Is there a rule for good usage as concerns applying a number word as "one", "two" and so on as opposed to Arabic numerals as "1342"? Is it, for instance, a rule that numbers less than 20 should be designated with number words and larger numbers with Arabic numerals?
I don't think there are strict rules about number vs numeral usage. The following tips may help:
Number versus numeral. First things first, what is the difference between a number and a numeral? A number is an abstract concept while a numeral is a symbol used to express that number. “Three,” “3” and “III” are all symbols used to express the same number (or the concept of “threeness”). One could say that the difference between a number and its numerals is like the difference between a person and her name.
Spell small numbers out. The small numbers, such as whole numbers smaller than ten, should be spelled out. That’s one rule you can count on. If you don’t spell numbers out it will look like you’re sending an instant message, and you want to be more formal than that in your writing.
No other standard rule: Experts don’t always agree on other rules. Some experts say that any one-word number should be written out. Two-word numbers should be expressed in figures. That is, they say you should write out twelve or twenty. But not 24.
A rule? Not necessarily, however, there are conventions you should look towards when choosing whether to write the numeric value of a number or its written form. Note that some of these do not apply to scientific papers, math papers, etc.
Below are some that I can think of off the top of my head:
Don't ever start a sentence with the numeric value of a number. It is almost never a good idea. If the sentence must start with a number that is possible to write - under ten letters - you should write it out. If this is not the case, try to reformat the sentence in a way that doesn't change its context/message.
Which leads to the second bullet: it's quite common to write the numeric value of a number if the number itself is under ten. Why? It's simple, writing the number eight is far less straining on the reader than writing say one-hundred-thousand-sixty-eight. See those dashes? Avoid them!
The convention above should be ignored when writing out centuries. Generally you should you be writing them out ex. eighteenth century.
DO NOT put two numbers right next to each other. It is bad practice and should generally be avoided at all cost.
If you're in need of a more credible source feel free to check out OwlPurdue for writing style/mechanics/conventions