What is a digital system? Is it only binary or decimal, octal as well?

[tggzzz]

I think it is clear that radiolistener is using the word "digital" in the same manner that Humpty Dumpty famously used the word "glory".

I doubt he has read the children's classic book that written >185(!) years ago, and which has many concepts that have entered several countries culture. Here's the relevant quote:

    “I don't know what you mean by 'glory,' " Alice said.
    Humpty Dumpty smiled contemptuously. "Of course you don't—till I tell you. I meant 'there's a nice knock-down argument for you!' "
    "But 'glory' doesn't mean 'a nice knock-down argument'," Alice objected.
    "When I use a word," Humpty Dumpty said, in rather a scornful tone, "it means just what I choose it to mean—neither more nor less."
    "The question is," said Alice, "whether you can make words mean so many different things."
    "The question is," said Humpty Dumpty, "which is to be master—that's all.”
https://www.goodreads.com/quotes/tag/humpty-dumpty

[radiolistener]

One useful definition of 'digital system' I like, is 'a system whose inputs, outputs, and internals have a finite number of valid discrete states'.

According to such definition, sound card output is digital, just because it has finite number of valid discrete states (for example 16777216 discrete amplitude states and 48000 states per second clock states)...

This leads to the first degree crude approximation of "digital ≃ discrete", "analog ≃ continuous".

Let's take for example a simple logic gate AND. It has input and output which is continuous in time.
So, using such "crude approximation" it leads to analog category for digital logic gate...

And another example with analog switch connected to some clock to enable its output at discrete points of time.
Since it's discrete in time, using such "crude approximation" it leads to digital category for signal which is pure analog...

[AndyBeez]

To the OP:

Digital means 'digitizing' the world into numbers.

Binary is a numeric system used in 'digitization'. Silicon based computers use Binary because they can only safely switch between the zero and the one state.

Binary is just another name for the base two number system. Hence, all number bases can be represented in any other number base. Octal is base 8, hexadecimal is base 16.

So a 'digital computer' could be a human with a pencil or an abacus. The only major difference is their calculations use a different number base. Often 10, 12, and 16.

Beware, the term Binary should not be confused with the term Boolean. To operate, both Boolean mathematics and logic require all values to be in base two, which is also binary.

[tggzzz]

One useful definition of 'digital system' I like, is 'a system whose inputs, outputs, and internals have a finite number of valid discrete states'.

According to such definition, sound card output is digital, just because it has finite number of valid discrete states (for example 16777216 discrete amplitude states and 48000 states per second clock states)...

The output of a sound card is indeed digital. The input is analogue. That's why the sound card is an "analogue to digital converter".

According to your peculiar misunderstanding, there are no such things as ADCs, only AACs.

This leads to the first degree crude approximation of "digital ≃ discrete", "analog ≃ continuous".

Let's take for example a simple logic gate AND. It has input and output which is continuous in time.
So, using such "crude approximation" it leads to analog category for digital logic gate...

The inputs and outputs of logic gates are indeed analogue.

The device that recieves the outputs interprets the analogue voltages (or currents) as digital signals.

If you doubt that, explain how a CMOS logic gate can be used as an amplifier[1], and have a look at how ECL/PECL logic gates work (in particular the voltages).

[1] e.g. a CD4007UB CMOS dual complementary pair plus inverter, with listed applications being extremely high-input impedance amplifiers, shapers, inverters, threshold detector, linear amplifiers, crystal oscillators https://www.ti.com/product/CD4007UB

And another example with analog switch connected to some clock to enable its output at discrete points of time.
Since it's discrete in time, using such "crude approximation" it leads to digital category for signal which is pure analog...

Oh, how naive and simplistic. Have a look at N-path filters, a.k.a. commutating capacitor filters

[TimFox]

Two electrical mechanisms that are truly digital, in decimal not binary:
The Strowger switch, used in telephone exchanges:  https://en.wikipedia.org/wiki/Strowger_switch
The Dekatron counter, using gas-discharge, counting directly and displaying the result:  https://en.wikipedia.org/wiki/Dekatron

[Nominal Animal]

One useful definition of 'digital system' I like, is 'a system whose inputs, outputs, and internals have a finite number of valid discrete states'.

According to such definition, sound card output is digital, just because it has finite number of valid discrete states (for example 16777216 discrete amplitude states and 48000 states per second clock states)...

The output of a DAC is still digital, but add a capacitor or a low-pass band-pass filter (often DC is blocked!), and you have an analog output, because it is continuous and no discrete states can be observed a priori in the output.

Let's take for example a simple logic gate AND. It has input and output which is continuous in time.
So, using such "crude approximation" it leads to analog category for digital logic gate...

No, it doesn't.  While the input and output are voltage levels, in normal use one range of voltages is considered "low", another "high", and the rest are indeterminate.  The logic is discrete, while the voltage levels used to convey that logic is continuous.

This is why it is so important to consider the system at the correct complexity level.  Any definition is useless if you only use its "holes" or weaknesses, instead of its strengths.  Definitions are tools, not some kind of absolute revelation of fundamental facts.

As tggzz explained, many logic gates can be used as an analogue component, too.  Thus, "discrete" or "continuous" also depends on the design, the intent, of the component and system.

And another example with analog switch connected to some clock to enable its output at discrete points of time.
Since it's discrete in time, using such "crude approximation" it leads to digital category for signal which is pure analog...

No, like I wrote: there are mixed systems, but we lack exact terminology for them.  I thought I made this clear with the triac mains dimmer with a switch: the states consist of both discrete (off) and a continuous range of states (duty cycle).  The clocked part is digital, because it has a finite number of states; but because of its construction, it can also be used with analog signals.  Thus, a mixed system.

The reason I prefer this definition over others is that it is useful.  Using that definition, it is very easy to branch into mathematics (discrete side to mathematical logic and various theories, continuous side to functions, calculus, transforms), physics (discrete vs. continuous systems, mechanics), electronics (especially the theoretical side), and even history of computing/calculation and esoteric systems like fluidics.  You can find "grasping points" in each using the terms in the definition.

Sure, if you stop nitpicking and use the definitions as I described, you can poke valid holes into it.  That does not make this definition any less valid in the above sense, because it is useful.  We haven't discarded Newton's Laws of Motion just because we discovered that they're not exactly correct in all situations; they're still a valid model within their known limitations.

[AndyBeez]

In the context of digitising analog series, has any definition included the uncertainty introduced by decimation and interpolation? A digital system observes at discrete intervals, but it cannot 'see' into the gaps inbetween.

Could a future computer technology operate in pure and infinite Fourier Transforms, rather than in bistate binary? But would it still be a 'digital' computer?

[tggzzz]

Two electrical mechanisms that are truly digital, in decimal not binary:
The Strowger switch, used in telephone exchanges:  https://en.wikipedia.org/wiki/Strowger_switch
The Dekatron counter, using gas-discharge, counting directly and displaying the result:  https://en.wikipedia.org/wiki/Dekatron

The world's oldest working computer is based on dekatrons and P.O. type 300 relays[1]. It is the Harwell Dekatron, a.k.a. WITCH. While many early computers represented numbers decimally, they used various encoding schemes to represent the ten digits using binary logic (e.g. BCD, bi-quinary and two-out-of-five codes). AFAIK the WITCH is the only computer that used decimal logic elements to implement arithmetic operations.

Presumably radiolistener will try to claim WITCH is analogue. Hopefully this clear real-world example that digital != binary won't cause him too much distress

Go to The National Museum of Computing and see it; well worth while. https://www.tnmoc.org/witch
And a video showing it working, reading instructions from paper tape and outputting results on a teletype: https://youtu.be/SYpPPIsxq64?t=660 https://youtu.be/SYpPPIsxq64?t=821

[1] I have both: the dekatrons in a geiger counter, and the relays from when I was a kid.

[TimFox]

The first computer I encountered was one sold by IBM for college use, their 1620.
https://www.ibm.com/ibm/history/exhibits/mainframe/mainframe_PP1620.html
The internal representation of numbers was BCD, using discrete-transistor logic.
(The keypunches installed next to the computer used 25L6GT tubes to drive the solenoids.)

I have also used electronic counters with Dekatrons in particle-counting lab exercises during the same era.

[TimFox]

In the context of digitising analog series, has any definition included the uncertainty introduced by decimation and interpolation? A digital system observes at discrete intervals, but it cannot 'see' into the gaps inbetween.

Could a future computer technology operate in pure and infinite Fourier Transforms, rather than in bistate binary? But would it still be a 'digital' computer?

True analog computation used to be a thing:  it worked well on "integro-differential" problems  https://en.wikipedia.org/wiki/Integro-differential_equation 
The continuous output of the computed answer could be seen on a CRT, or traced with an analog pen plotter.

When visiting a Boeing facility, I saw their house newsletter, which had lots of Boeing history (including photos of aircraft).
One issue showed a Boeing engineer, ca. 1950, operating an analog computer, properly dressed in white shirt and tie.
To his left was a tall rack cabinet, holding the electronics.  To his right was a blackboard, identifying the knobs on the computer with the terms of his equations.
The caption, written by a younger person, stated that the computer generated so much heat that 400 vacuum tubes were required to cool it.
In the next issue, there were letters from retired Boeing employees about that comment.

[wasedadoc]

In the context of digitising analog series, has any definition included the uncertainty introduced by decimation and interpolation? A digital system observes at discrete intervals, but it cannot 'see' into the gaps inbetween.

That's what Nyquist is all about. If the signal is bandlimited limited with none above half the sampling frequency, the signal is truly and completely characterised by the samples.  The original signal, including all the portions between the samples can in theory be recovered with 100% accuracy.

If you decimate the samples that is just equivalent to reducing the original sample rate by the corresponding ratio.

Note that all of the above is applicable to the analogue samples.  Quantising by for example an A to D converter introduces additional considerations.

[TimFox]

Yes, "Nyquist" is about discrete-time sampling.  "Quantization error" is about discrete-level quantizing.  Normal analog-to-digital conversion systems involve both types of error.

[CatalinaWOW]

The first computer I encountered was one sold by IBM for college use, their 1620.
https://www.ibm.com/ibm/history/exhibits/mainframe/mainframe_PP1620.html
The internal representation of numbers was BCD, using discrete-transistor logic.
(The keypunches installed next to the computer used 25L6GT tubes to drive the solenoids.)

I have also used electronic counters with Dekatrons in particle-counting lab exercises during the same era.

Yes.  And the 1620 did multiplication and division, not by multiplying those BCD numbers "digitally", but using a look up table identical in principal to the times table used to teach children arithmetic.  Certainly digital, but not boolean.

[TimFox]

Back in the 1970s, Electronic Design magazine ran a series of articles about how to do digital arithmetic with BCD-represented values.

[tggzzz]

The first computer I encountered was one sold by IBM for college use, their 1620.
https://www.ibm.com/ibm/history/exhibits/mainframe/mainframe_PP1620.html
The internal representation of numbers was BCD, using discrete-transistor logic.
(The keypunches installed next to the computer used 25L6GT tubes to drive the solenoids.)

I have also used electronic counters with Dekatrons in particle-counting lab exercises during the same era.

I used an Elliott 803, also working at the TNMoC.

Distinguishing features: bit serial, 39-bit words, binary, magnetic film with sprocket holes - and a seminal Algol-60 compiler by CAR (Tony) Hoare. He also created Quicksort, CSP, and the null pointer (his self-acknowledged billion dollar mistake!), etc, etc,

[RJSV]

   Firstly,  I think it's a falacy to bring 'binary' zeros and ones into the definition, where those two symbols are specific examples; I.E. 'A digital system will often use zero and one.'

   Digital means that your (analog) value is resolved, to discrete digits, and that analog level test often uses a slight bit of hysteresis.  The key is understanding the term 'discrete', as being a confident / assured single result.  It maybe sounds silly, but in a number result like '345' each digit, like a '3', can be interpreted as 'not 2 and not 4', and as discrete from those surrounding numbers.
   Thus, a system could be, for example; in base 7, using a multiple state chemical compound.  The various chemical 'states' are discrete from one another, (in whatever sensing method used).  It's the testing against some level or state that gives it 'digital' quality.

Um,...kinda underperformed, on that answer!

[BeBuLamar]

   Firstly,  I think it's a falacy to bring 'binary' zeros and ones into the definition, where those two symbols are specific examples; I.E. 'A digital system will often use zero and one.'

   Digital means that your (analog) value is resolved, to discrete digits, and that analog level test often uses a slight bit of hysteresis.  The key is understanding the term 'discrete', as being a confident / assured single result.  It maybe sounds silly, but in a number result like '345' each digit, like a '3', can be interpreted as 'not 2 and not 4', and as discrete from those surrounding numbers.
   Thus, a system could be, for example; in base 7, using a multiple state chemical compound.  The various chemical 'states' are discrete from one another, (in whatever sensing method used).  It's the testing against some level or state that gives it 'digital' quality.

Um,...kinda underperformed, on that answer!

That is why the Frechn calls it numerique.

[radiolistener]

The output of a sound card is indeed digital. The input is analogue. That's why the sound card is an "analogue to digital converter".

Can you please explain why any soundcard uses DAC (Digital to Analog Converter) on it's output?

In short, you're claiming that analog output of DAC is digital. If output is digital, then please explain what is the reason for DAC here?

Maybe it will help you to understand your mistake... 

[wasedadoc]

The output of a sound card is indeed digital. The input is analogue. That's why the sound card is an "analogue to digital converter".

Can you please explain why any soundcard uses DAC (Digital to Analog Converter) on it's output?

In short, you're claiming that analog output of DAC is digital. If output is digital, then please explain what is the reason for DAC here?

Maybe it will help you to understand your mistake... 

Crossed wires!  (that is a figure of speech, not a literal statement.)

A soundcard can be digital in converted to analogue out or analogue in converted to digital out or can do both of those conversions.  radiolistener is thinking of the first of those three whereas tggzzz's sentence is about the second.

[radiolistener]

The output of a DAC is still digital, but add a capacitor or a low-pass band-pass filter (often DC is blocked!), and you have an analog output, because it is continuous and no discrete states can be observed a priori in the output.

No, it is not digital and cannot be processed with digital input of some circuit. Just because it uses non binary signal levels.

If you want to refute this, then please provide example of digital input which can handle analog signal from DAC (with no using ADC for converting from analog to digital and no using hacks like using digital circuit to perform analog to digital conversion, like using time-to-digital conversion with some RC, etc)

Regarding to filter requirement, even without filter it is not digital, because has more than 2 signal levels. The filter is required for different purpose. It is required just to remove aliases which appears during digital-to-analog conversion. And this is why this filter is named "antialiasing filter".

[radiolistener]

A soundcard can be digital in converted to analogue out or analogue in converted to digital out or can do both of those conversions.  radiolistener is thinking of the first of those three whereas tggzzz's sentence is about the second.

Just to simplify things for better understanding and avoid confusion. I'm talking about output of sound card which don't have input, only output.

This output is analog, and this is why there is digital to analog converter on sound card output.
It needs to convert digital signals (binary data) to analog signal on soundcard output.

[radiolistener]

Back in the 1970s, Electronic Design magazine ran a series of articles about how to do digital arithmetic with BCD-represented values.

Yes, such arithmetic is digital, because it deals with digital data which is binary.
As you probably know BCD means Binary Coded Decimal

[gf]

The output of a DAC is still digital, but add a capacitor or a low-pass band-pass filter (often DC is blocked!), and you have an analog output, because it is continuous and no discrete states can be observed a priori in the output.

No, it is not digital and cannot be processed with digital input of some circuit. Just because it uses non binary signal levels.

Wikipedia's definition of Digital Signal is

A digital signal is a signal that represents data as a sequence of discrete values; at any given time it can only take on, at most, one of a finite number of values.
...
Digital signals having more than two states are occasionally used; circuitry using such signals is called multivalued logic. For example, signals that can assume three possible states are called three-valued logic.

A "digital PAM signal with five levels" is also given as an example and plotted in a diagram.
If this PAM signal is considered "digital", then the usual quantized staircase output of a DAC can be considered "digital", too.

[radiolistener]

No, it doesn't.  While the input and output are voltage levels, in normal use one range of voltages is considered "low", another "high", and the rest are indeterminate.  The logic is discrete, while the voltage levels used to convey that logic is continuous.

I talked about time continuity, not level.

Regarding to the level, yes digital logic is discrete, because it works with digital signals which is binary. And binary is discrete.

If you want to talk about digital logic which is not binary, then provide example of such circuit, but this circuit should be digital. You can't find it, because digital circuit requires digital signals which is binary...

And if some circuit works with non binary signals you will find that it uses analog-to-digital converter to convert analog signal to digital, process it in digital circuit and then convert back to analog with digital-to-analog converter.

There is no way to process non binary signals with digital circuit without ADC and DAC, just because digital circuit works with binary signals only. And non binary signals are not compatible with digital circuit, so you're needs to convert it from analog to digital, then process in digital circuit and then convert it back from digital to analog. 

In simple words, non binary signal always require analog component to convert it from analog to digital code which is binary and from digital code to analog. Because non binary signals are not digital, they are analog.

Even if signal is discrete in amplitude and discrete in time, but their amplitude is not binary, it still analog and requires ADC and DAC in order to process it with digital circuit.

[TimFox]

An ancient non-binary, often decimal, digital system is the abacus or soroban.