A Brief Description of the Question:
Why was the Quran not compiled as a book while the Prophet was alive?
The sending down of the Quran was completed when the Prophet was alive; similarly, it was recorded and memorized when the Prophet (pbuh) was alive.
As the Quran was revealed verse by verse or chapter by chapter, the Prophet would summon one of the revelation scribes and dictated them. The prophet would tell him to which chapter to add the new verse or verses and after which verses. However, the revelations were not sent chronologically; they were sent based on the style of the Quran peculiar to it. All of the verses and chapters that were revealed were recorded in the presence of the Prophet (pbuh).
The revelation scribes also wrote a copy for themselves. Thus, the verses were recorded and written safely. The verses that had been written were recited in the presence of the Prophet and the mistakes, if there were any, were corrected. The first copy of the verses that were written was kept in the house of the Prophet (pbuh). All of the written copies were kept in the house of the Prophet (pbuh).
There was not enough paper at that time; so the revelations were written on tanned leathers, wooden tablets, camel scapulas, white flat stones, branches of dates, pieces of paper and porcelain. The order of the verses was determined by definite orders and decrees. Hz. Jibril (Gabriel) would say to the Prophet (pbuh),"Put this verse in such and such chapter."and the Prophet (pbuh) would order and arrange the verses accordingly.
According to many Islamic scholars, "The arrangement of the verses in the chapters was done by the Messenger of Allah (pbuh) based on the description of Hz. Jibril." The arrangement of the chapters was made based on revelation like the arrangement of the verses. As a matter of fact, the arrangement of the Quran, which was a work of revelation in all aspects, had to be done based on revelation.
For, the whole Quran was memorized by many Companions at the time of the Prophet (pbuh).
There were many people who had memorized the whole Quran then. The Companions read it based on a certain order. It is also stated in the Quran that the Quran was arranged by the Prophet (pbuh):"It is for Us to collect it and to promulgate it: But when We have promulgated it, follow thou its recital (as promulgated)." (al-Qiyama,75/17-18)The Quran was not a hardback during the time of the Prophet (pbuh) as it is today. It was not necessary then. For, the revelation had just ended and the Prophet (pbuh) was alive.
The martyrdom of more than seventy Muslims who had memorized the Quran during the Battle of Yamama after the death of the Prophet (pbuh) and similar incidents made it necessary for the Quran to be compiled as a hardback.
During the caliphate of Hz. Abu Bakr, the duty of compiling the Quran as a single book was given to Zayd bin Thabit. Hz. Zayd collected the verses of the Quran written on various materials and put them together. This Quran, which was collected like that, was called "Mushaf".