Chapter 1a: binary, hex, Two's Complement numbers

Wow there's so much content per class. This entire chapter took me like 3h to go through. Here's the first half ish of it
Managing complexityA lot goes on in electronics. When designing, we use various tools to manage the complexity.
Abstraction: hide details when they aren't important. For this class, the levels go physics => devices (transistors, diodes) => analog circuits (amplifiers, filters) => digital circuits (gates) => logic (adders, memory) => micro-architecture => architecture => operating systems => application software, as seen in diagram below.
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Discipline: intentionally restrict design choices. For digital design, intentionally using only discrete subset of voltages.
Hierarchy: recursively subdividing system
Modularity: each piece having no side effects, being self contained, having well-defined interfaces with other pieces
Regularity: interchangeable, standardized parts
Binary and hex numbersBinary numbers are base 2. Hex numbers are base 16, which is convenient because each hex digit can represent four binary digits. Hex numbers use 0-9 and A-F to represent values from 0-15 in a single digit. For example, 
 4AF_{16} = 010010101111_2 4AF16=0100101011112 4AF_{16} = 010010101111_2 4AF16​=0100101011112​
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They can also be written without subscripts as 0x4AF = 0b010010101111.
Amounts of information
Bit: one binary digit
Byte: 8 bits
Nibble: 4 bits or one hex digit
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The most significant digit is the leftmost, the least significant is the rightmost.
An N-bit binary number has a range of 
 [0, 2^N - 1] [0,2N−1] [0, 2^N - 1] [0,2N−1]
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A useful approximation is that 
 2^{10} \approx 10^3, 2^{20} \approx 10^6, 2^{30} \approx 10^9 210≈103,220≈106,230≈109 2^{10} \approx 10^3, 2^{20} \approx 10^6, 2^{30} \approx 10^9 210≈103,220≈106,230≈109
 and so on.
Signed binary numbersThere are two systems for signed binary numbers.
Sign/magnitude has some problems. In this system, the leftmost digit either represents a positive (0) or negative (1) sign. For example, 6 = 0110, -6 = 1110.
The range of an N bit sign/magnitude number is 
 [-2^{N-1} - 1, 2^{N-1} - 1] [−2N−1−1,2N−1−1] [-2^{N-1} - 1, 2^{N-1} - 1] [−2N−1−1,2N−1−1]
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The problems are that both +0 and -0 exist (0000, 1000) and that the numbers can't be easily added.
Two's complement solves these problems. In this system the leftmost digit has a value of 
 -2^{N-1} −2N−1 -2^{N-1} −2N−1
. e.g. in a 4-bit two's complement number, the digits represent  -8 \: 4 \: 2 \: 1 −8 4 2 1 -8 \: 4 \: 2 \: 1 −8421
. 6 = 0110, -6 = 1010.
The range of an N bit 2's complement number is 
 [-2^{N-1}, 2^{N-1}-1] [−2N−1,2N−1−1] [-2^{N-1}, 2^{N-1}-1] [−2N−1,2N−1−1]
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It's easy to invert a two's complement number: take the "two's complement" by inverting each digit, then adding 1. For example, starting with 6 = 0110, -6 = 1001 + 1 = 1010.
When adding 2's complement numbers, discard the overflow. For example, 6 + -6 = 0110 + 1010 = 10000 = 0.
To subtract 2's complement numbers, invert one of them and then add.
To extend 2's complement numbers, add all digits with the signed value. For example, 0011 becomes 00000011 and 1011 becomes 11111011 to have the same value in 8 bits.
To extend unsigned binary numbers, just add 0.
Chapter 1a: binary, hex, Two's Complement numbers

Managing complexity

Binary and hex numbers

Signed binary numbers

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E85 Digital Design
