color{black}{} & \color{black}{1} & \color{black}{0} & \color{black}{0} & \color{black}{1 } & \color{black}{1} & \color{black}{1} & \color{black}{0} & \color{black}{1} \\ |
| \color{white}{\text{index}:} & \color{white}{i} & \color{white}{7} & \color{white}{6} & \color{white}{5} & \color{white}{4 } & \color{white}{3} & \color{white}{2} & \color{black}{1} & \color{black}{0} \\ | \color{white}{\text{index}:} & \color{white}{i} & \color{white}{7} & \color{white}{6} & \color{white}{5} & \color{white}{4 } & \color{white}{3} & \color{white}{2} & \color{white}{1} & \color{white}{0} \\ |
| \color{white}{\text{place factor}:} & \color{white}{B^i} & \color{white}{2^7} & \color{white}{2^6} & \color{white}{2^5} & \color{white}{2^4} & \color{white}{2^3} & \color{white}{2^2} & \color{white}{2^1} & \color{white}{2^0} \\ | \color{white}{\text{place factor}:} & \color{white}{B^i} & \color{white}{2^7} & \color{white}{2^6} & \color{white}{2^5} & \color{white}{2^4} & \color{white}{2^3} & \color{white}{2^2} & \color{white}{2^1} & \color{white}{2^0} \\ |
| \color{white}{} & \color{white}{} & \color{white}{128} & \color{white}{64} & \color{white}{32} & \color{white}{16} & \color{white}{8} & \color{white}{4} & \color{white}{2} & \color{white}{1} \\ | \color{white}{} & \color{white}{} & \color{white}{128} & \color{white}{64} & \color{white}{32} & \color{white}{16} & \color{white}{8} & \color{white}{4} & \color{white}{2} & \color{white}{1} \\ |
| \color{white}{\text{digits }:} & \color{white}{z_i} & \color{white}{1} & \color{white}{0} & \color{white}{0} & \color{white}{1 } & \color{white}{1} & \color{black}{1} & \color{black}{0} & \color{black}{1} \\ | \color{white}{\text{digits }:} & \color{white}{z_i} & \color{white}{1} & \color{white}{0} & \color{white}{0} & \color{white}{1 } & \color{white}{1} & \color{white}{1} & \color{white}{0} & \color{white}{1} \\ |
| \color{white}{\text{place value}:} & \color{white}{z_i \cdot B^i} & \color{white}{128} & \color{white}{0} & \color{white}{0} & \color{white}{16} & \color{white}{8} & \color{white}{4} & \color{white}{0} & \color{white}{1} \\ | \color{white}{\text{place value}:} & \color{white}{z_i \cdot B^i} & \color{white}{128} & \color{white}{0} & \color{white}{0} & \color{white}{16} & \color{white}{8} & \color{white}{4} & \color{white}{0} & \color{white}{1} \\ |
| \color{white}{\text{result}:} & \color{white}{\sum_i{ z_i \cdot B^i }} & & & \color{white}{157} \\ | \color{white}{\text{result}:} & \color{white}{\sum_i{ z_i \cdot B^i }} & & & \color{white}{157} \\ |
| \begin{align*} | \begin{align*} |
| \begin{smallmatrix} | \begin{smallmatrix} |
| \color{black}{\text{numeral}:} & \color{black}{} & \color{black}{2} & \color{black}{6} & \color{black}{5} & \color{black}{8.} & \color{black}{4} & \color{black}{7} \\ | \color{black}{\text{numeral}:} & \color{black}{} & \color{black}{1} & \color{black}{0} & \color{black}{0} & \color{black}{1 } & \color{black}{1} & \color{black}{1} & \color{black}{0} & \color{black}{1} \\ |
| \color{blue }{\text{index}:} & \color{blue }{i} & \color{blue }{3} & \color{blue }{2} & \color{blue }{1} & \color{blue }{0 } & \color{blue }{-1} & \color{blue }{-2} \\ | \color{blue}{\text{index}:} & \color{blue }{i} & \color{blue }{7} & \color{blue }{6} & \color{blue }{5} & \color{blue }{4 } & \color{blue }{3} & \color{blue }{2} & \color{blue }{1} & \color{blue }{0} \\ |
| \color{white}{\text{place factor}:} & \color{white}{B^i} & \color{white}{10^3} & \color{white}{10^2} & \color{white}{10^1} & \color{white}{10^0} & \color{white}{10^{-1}} & \color{white}{10^{-2}} \\ | \color{white}{\text{place factor}:} & \color{white}{B^i} & \color{white}{2^7} & \color{white}{2^6} & \color{white}{2^5} & \color{white}{2^4} & \color{white}{2^3} & \color{white}{2^2} & \color{white}{2^1} & \color{white}{2^0} \\ |
| \color{white}{} & \color{white}{} & \color{white}{1000} & \color{white}{100 } & \color{white}{10 } & \color{white}{1 } & \color{white}{0.1 } & \color{white}{0.01 } \\ | \color{white}{} & \color{white}{} & \color{white}{128} & \color{white}{64} & \color{white}{32} & \color{white}{16} & \color{white}{8} & \color{white}{4} & \color{white}{2} & \color{white}{1} \\ |
| \color{white}{\text{digits }:} & \color{white}{z_i} & \color{white}{2} & \color{white}{6} & \color{white}{5} & \color{white}{8 } & \color{white}{4} & \color{white}{7} \\ | \color{white}{\text{digits }:} & \color{white}{z_i} & \color{white}{1} & \color{white}{0} & \color{white}{0} & \color{white}{1 } & \color{white}{1} & \color{white}{1} & \color{white}{0} & \color{white}{1} \\ |
| \color{white}{\text{place value}:} & \color{white}{z_i \cdot B^i} & \color{white}{2000} & \color{white}{600} & \color{white}{50} & \color{white}{8 } & \color{white}{0.4} & \color{white}{0.07} \\ | \color{white}{\text{place value}:} & \color{white}{z_i \cdot B^i} & \color{white}{128} & \color{white}{0} & \color{white}{0} & \color{white}{16} & \color{white}{8} & \color{white}{4} & \color{white}{0} & \color{white}{1} \\ |
| \color{white}{\text{result}:} & \color{white}{\sum_i{ z_i \cdot B^i }} & & & \color{white}{2658.47} \\ | \color{white}{\text{result}:} & \color{white}{\sum_i{ z_i \cdot B^i }} & & & \color{white}{157} \\ |
| \end{smallmatrix} | \end{smallmatrix} |
| \end{align*} | \end{align*} |
| \begin{align*} | \begin{align*} |
| \begin{smallmatrix} | \begin{smallmatrix} |
| \color{black}{\text{numeral}:} & \color{black}{} & \color{black}{2} & \color{black}{6} & \color{black}{5} & \color{black}{8.} & \color{black}{4} & \color{black}{7} \\ | \color{black}{\text{numeral}:} & \color{black}{} & \color{black}{1} & \color{black}{0} & \color{black}{0} & \color{black}{1 } & \color{black}{1} & \color{black}{1} & \color{black}{0} & \color{black}{1} \\ |
| \color{black}{\text{index}:} & \color{black}{i} & \color{black}{3} & \color{black}{2} & \color{black}{1} & \color{black}{0 } & \color{black}{-1} & \color{black}{-2} \\ | \color{black}{\text{index}:} & \color{black}{i} & \color{black}{7} & \color{black}{6} & \color{black}{5} & \color{black}{4 } & \color{black}{3} & \color{black}{2} & \color{black}{1} & \color{black}{0} \\ |
| \color{blue }{\text{place factor}:} & \color{blue }{B^i} & \color{blue }{10^3} & \color{blue }{10^2} & \color{blue }{10^1} & \color{blue }{10^0} & \color{blue }{10^{-1}} & \color{blue }{10^{-2}} \\ | \color{blue}{\text{place factor}:} & \color{blue }{B^i} & \color{blue }{2^7} & \color{blue }{2^6} & \color{blue }{2^5} & \color{blue }{2^4} & \color{blue }{2^3} & \color{blue }{2^2} & \color{blue }{2^1} & \color{blue }{2^0} \\ |
| \color{white}{} & \color{white}{} & \color{white}{1000} & \color{white}{100 } & \color{white}{10 } & \color{white}{1 } & \color{white}{0.1 } & \color{white}{0.01 } \\ | \color{white}{} & \color{white}{} & \color{white}{128} & \color{white}{64} & \color{white}{32} & \color{white}{16} & \color{white}{8} & \color{white}{4} & \color{white}{2} & \color{white}{1} \\ |
| \color{white}{\text{digits }:} & \color{white}{z_i} & \color{white}{2} & \color{white}{6} & \color{white}{5} & \color{white}{8 } & \color{white}{4} & \color{white}{7} \\ | \color{white}{\text{digits }:} & \color{white}{z_i} & \color{white}{1} & \color{white}{0} & \color{white}{0} & \color{white}{1 } & \color{white}{1} & \color{white}{1} & \color{white}{0} & \color{white}{1} \\ |
| \color{white}{\text{place value}:} & \color{white}{z_i \cdot B^i} & \color{white}{2000} & \color{white}{600} & \color{white}{50} & \color{white}{8 } & \color{white}{0.4} & \color{white}{0.07} \\ | \color{white}{\text{place value}:} & \color{white}{z_i \cdot B^i} & \color{white}{128} & \color{white}{0} & \color{white}{0} & \color{white}{16} & \color{white}{8} & \color{white}{4} & \color{white}{0} & \color{white}{1} \\ |
| \color{white}{\text{result}:} & \color{white}{\sum_i{ z_i \cdot B^i }} & & & \color{white}{2658.47} \\ | \color{white}{\text{result}:} & \color{white}{\sum_i{ z_i \cdot B^i }} & & & \color{white}{157} \\ |
| \end{smallmatrix} | \end{smallmatrix} |
| \end{align*} | \end{align*} |
| \begin{align*} | \begin{align*} |
| \begin{smallmatrix} | \begin{smallmatrix} |
| \color{black}{\text{numeral}:} & \color{black}{} & \color{black}{2} & \color{black}{6} & \color{black}{5} & \color{black}{8.} & \color{black}{4} & \color{black}{7} \\ | \color{black}{\text{numeral}:} & \color{black}{} & \color{black}{1} & \color{black}{0} & \color{black}{0} & \color{black}{1 } & \color{black}{1} & \color{black}{1} & \color{black}{0} & \color{black}{1} \\ |
| \color{black}{\text{index}:} & \color{black}{i} & \color{black}{3} & \color{black}{2} & \color{black}{1} & \color{black}{0 } & \color{black}{-1} & \color{black}{-2} \\ | \color{blue}{\text{index}:} & \color{black}{i} & \color{black}{7} & \color{black}{6} & \color{black}{5} & \color{black}{4 } & \color{black}{3} & \color{black}{2} & \color{black}{1} & \color{black}{0} \\ |
| \color{blue }{\text{place factor}:} & \color{blue }{B^i} & \color{blue }{10^3} & \color{blue }{10^2} & \color{blue }{10^1} & \color{blue }{10^0} & \color{blue }{10^{-1}} & \color{blue }{10^{-2}} \\ | \color{blue }{\text{place factor}:} & \color{blue }{B^i} & \color{blue }{2^7} & \color{blue }{2^6} & \color{blue }{2^5} & \color{blue }{2^4} & \color{blue }{2^3} & \color{blue }{2^2} & \color{blue }{2^1} & \color{blue }{2^0} \\ |
| \color{blue }{} & \color{blue }{} & \color{blue }{1000} & \color{blue }{100 } & \color{blue }{10 } & \color{blue }{1 } & \color{blue }{0.1 } & \color{blue }{0.01 } \\ | \color{white}{} & \color{blue }{} & \color{blue }{128} & \color{blue }{64} & \color{blue }{32} & \color{blue }{16} & \color{blue }{8} & \color{blue }{4} & \color{blue }{2} & \color{blue }{1} \\ |
| \color{white}{\text{digits }:} & \color{white}{z_i} & \color{white}{2} & \color{white}{6} & \color{white}{5} & \color{white}{8 } & \color{white}{4} & \color{white}{7} \\ | \color{white}{\text{digits }:} & \color{white}{z_i} & \color{white}{1} & \color{white}{0} & \color{white}{0} & \color{white}{1 } & \color{white}{1} & \color{white}{1} & \color{white}{0} & \color{white}{1} \\ |
| \color{white}{\text{place value}:} & \color{white}{z_i \cdot B^i} & \color{white}{2000} & \color{white}{600} & \color{white}{50} & \color{white}{8 } & \color{white}{0.4} & \color{white}{0.07} \\ | \color{white}{\text{place value}:} & \color{white}{z_i \cdot B^i} & \color{white}{128} & \color{white}{0} & \color{white}{0} & \color{white}{16} & \color{white}{8} & \color{white}{4} & \color{white}{0} & \color{white}{1} \\ |
| \color{white}{\text{result}:} & \color{white}{\sum_i{ z_i \cdot B^i }} & & & \color{white}{2658.47} \\ | \color{white}{\text{result}:} & \color{white}{\sum_i{ z_i \cdot B^i }} & & & \color{white}{157} \\ |
| \end{smallmatrix} | \end{smallmatrix} |
| \end{align*} | \end{align*} |
| \begin{align*} | \begin{align*} |
| \begin{smallmatrix} | \begin{smallmatrix} |
| \color{black}{\text{numeral}:} & \color{black}{} & \color{black}{2} & \color{black}{6} & \color{black}{5} & \color{black}{8.} & \color{black}{4} & \color{black}{7} \\ | \color{black}{\text{numeral}:} & \color{black}{} & \color{black}{1} & \color{black}{0} & \color{black}{0} & \color{black}{1 } & \color{black}{1} & \color{black}{1} & \color{black}{0} & \color{black}{1} \\ |
| \color{black}{\text{index}:} & \color{black}{i} & \color{black}{3} & \color{black}{2} & \color{black}{1} & \color{black}{0 } & \color{black}{-1} & \color{black}{-2} \\ | \color{blue}{\text{index}:} & \color{black}{i} & \color{black}{7} & \color{black}{6} & \color{black}{5} & \color{black}{4 } & \color{black}{3} & \color{black}{2} & \color{black}{1} & \color{black}{0} \\ |
| \color{black}{\text{place factor}:} & \color{black}{B^i} & \color{black}{10^3} & \color{black}{10^2} & \color{black}{10^1} & \color{black}{10^0} & \color{black}{10^{-1}} & \color{black}{10^{-2}} \\ | \color{black}{\text{place factor}:} & \color{black}{B^i} & \color{black}{2^7} & \color{black}{2^6} & \color{black}{2^5} & \color{black}{2^4} & \color{black}{2^3} & \color{black}{2^2} & \color{black}{2^1} & \color{black}{2^0} \\ |
| \color{black}{} & \color{black}{} & \color{black}{1000} & \color{black}{100 } & \color{black}{10 } & \color{black}{1 } & \color{black}{0.1 } & \color{black}{0.01 } \\ | \color{white}{} & \color{black}{} & \color{black}{128} & \color{black}{64} & \color{black}{32} & \color{black}{16} & \color{black}{8} & \color{black}{4} & \color{black}{2} & \color{black}{1} \\ |
| \color{blue }{\text{digits }:} & \color{blue }{z_i} & \color{blue }{2} & \color{blue }{6} & \color{blue }{5} & \color{blue }{8 } & \color{blue }{4} & \color{blue }{7} \\ | \color{blue }{\text{digits }:} & \color{blue }{z_i} & \color{blue }{1} & \color{blue }{0} & \color{blue }{0} & \color{blue }{1 } & \color{blue }{1} & \color{blue }{1} & \color{blue }{0} & \color{blue }{1} \\ |
| \color{white}{\text{place value}:} & \color{white}{z_i \cdot B^i} & \color{white}{2000} & \color{white}{600} & \color{white}{50} & \color{white}{8 } & \color{white}{0.4} & \color{white}{0.07} \\ | \color{white}{\text{place value}:} & \color{white}{z_i \cdot B^i} & \color{white}{128} & \color{white}{0} & \color{white}{0} & \color{white}{16} & \color{white}{8} & \color{white}{4} & \color{white}{0} & \color{white}{1} \\ |
| \color{white}{\text{result}:} & \color{white}{\sum_i{ z_i \cdot B^i }} & & & \color{white}{2658.47} \\ | \color{white}{\text{result}:} & \color{white}{\sum_i{ z_i \cdot B^i }} & & & \color{white}{157} \\ |
| \end{smallmatrix} | \end{smallmatrix} |
| \end{align*} | \end{align*} |
| ----> | ----> |
| 5. calculate the place value \\ $\quad$ | 5. calculate the place value \\ $\quad$ |
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| |
| \begin{align*} | \begin{align*} |
| \begin{smallmatrix} | \begin{smallmatrix} |
| \color{black}{\text{numeral}:} & \color{black}{} & \color{black}{2} & \color{black}{6} & \color{black}{5} & \color{black}{8.} & \color{black}{4} & \color{black}{7} \\ | \color{black}{\text{numeral}:} & \color{black}{} & \color{black}{1} & \color{black}{0} & \color{black}{0} & \color{black}{1 } & \color{black}{1} & \color{black}{1} & \color{black}{0} & \color{black}{1} \\ |
| \color{black}{\text{index}:} & \color{black}{i} & \color{black}{3} & \color{black}{2} & \color{black}{1} & \color{black}{0 } & \color{black}{-1} & \color{black}{-2} \\ | \color{blue}{\text{index}:} & \color{black}{i} & \color{black}{7} & \color{black}{6} & \color{black}{5} & \color{black}{4 } & \color{black}{3} & \color{black}{2} & \color{black}{1} & \color{black}{0} \\ |
| \color{black}{\text{place factor}:} & \color{black}{B^i} & \color{black}{10^3} & \color{black}{10^2} & \color{black}{10^1} & \color{black}{10^0} & \color{black}{10^{-1}} & \color{black}{10^{-2}} \\ | \color{black}{\text{place factor}:} & \color{black}{B^i} & \color{black}{2^7} & \color{black}{2^6} & \color{black}{2^5} & \color{black}{2^4} & \color{black}{2^3} & \color{black}{2^2} & \color{black}{2^1} & \color{black}{2^0} \\ |
| \color{black}{} & \color{black}{} & \color{black}{1000} & \color{black}{100 } & \color{black}{10 } & \color{black}{1 } & \color{black}{0.1 } & \color{black}{0.01 } \\ | \color{white}{} & \color{black}{} & \color{black}{128} & \color{black}{64} & \color{black}{32} & \color{black}{16} & \color{black}{8} & \color{black}{4} & \color{black}{2} & \color{black}{1} \\ |
| \color{black}{\text{digits }:} & \color{black}{z_i} & \color{black}{2} & \color{black}{6} & \color{black}{5} & \color{black}{8 } & \color{black}{4} & \color{black}{7} \\ | \color{black}{\text{digits }:} & \color{black}{z_i} & \color{black}{1} & \color{black}{0} & \color{black}{0} & \color{black}{1 } & \color{black}{1} & \color{black}{1} & \color{black}{0} & \color{black}{1} \\ |
| \color{blue }{\text{place value}:} & \color{blue }{z_i \cdot B^i} & \color{blue }{2000} & \color{blue }{600} & \color{blue }{50} & \color{blue }{8 } & \color{blue }{0.4} & \color{blue }{0.07} \\ | \color{blue }{\text{place value}:} & \color{blue }{z_i \cdot B^i} & \color{blue }{128} & \color{blue }{0} & \color{blue }{0} & \color{blue }{16} & \color{blue }{8} & \color{blue }{4} & \color{blue }{0} & \color{blue }{1} \\ |
| \color{white}{\text{result}:} & \color{white}{\sum_i{ z_i \cdot B^i }} & & & \color{white}{2658.47} \\ | \color{white}{\text{result}:} & \color{white}{\sum_i{ z_i \cdot B^i }} & & & \color{white}{157} \\ |
| \end{smallmatrix} | \end{smallmatrix} |
| \end{align*} | \end{align*} |
| \begin{align*} | \begin{align*} |
| \begin{smallmatrix} | \begin{smallmatrix} |
| \color{black}{\text{numeral}:} & \color{black}{} & \color{black}{2} & \color{black}{6} & \color{black}{5} & \color{black}{8.} & \color{black}{4} & \color{black}{7} \\ | \color{black}{\text{numeral}:} & \color{black}{} & \color{black}{1} & \color{black}{0} & \color{black}{0} & \color{black}{1 } & \color{black}{1} & \color{black}{1} & \color{black}{0} & \color{black}{1} \\ |
| \color{black}{\text{index}:} & \color{black}{i} & \color{black}{3} & \color{black}{2} & \color{black}{1} & \color{black}{0 } & \color{black}{-1} & \color{black}{-2} \\ | \color{blue}{\text{index}:} & \color{black}{i} & \color{black}{7} & \color{black}{6} & \color{black}{5} & \color{black}{4 } & \color{black}{3} & \color{black}{2} & \color{black}{1} & \color{black}{0} \\ |
| \color{black}{\text{place factor}:} & \color{black}{B^i} & \color{black}{10^3} & \color{black}{10^2} & \color{black}{10^1} & \color{black}{10^0} & \color{black}{10^{-1}} & \color{black}{10^{-2}} \\ | \color{black}{\text{place factor}:} & \color{black}{B^i} & \color{black}{2^7} & \color{black}{2^6} & \color{black}{2^5} & \color{black}{2^4} & \color{black}{2^3} & \color{black}{2^2} & \color{black}{2^1} & \color{black}{2^0} \\ |
| \color{black}{} & \color{black}{} & \color{black}{1000} & \color{black}{100 } & \color{black}{10 } & \color{black}{1 } & \color{black}{0.1 } & \color{black}{0.01 } \\ | \color{white}{} & \color{black}{} & \color{black}{128} & \color{black}{64} & \color{black}{32} & \color{black}{16} & \color{black}{8} & \color{black}{4} & \color{black}{2} & \color{black}{1} \\ |
| \color{black}{\text{digits }:} & \color{black}{z_i} & \color{black}{2} & \color{black}{6} & \color{black}{5} & \color{black}{8 } & \color{black}{4} & \color{black}{7} \\ | \color{black}{\text{digits }:} & \color{black}{z_i} & \color{black}{1} & \color{black}{0} & \color{black}{0} & \color{black}{1 } & \color{black}{1} & \color{black}{1} & \color{black}{0} & \color{black}{1} \\ |
| \color{black}{\text{place value}:} & \color{black}{z_i \cdot B^i} & \color{black}{2000} & \color{black}{600} & \color{black}{50} & \color{black}{8} & \color{black}{0.4} & \color{black}{0.07} \\ | \color{black}{\text{place value}:} & \color{black}{z_i \cdot B^i} & \color{black}{128} & \color{black}{0} & \color{black}{0} & \color{black}{16} & \color{black}{8} & \color{black}{4} & \color{black}{0} & \color{black}{1} \\ |
| \color{blue }{\text{result}:} & \color{blue }{\sum_i{ z_i \cdot B^i }} & & & \color{blue }{2658.47} \\ | \color{blue }{\text{result}:} & \color{blue }{\sum_i{ z_i \cdot B^i }} & & & \color{blue }{157} \\ |
| \end{smallmatrix} | \end{smallmatrix} |
| \end{align*} | \end{align*} |