Write or write down

1. I'll write down your phone number so I won't forget it.
2. You can write your name here.
3. Write about the importance of the treaty between...

CB

Oh, does it mean that both of these sentences are wrong?

Write down the importance of the treaty between...
Write on the importance of the treaty between...

Tom

Write on wasn't in your original post. On is often used instead of about, so Write on the importance of the treaty... is fine.

Another example:
He wrote a book on/about mediaeval music.

CB

Thanks a lot! And one last question.

Which one of these is better? Or are they the same?

Write a few lines on the importance of...

Write down a few lines on the importance of...

Tom

As I have shown with an example, write down is not right in this context. A student taking an exam does not write down anything to remember it later when he gets home.

CB

Thanks once again!

When I analyzed my confusion about the use of "write down", it traced back to my dictionary definition of this phrase; Essential Dictionary for American Students defines the phrase "write down" as "express".

For example: Write down one's thoughts, Write down one's memoirs...

....but then you are r