sono bangou wo memo ni kakimasita.

その means that, and it modifies the noun that follows it.

So:

• その番号 = that number

This is part of the こ・そ・あ・ど set:

• この = this ... (near the speaker)
• その = that ... (near the listener, or already mentioned)
• あの = that ... over there / that ... we both know about
• どの = which ...

In this sentence, その is directly attached to 番号, so you do not put a particle between them.

を marks the direct object of the verb.

The verb here is 書きました = wrote.
What was written? その番号.

So:

• その番号を = that number as the thing being written

A very literal breakdown is:

• その番号を = that number
• メモに = on/in a memo or note
• 書きました = wrote

So the sentence structure is basically:

[thing written] + を + [place/target of writing] + に + 書きました

Here, に shows the place/target where something is written.

So:

• メモに書く = to write on/in a note
• ノートに書く = to write in a notebook
• 紙に書く = to write on paper

In this sentence, メモに means something like:

• in a memo
• on a note
• in my notes

The exact English translation can vary depending on context, but the Japanese idea is the place or medium where the writing goes.

Because that would mean something different.

• メモを書く = to write a memo / write a note
• メモに書く = to write something on/in a memo

In your sentence, the thing being written is その番号.
So the sentence says:

• I wrote that number on/in a memo

If you said メモを書きました, the memo itself would be the thing you wrote.

書きました is the polite past affirmative form of 書く (to write).

Breakdown:

• dictionary form: 書く
• polite non-past: 書きます
• polite past: 書きました

So 書きました means:

• wrote
• have written (depending on context)

It is a normal polite form used in everyday speech.

Yes, that is a very natural English translation.

Japanese does not need a separate word for down here.
English often says:

• write the number down

But Japanese simply uses 書く (to write) and lets the context do the rest.

So these are all possible natural translations:

• I wrote that number down.
• I wrote that number in my notes.
• I wrote that number on a memo.

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

So even though English usually needs I, Japanese often does not say it.

This sentence could be understood as:

• I wrote that number down
• He wrote that number down
• She wrote that number down

depending on the situation.

If you wanted to make the subject explicit, you could say:

• 私はその番号をメモに書きました。 = I wrote that number down.

But in normal conversation, 私は is often omitted.

Not completely. Japanese word order is more flexible than English, as long as the particles stay correct.

These are all possible:

• その番号をメモに書きました。
• メモにその番号を書きました。

Both are grammatical.

The basic rule is that the verb usually comes at the end, and the particles show each word’s role.

Still, その番号をメモに書きました is a very natural neutral order.

番号 means number, usually in the sense of an identifying number.

Examples:

• phone number
• ticket number
• reference number
• room number

It is often used for numbers that identify something, rather than just any number in math.

So in context, その番号 likely means something like:

• that phone number
• that reference number
• that number you mentioned

It comes from English memo, but in Japanese メモ often means a note or written reminder, and it is very common in everyday speech.

Depending on context, メモ can mean:

• a memo
• a note
• notes

So メモに書きました can sound more natural in English as:

• I wrote it in my notes
• I wrote it on a note

rather than always translating it mechanically as memo.

Because その is not a noun by itself here. It is a modifier directly attached to 番号.

So:

• その番号 = that number

This works like English that number, not like that standing alone.

Particles attach to the whole noun phrase:

• その番号を

not to その by itself.

Normally, に is the natural particle here.

With 書く, に is commonly used to mark the place or surface where something is written:

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

Using で here would not sound natural for the same meaning.
Using へ would also not fit, because へ is mainly for direction.

So for this sentence, メモに書きました is the correct and natural choice.

Yes. Japanese に here does not force a single English preposition.

Depending on context, it could be translated as:

• on a note
• in a note
• in my notes
• on a memo pad

English chooses the most natural wording based on the situation.
Japanese simply marks メモ as the place/medium where the writing was made.

They are usually added only for learners.

Normal Japanese writing would be:

その番号をメモに書きました。

Japanese is not normally written with spaces between words.
Textbooks and beginner materials often insert spaces to make the sentence structure easier to see.