Uncertainty and infinity are related but different concepts.
On the one hand, uncertainty is a concept that expresses a state of modality, or whether or not knowledge or information is complete or certain. This is related to such concepts as probability and determinism. Infinity is a concept related to cardinality and ordinality, that is to say counting and ordering.
Precision is a concept that bridges the two concepts, by expressing the modality of a numerical quantity. The square root of two serves as a perfect example to explain how modality, precision, and quantity are related.
In the abstract, the square root of two is a number that is defined as an operation performed on a natural number. In essence, mathematical philosophy states that the square root of two is irrational, that is to say cannot be expressed as a ratio of two integers, and any attempt to find it therefore must be a non-terminating, non-repeating decimal. (Note non-terminating, repeating decimals can be expressed as fractions.)
However, in practical application, say one is building a truss for a roof whereby the angle at the ridgeboard is 45 degrees, one needs to actually cut rafters and the bottom chord, and so having a length of lumber and cutting the square root of two needs to, and can be done. Does it make a difference of discussing the root of 2 as a mathematical object and a practical measurement with a ruler with limited precision? Absolutely. It is frequently said that an engineer is a mathematician that rounds off numbers, and in computer science the misconstruction of circuitry to conduct floating-point operations can lead to errors in computation.
This is where the concept of precision comes in. The precision of a measurement is a quantity that expresses a degree of certainty about the measurement. If the chord needs to be cut to a relative length of square root of two, the square root of two can be expressed to some degree or number of decimal places. For instance, if the rafters are 1.000 meter, then the chord can be cut to a precision of 1.4 meters or 1.41 meters or even 1.414 meters. The more precise the measurement, the more certainty that the cut will function in the system.
To answer the second part of your question, yes, certainty and discreteness are related, because the more precise a measurement, the more information you possess. Small differences in measurement can lead to big differences in the outcome of physical systems that are constructed. That's why NASA uses multiple teams all of whom do calculations with as much precision as possible and sometimes averages their results.
As all measurements have a degree of precision, they by definition also have a degree of imprecision. The question of how much precision is needed or possible is a subject that is covered under the debate over the nature of infinity, with some people rejecting it (no such thing as infinity), some people admitting it occurs in a practical, physical context and is limited (potential infinity), and yet others believing infinity is actual characteristic of the universe itself (actual infinity).