# Is there a mathematical process resembling the terms "digital"/"discrete" and "analog"/"continuous"?

I always had trouble understanding the terms "digital" and "analog"; Wikipedia and various Q&A sessions didn't contain explanations I found clear.

I understood that these terms aren't well defined in Computer Science or Physics literature but perhaps only in the "signal process" engineering field, hence not based on some formal logic theory, but it might be possible to represent them in a mathematical process (as with Continuous or discrete variables).

It might be grasped absurd but I was thinking that *addition* of *natural numbers* as with `1+1`

is a "digital"/"discrete" process and that *multiplication* of *natural numbers* as with `3*2`

is an "analog"/"continuous" process (because of the continuous addition of 2, three times).

**Is there a mathematical process resembling the terms "digital"/"discrete" and "analog"/"continuous"?**

## 3 answers

I always had trouble understanding the terms "digital" and "analog"

A digital signal is intended to indicate one of a finite set of discrete states. The number of discrete states is usually two for implementation simplicity.

A two-valued digital signal is either high or low, on or off, etc. This is what gives rise to binary number representations in a computer. Each bit (binary digit) can be nicely expressed by a single two-valued digital signal.

An analog signal has a continuum of values. In theory, an analog signal can express an infinity of values. However, in reality the meaningfully distinguishable values are limited by the signal to noise ratio, and how fast you want to resolve a new value (this was quantified by Shannon and Nyquist).

Both digital (discrete) and analog (continuous) signals and actuators are around us in common usage. A normal light switch, for example, is discrete in that it is either on or off. There aren't any valid in-between state. A sliding dimmer, on the other hand, could be continuous.

Typical geared transmissions in a car are discrete. You can't put one in 2½ gear, for example. Water faucets are usually continuous. Even small movements of the faucet cause different water flows, and it's perfectly fine to be at any intermediate settings between off and maximum flow.

I was thinking that addition of natural numbers as with 1+1 is a "digital"/"discrete" process

Yes.

and that multiplication of natural numbers as with 3*2 is an "analog"/"continuous" process

No. It's still discrete. Integers are inherently discrete, while real numbers are continuous. The multiplication of any two integers always yields another integer.

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Is there a mathematical process resembling the terms "digital"/"discrete" and "analog"/"continuous"?

Rounding (down, or up, or to the nearest integer) is he mathematical process which discretizes value.

Multiplication by the Ш-function (Dirac comb) discretizes time.

I always had trouble understanding the terms "digital" and "analog"...

I'll can reply with a quote from a lecture by Prof. Donald Cox :

There is no such thing as digital signal. There is only digital information imposed on an analog signal. "Digital signal" is just a shorthand.

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Discrete sets, as ℕ ℤ and ℚ, are in bijection within themselves, and the number of elements in them is Aleph (א) sub zero.

Continuum sets as ℝ or 𝕀 are not in bijection to the former ℕ ℤ ℚ, and the number of elements in them is Aleph (א) sub one.

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