The past form(-ta form) of 書く is 書いた(plain form) and 書きました(polite form).
It would be possible to think that the function of the -te form 書いて is partially overlapping with the function of the present participle "writing" in a sense, but the concepts of the two forms in each language (i.e., -te form in Japanese and present participle in English) are different. The following wikipedia page would be helpful for verb conjugations.
I would like to add upon what Toritoribe has said.
One of the most difficult parts of studying Japanese is there are many levels of politeness. There are about NINE levels of politeness in Japanese. Let’s take a closer look at two of them, the “polite” form and the “plain” form”. Let’s use 書 (pronounced か and meaning “write”) in these examples.
does not write 書きません
did not write 書かなかったです
is writing 書いています
is not writing 書いていません
Please write. 書いてください。
Please do not write. 書かないでください。
Let’s write. 書きましょう。
does not write 書かない
did not write 書かなかった
is writing 書いている
is not writing 書いていない
Please write 書いて。
Please do not write. 書かないで。
the book that I wrote (dependent clause) 書いた本 (kaita hon)
Please note that 書いて, an example you have used previously, is the plain command form (“Please write.”)
Also please note that, when in doubt, you should always use the polite form rather than the plain form.
As I mentioned previously, learning Japanese verb tenses (I call them “verb tenses” not “verb conjugations”) is a lot of hard work. You may want to take a look at a book called 501 Japanese Verbs. It takes one verb per page and goes through all the tenses of that verb (and because there are so many of them, one page is needed for each verb.)
Well, I want to make clear for learners that the concept of "verb conjugation (動詞活用)" is different from "tense (時制)." For example, many inflections like the -te form or -masu stem(-i form) don't have its tense. The -te form has many functions, as explained concisely in the wikipedia page I linked in my previous post, so "light command/soft imperative" is just one of the meanings even if it's put at the end of a sentence.
Grammatically, 書いています express different aspect from 書きます, not tense. Similarly, 書いて as a light command expresses a mood, not tense, and the passive form is a voice, not tense. For instance, 書いています vs. 書いていました or 書いている vs. 書いていた are different tenses.
I've never heard "nine levels of politeness." If it refers to formal, informal I, II, III or like that mentioned in the book, I don't agree with it.
e.g. 乾かす kawakasu
I 乾かせ kawakase
II 乾かしなさい kawakashinasai
III 乾かしてください kawakashitekudasai
I don't think 乾かしてください is informal, and there are many other expressions used for order/imperative like followings.
Japanese verbs conjugate relatively regularly. There are only two irregular verbs, する and 来る. So, if you know how to make -masu stem or -te form of verbs, you would be able to get each expression listed as "Imprative" in the book without remembering all the expressions for every verb exhaustively. Whether it's 可愛がる or 乾かす, the expression the book called "FORMAL" is "お + -masu stem + なさいませ" after all.
可愛がる kawaigaru --> u is changed to i --> 可愛がり kawaigari(-masu stem)
--> お可愛がりなさいませ o kawaigari nasai mase
乾かす kawakasu --> u is changed to i --> 乾かし kawakashi (-masu stem)
--> お乾かしなさいませ o kawakashi nasai mase
(Note: These are examples of godan verbs. Ichidan verbs conjugate differently.)
Furthermore, it's inappropriate to call the -te form "gerund." Some textbooks use this term, but it often confuses learners. Refer to the following post for details.
Probably a linguistic curiosity rather than of any practicality. Usually when one combines the causative form and the passive form, the causative inflection is attached first to the root, then the passive. My question is, can you switch the process? e.g. 食べる, 食べさせられる, 食べられさせる. Also, if this...
In the early stage of learning, I think it's OK just to remember how verbs conjugate, especially basic forms like -te from, -masu stem(called i-form in the wikipedia), past from(perfective form), -nai stem(negative form) or potential form. It's not necessary to remember all the functions of each form, or the expressions using these forms like the ones written in the book.