yoyakubangou wo wasurenai you ni, tetyou ni kaite okimasu.

It means so that I don’t forget or in order not to forget.

The pattern is:

• dictionary form + ように
• negative form + ように

So here:

• 忘れない = do not forget / won’t forget
• 忘れないように = so as not to forget

In this sentence, it gives the purpose of the main action:

• 予約番号を忘れないように、手帳に書いておきます。
• I’ll write the reservation number in my planner so that I don’t forget it.

Yes. よう is a separate word, and in this pattern ように works like so that or in such a way that.

You do not usually translate よう literally here. It is part of a set grammar pattern:

• 忘れないようにする = make sure not to forget
• 見えるようにする = make it so that it can be seen
• 聞こえるように話す = speak so that it can be heard

So in your sentence, 忘れないように is a very natural purpose expression.

Both can sometimes be translated as in order to, but they are not used in exactly the same way.

A useful rule is:

• ために is often used when the goal is something more direct and intentional
• ように is often used when the goal involves something less controllable, like remembering, being able to do something, or avoiding forgetting

Since forgetting is not something you directly control moment by moment, 忘れないように sounds very natural.

So:

• 予約番号を忘れないように、手帳に書いておきます。 = very natural

You may also hear 忘れないために, and it can make sense, but ように is especially common for this kind of idea.

～ておきます means do something in advance, do something beforehand, or sometimes do something and leave it that way.

So:

• 書く = to write
• 書いておく = to write down in advance / to write and keep it there for later
• 書いておきます = polite form of 書いておく

In this sentence, the nuance is:

• I’ll write it down now so it will be ready/useful later

That is stronger than just saying I write it.

書きます simply means write.

書いておきます adds the idea of preparation:

• 手帳に書きます = I’ll write it in my planner
• 手帳に書いておきます = I’ll write it in my planner in advance so I have it later

So ～ておきます shows that the speaker is doing it as a precaution or preparation.

In this sentence, that matches the idea of not forgetting very well.

Grammatically, 予約番号を goes with 忘れない:

• 予約番号を忘れない = not forget the reservation number

That whole part becomes:

• 予約番号を忘れないように = so that I don’t forget the reservation number

Then the main clause is:

• 手帳に書いておきます = I’ll write it down in my planner

The object of 書いておきます is omitted because it is obvious from context: it is also the reservation number.

So in English terms, Japanese is doing something like:

• So that I don’t forget the reservation number, I’ll write it in my planner.

Because に marks the place/surface/target where something is written.

• ノートに書く = write in a notebook
• 紙に書く = write on paper
• 手帳に書く = write in a planner

Here, に shows the destination or location of the written content.

If you used で, it would usually mark the means or the place where an action happens, which is not what you want here.

So:

• 手帳に書く = correct and natural
• 手帳で書く = unnatural in this meaning

Because Japanese often leaves out the subject when it is clear from context.

In English, you usually need to say:

• I’ll write it down...

In Japanese, if the speaker is obviously talking about their own action, the subject is often omitted.

So this sentence naturally implies:

• I will write it down in my planner so that I don’t forget the reservation number.

This omission is extremely common in Japanese.

Because ように attaches to the plain form of the verb, including the plain negative form.

So:

• 忘れる = plain dictionary form
• 忘れない = plain negative form

And then:

• 忘れるように = usually would mean so that someone forgets or as if forgetting, depending on context
• 忘れないように = so that someone does not forget

The sentence needs the negative idea, so 忘れない is the correct form.

Yes. 書いておきます is polite because it uses ます.

The casual form is:

• 書いておく

In everyday speech, this is often contracted to:

• 書いとく

So you might hear:

• 予約番号を忘れないように、手帳に書いとく。

That is a casual spoken version of the same idea.

The comma shows a pause and makes the sentence easier to read.

It separates:

• the purpose part: 予約番号を忘れないように
• the main action: 手帳に書いておきます

Japanese commas are often more flexible than English commas, so the sentence could also appear without one. But with a longer opening phrase like this, the comma is very natural.