What is the correct number of digits to express a measurement

It completely depends on what you are measuring, what instrument you are using, and the uncertainty associated with your measurement.

Explanation:

When expressing a measurement, there is always going to be a part that you know, and a part that you guess. They both make up known numbers collectively called significant figures (commonly shortened to "sig figs"). For instance, imagine you are measuring the length of a pencil. Let's say it is a typical ruler with divisions of 0.1 cm (picture for reference). You measure the pencil, and to your eyes it looks like it is about 9 cm and 5 small tick marks (so, 0.5 cm), so we'll call that 9.5 cm. Now, this is a known value, so you are certain that it is a least 9.5 cm. However, it may be true that there is one more decimal place that can be known. The general rule is that you want to add one more uncertain digit to the rightmost side of the number, and how many digits depends on the divisions of the instrument. Our ruler has divisions of 0.1 cm (the small tick marks), and so we will go one more place out, and add another digit. Now, let's say our eyes are pretty good, so let's say it's 9.5 cm, and then add that one uncertain digit, a 0 to the smallest division of 0.1, so, 0.10. The final measurement we record is 9.50 cm. Now, it may seem like we are simply guessing or even being unreasonable in our assessment of how long this pencil is. I can promise you that not knowing the actual measurement of this pencil will not make a spaceship crash because of a specification that was not the "right measurement". For those life-or-death measurements, much more precise measurement machines are used; this is just a pencil. Also, it would be unreasonable to say that it is 9.500000 cm or 9.500001 cm. We do not have that kind of certainty with this ruler, only divisions of 0.1 cm, so we can only estimate out to one more decimal place.

In short, find the smallest division of a ruler or whatever device (like a chemical beaker) you are using, and then add one uncertain digit to the end of that. If it's 0.1 cm, then add one digit to that. (example: 5.6 --> 5.63; 3 because that is what it looks like on the ruler). If it's 1 cm, then add an extra .1, so the measurement is 1.1 cm. And of course, it's not always cm you are using, that was just an example. Best of luck!

it can be observed without changing the identity of the object.

it depends on what substance you're talking about as well as many other factors. in quantum mechanics, the observation of particles affects the observers ability to measure certain properties. for example: you can know the velocity of a particle, but you can't know it's position; or you can know it's position, but not it's velocity. it's like trying to find out where a grain of sand is by throwing other grains of sand at it. once you find out where it is, it's been moved. it's also like trying to find an answer to the question of whether or not a tree falling in a forest makes a sound if nobody is there to hear it. paradoxically, the only way to find out is to be there and hear it.