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introduction_to_digital_systems:number_systems [2023/05/08 16:27]
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-====== 2. Number Systems ======+====== 2 Number Systems ======
  
 ===== 2.1 Types of Number Systems ===== ===== 2.1 Types of Number Systems =====
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                +672 \\                +672 \\
 \hline  \hline 
-1\overset{\color{red}{2}}{1}\overset{\color{red}{1}}{4}\overset{}{2}+1\overset{\color{red}{2}}{2}\overset{\color{red}{1}}{4}\overset{}{2}
 \end{align*} \end{align*}
  
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   * [[https://calculator.name/baseconvert/decimal/hexadecimal/|Conversion tool from decimal to hexadecimal]], this tool shows also the steps and can be used vice versa   * [[https://calculator.name/baseconvert/decimal/hexadecimal/|Conversion tool from decimal to hexadecimal]], this tool shows also the steps and can be used vice versa
-  * The [[https://www.omnicalculator.com/math/binary-fraction|OmniCalculator]] can also calculate binary fractions+  * The [[https://www.omnicalculator.com/math/binary-fraction|OmniCalculator]] can also calculate binary fractions and comes with another explaination of the manual process. 
   * A short video on the "[[https://www.youtube.com/watch?v=_s5RFgd59ao&ab_channel=Numberphile|Everything Formula]]" which prints all possible output and is directly related to binary encoding.   * A short video on the "[[https://www.youtube.com/watch?v=_s5RFgd59ao&ab_channel=Numberphile|Everything Formula]]" which prints all possible output and is directly related to binary encoding.
  
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 </WRAP></WRAP></panel> </WRAP></WRAP></panel>
  
-<panel type="info" title="Exercise 2.3.9. One's Complement"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>+<panel type="info" title="Exercise 2.3.9. Two's Complement"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
  
-In the simulation in <imgref pic70> the one's complement of a 4bit value is shown. \\ +In the simulation in <imgref pic70> the two's complement of a 4bit value is shown. \\ 
-On initialization the value is the following:  +On initialization, the value is the following: 
-  * The value for $A$ is $1010_2 = 10_{10}$. The decimal value ''10'' is also shown in the first display from the left in the circuit.  +  * The value for variable $A$ is $1010_2 = 10_{10}$. The decimal value ''10'' is also shown in the first display from the left in the circuit.  
-  * The value for $B$ is $1110_2 = 14_{10}$. The decimal value ''14'' is also shown in the second display from the left in the circuit.  +  * The value for variable $B$ is $1110_2 = 14_{10}$. The decimal value ''14'' is also shown in the second display from the left in the circuit.  
-  * The addition $A+B$ leads to $S = 1000_2 = 8_{10}$. The decimal value ''10'' is also shown in the rightmost display in the circuit.  +  * The addition $A+B$ leads to $S = 1000_2 = 8_{10}$. The decimal value ''10'' is also shown on the rightmost display in the circuit.  
-  * There is another value in the second display from the right. This is called $B'$ and it is equal to $0010_2 = 2_{10}$.+  * There is another value in the second display from the right. This is called $B'$, and it is equal to $0010_2 = 2_{10}$.
    
 <WRAP><well> <WRAP><well>
-<imgcaption pic70|One's Complement></imgcaption> \\ +<imgcaption pic70|Two's Complement></imgcaption> \\ 
-{{url>https://www.falstad.com/circuit/circuitjs.html?hideSidebar=true&ctz=CQAgzCAMB0l3BOJyWoSALNATAVmwIyQAc28kAbNgOzrbEi6OSMCmAtAQQFBnUjswYCiAIYMAsPlHSJAQUi9I-QcNH1J0gmUwg5PPpLVdsm09pbzsSlUJEIRgrZAnyw3AB4gEppkQkMEgQMAMqeAtgYDI7YauhBDABC4YIYLLYaRLrBeik05gQRuH6u4CwAOgAOedSmGOjseEyyZSBV3BgEDWCmDlDeIoUY3ADuAsQM2JHjk2lQ3ACyIBOMIitzptjQuIsgtat7dSyb27v+B+cbIFs7SxYXOlc33AAyESu4MevH-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-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-CIMJoS2YhPLAZAJRkVz5RXGZLxOYowczScNKc7MRBeSiWzfUvYEmiHyYMTMig7jlLKUad0zwljFP6E8U4Sw4n0zqScW4eSXQiBaTsLw7BcCuBUBgZoYoQwCBSLQTAdIKBXwUS4kAAAKbRXg0aYFMGAIYTBFEAEpuBAA noborder}}+{{url>https://www.falstad.com/circuit/circuitjs.html?ctz=CQAgzCAMB0l3BOJyWoSALNATAVmwIyQAc28kAbNgOzrbEi6OSMCmAtAQQFBnUjswYCiAIYMAsPlHSJAQUi9I-QcNH1J0gmUwg5PPpLVdsm09pbzsSlUJEIRgrZAnyw3AB4gEppkQkMEgQMAJqeAtgYDI7YauhBDABC4YIYLLYaRLrBeik05gQRuH6u4CwAOgAOedSmGOjseEyyZSBV3BgEDWCmDlDeIoUY3ADuAsQM2JHjk2lQ3ACyoi6MgytzptjQuIsgE6t7gSyb27sWB+cbIFs7SwRquINqVzfcADIR+4+fR9eiIAAzACGABsAM6sRjzD6CFbfWGuY79YHgyFMRQwuwHVQiK4sFEQqEYyQ6eE9SxIwoEtHzQyNOZiCSNb4EFqJRR0yIsbQMZmDHQSRIGZQRBnaIr8ywgRLWTlzPp85alRLuGFTCTw9WYSmA0GE9HvUUsTVzPG61FEw2NEjYsi-czm-XQgRdXyOV3av5UvU0xRjRpfGL7YgsP2SOGOMArWrzf3k7HxigSMMBhghowidMprHpnGicSxjPqXlYmPZtQxhEgMujEksJNFhvlkQNvMmQt5hVY77N-NMrGEETZ8Vc8bHOYp-ajwQ6Ue9rVtjS99t52KmbMaad9Oe19jbuZ7kR2jtT6bsdNayfHG2H642q+iHwulYelP+R+mTgrHf+9-TgcTruAEsJ2Q5AZuB4Dj0HYDme8Y-kWg6SCOgH+qO9AgUhUbJru6H7I00zYR2a5lEy6F9Cmo5EfSLAUbu77UVRKyUdB1F9ERlFip+5FgWhGz4aOPIngwjGEcxuEECI1HphxuEMgWNHXFmcm0TEcw9ipICtoJ4qUdM2lzE2mmVseNZ8dyGiKWZRp7Jh8k4b+T65qOylodMmr6Q5EQ2pqNrWZwT7wh6ymdA0o4KsefRDPR35BSssn+hecUSHgvFjtclbbv5U74Re+wpuxKy3vkwlKby+yyaFRZdisUWYB0XSHAM6iqf8wxjPcIgsmorktRcOjWe+LIrEZHU9SwwahrWk1NYyhb7EJ+zWUt5gDeutbvg2+yjU1W0MO2Yb7H0gbzftBZTstkx7cWp2iOKU4aWMU5RE1dGGAtGgxp1ugAMrCvwC0Fl9LQ-RyIoLeKX0CiAP2yuDDB9F96ww+4cabDGF3rU9V0BMcl3XN8MYlYdszRCly149x5NY6KGVUwTaX0gzJLMxu1xnpM0xvkeB6SXTHZ8yVRPZezTKTDWdJCQqc3voK7iS5k4qcAWlzSnDKhCUJytBNDQo2C6mQKTLbKKAAhCAACSAB2ABurAAE4AC4O-0AAaYAADSu9gXsEF7LAAPTux7Qc+0HftByw1t207LvkOUACOXBcOUVvwAQifJ5AqdwEQmdEBnaeQCnSdENnadl5nxeF8XxdV3nFd50nuc58nVdwKnZfZwn1eF2XGc91w5cF93td99Xie18Plc9xPVtDwPufT036ed23s8d1bSi8lLjh9HV0cO879tEv0ihdI46YeuwX1PhA7Us9OguoeM-MNsT3DmwACiCACuYKFBYMHb2vt-YgBCJ7EIPsQh+xCCwH+-9AHwEni3IuddZ4NyXiglORcS5T1bh3QedcK7oOruXMhk826NwHl3TuyDm7EP7vXYetd67jxLkPaehCF5r3QVnNeTdk593XvHLex5zytXYC9OqCCAGn2NOECgQxqyFH4AkNoCdFGFFUCo8M2QGCJxSBgSM2iwAqA0BIKMAgUhgEgiBHovIaD6NyF4RoGQDYRHiKIJIABycIT4ZxKk8c4sIaNmqBMeizVs0EDr0RbP2UwO1AllmrDTQJWtoL+WgrmGJBYwx8xyaYXqnAuruniYWY0Q5UmFj5iyTMU07ilNELiJErwliVP6C8U4SwImRjqK07pLomklM9CcHYrjcCuBUBgZoUorHsBSLQTATIKD3zUd4kAAAKQxXhsC4lMGAIYTB1EAEpwIfi0kEJ8YYGwegbEZJY9yJANhemM7gQA noborder}}
 </well></WRAP> </well></WRAP>
  
   - Why is ''10'' + ''14'' = ''8''? What happens to other values? \\ The input values ''A'' and ''B'' can be changed by clicking on the bit values.    - Why is ''10'' + ''14'' = ''8''? What happens to other values? \\ The input values ''A'' and ''B'' can be changed by clicking on the bit values. 
   - Try to analyze how ''B''' (shown in the brackets) is derived from ''B''.    - Try to analyze how ''B''' (shown in the brackets) is derived from ''B''. 
-  - Have a look at the wiki page of the {{wp>Ones' complement}} to understand how negative numbers are represented in a microcontroller.+  - Have a look at the wiki page of the {{wp>Two's complement}} to understand how negative numbers are represented in a microcontroller. \\ What does the following representation show? \\ {{drawio>introduction_to_digital_systems:SignedIntCircle.svg}}
   - Imagine that you have to rescue data from an old storage device. The interesting bits are given by the boxed area in <imgref picex01>. The bytes are LSB 0 oriented.    - Imagine that you have to rescue data from an old storage device. The interesting bits are given by the boxed area in <imgref picex01>. The bytes are LSB 0 oriented. 
     - What are these values in an unsigned integer?     - What are these values in an unsigned integer?
-    - What are these values in signed integers based on the ones' complement? +    - What are these values in signed integers based on the Two's complement?