When and why should you write numbers less than 10 in words?

It depends. I'd for example say "the two above equations show that the value is 5". If it is a number (result, a quantity like "5 [km]", ...) I use numerals; for "narrative" numbers up to ten or so, like "two equations" or "nine people", even up to twenty, I'd write them out. If the number comes from an equation or will be used in one, or be somehow be combined with others, I'd prefer numerals. But also consider that text is to be read as text, not as a mixture of text, symbols, and numbers. In case of doubt, write it out. Even "a thousand repetitions of the experiment...".

In absence of some style guide (or red pencil by your advisor), don't worry too much.

Follow the style guide and the authors' guide for your publication. Almost all such guides for academic publications include guidelines for spelling, presenting numbers, and referencing.

If in doubt, quickly scan through recent documents from the same series (so, if it's a thesis, flick through some recent theses; if it's a paper in a particular journal, have a quick read of that journal), and reproduce their formatting.

After that, if still in doubt, ask your supervisor, advisor or editor.

Many style manuals will have specific requirements regarding when to express a number as numerals or in words.

APA Style

The APA version 6 style manual has an entire section starting from 4.31 on how to present numbers in text.

The manual does include a general rule that numbers 10 and above should be expressed as numerals, but there are a wide range of exceptions.

Should be expressed as numerals

• Any number in an abstract
• Any number immediately preceding a unit of measurement (e.g. 3 kg)
• Any number representing a statistical operation (e.g., 4 times as many people)
• times, dates, ages, scores, and points, exact quantities of money, numerals as numerals
• numbers that indicate a place in a series (e.g., page numbers, Table 1, Table 2, etc.)

There are also further exceptions for when numbers should be expressed in words

• Numbers at the start of a sentence or heading
• Common fractions
• Where the usage is common

Why have the general rule to express numbers below 10 in words?

Compare the following:

• I had 2 apples.
• I had two apples.

and

• I had one-hundred and thirty-two choices.
• I had 132 choices

A few possible principles:

• Numbers less than 10 are easy to read as words; Large numbers are difficult to read as words and easy to read as numerals.
• In general, conventions make life easier for writer and reader.
• There may be a slight disruption to the look of and reading of prose when interspersing numbers with words.
• A few numbers, particularly "1" (which looks a bit like "I") might introduce slight ambiguities.

I don't think it is an unnecessary complication, as long as you remember to be consistent. But it may not necessarily be appropriate for the document you're writing.

In English non-technical contexts, the style is often to write out numbers from one to nine as words, and then from 10 onwards as numbers (this is style used by the Associated Press and in some cases the Chicago Manual of Style). The Chicago Manual of Style also alternatively prescribes writing out numbers from one to one hundred as words. In both style guides, numbers that begin a sentence should be written out as words and Chicago specifies that, if this is an unwieldy solution, then the sentence can be rephrased. The webpage Numbers: Spell Out or Use Numerals? (Number Style 101) provides a useful guide, which goes into greater detail on good style for numbers in general documents.

However, you're likely writing a technical document, and as such, the guidelines are different, suggesting that numbers should be written with digits much more, with David A. McMurrey, author of Processes in Technical Writing, stating: 'You should use numerals, not words, when the number is a key value, an exact measurement value, or both. For example, in the sentence “Our computer backup system uses 4 mm tape” the numeral is in order.' The webpage which was the source of this quote provides helpful guidelines.

I write numbers as words when they are used descriptively as part of the narrative, and with digits when they related to technical constants.

For example: I have two conclusions to make. First, the coefficient of 1 for reward is the best one to use. Second, the algorithm takes 34 iterations to converge.

But that's just my personal rule.

I have been taught, without justification, to write integer positive numbers from 1 up to and including 10 in words, and all the rest using numerical symbols.

But this was a general rule, when I was learning my (native) language as a youngster. Nowadays, a sub-rule also appears somehow fitting to my eyes: the distinction between whether the number is used to count the cardinality of a collection, or to inform about the magnitude of an entity.

I would write "Let two firms engaged in oligopolistic competition" rather than "let 2 firms..." (this is "counting the cardinality of a collection"). I would also write "Let fifty firms..." rather then "Let 50 firms"

But, I would write "the value of this parameter is set to 6", rather than "... is set to six".