Electronics > Beginners
Why binary is represented by two bits 0 and 1 and not three bits?
golden_labels:
IDEngineer has given the first half of the answer. The second is: the Boolean algebra with related concepts. While the Boolean algebra was not getting much attention before second quarter of 20th century, it was quickly recognized as a perfect tool for the job. It is deeply connected with prepositional logic, finds its place in multiple branches of mathematics, uses a tiny number of elements in implementation(1), allows super easy synthesis of circuits.(2)(3) and everyone knew how to prove the circuits to be right.
Trivia: there also were negabinary computers. Anecdotal story is that the guys misunderstood a joke and implemented the machine, approaching the subject with complete seriousness. :D
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(1) Any expression can be represented with only NANDs or only NORs, in the past also only a set of AND/OR/NOT gates. Each of those are trivial and robust circuits.
(2) E.g. De Morgan’s law quickly turns any messy expression into simple, conventional form of OR-of-ANDs or AND-of-ORs, which is a circuit consisting of at most 3 layers of digital gates (optional NOTs, a bunch of ANDs, a single multi-input OR).
(3) Historically Karnaugh maps have been used, and those can be solved by children in kindergarten.
paulca:
--- Quote from: golden_labels on October 29, 2019, 09:06:13 am ---Any expression can be represented with only NANDs
--- End quote ---
In uni we were given an assignment to build a logic circuit to run a 7 segment display from a 4 bit binary bus input. We were only allowed to use NAND gates. Was quite challenging and quite interesting to do. Think it ended up being about 50 gates.
These days you just google it.
tggzzz:
--- Quote from: golden_labels on October 29, 2019, 09:06:13 am ---IDEngineer has given the first half of the answer. The second is: the Boolean algebra with related concepts. While the Boolean algebra was not getting much attention before second quarter of 20th century, it was quickly recognized as a perfect tool for the job. It is deeply connected with prepositional logic, finds its place in multiple branches of mathematics, uses a tiny number of elements in implementation(1), allows super easy synthesis of circuits.(2)(3) and everyone knew how to prove the circuits to be right.
Trivia: there also were negabinary computers. Anecdotal story is that the guys misunderstood a joke and implemented the machine, approaching the subject with complete seriousness. :D
____
(1) Any expression can be represented with only NANDs or only NORs, in the past also only a set of AND/OR/NOT gates. Each of those are trivial and robust circuits.
(2) E.g. De Morgan’s law quickly turns any messy expression into simple, conventional form of OR-of-ANDs or AND-of-ORs, which is a circuit consisting of at most 3 layers of digital gates (optional NOTs, a bunch of ANDs, a single multi-input OR).
(3) Historically Karnaugh maps have been used, and those can be solved by children in kindergarten.
--- End quote ---
That's too simplistic. The key is in the phrase in your footnote: "robust circuits", even if they weren't trivial.
For the first quarter century of logic-based machines, up until the 60s, active devices were very expensive (cost, power, volume) and could only be used very sparingly. To illustrate that, some commercially important general purpose computers were serial machines, i.e. the data paths were one bit wide.
So why weren't semi-analogue techniques (e.g. dividing by 10 with three active devices) used more frequently?
Shannon's theorm provides a big clue.
mariush:
because i didn't see someone else linking to them
wraper:
--- Quote from: Raj on October 25, 2019, 09:04:17 am ---Btw cheap Solid state hard drives do store more than just on/off or 0/1 per cell, they are like 0/1/2/3 per cell.
--- End quote ---
Almost all SSD. Even expensive industrial stuff is usually MLC, nowadays often even TLC. Cheap stuff is QLC with 4 bits per cell (16 voltage levels). SLC (1 bit per cell) is very niche nowadays.
--- Quote ---They do have additional circuitry next to the storage chip to convert them back into zeros/ones.
--- End quote ---
:palm: ALL SSD have controller chip. And it does not convert anything. NAND flash chip outputs binary data.
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