These nine [berachot of the Amida] of Rosh Hashana correspond to what? Rabbi Yitzchak from Kartignin said: They correspond to the nine references to Hashem that Chana said in her prayers, as it was said: It was on Rosh Hashana that Sarah, Rachel, and Chana were remembered [with the children they prayed for]. Ein Ayah:
On Rosh Hashana, all the world’s inhabitants pass before Hashem like b’nei maron, which [according to one explanation in Rosh Hashana 18a] means like a flock of sheep. This illustrates the matter of Divine Providence on an individual basis, which exists beyond the Divine Providence that relates to a species or is general Providence.
The number ten represents an entire group as a whole, as this is the lowest number that is considered a community. Therefore, the most appropriate number to represent one relating to individuals in a group as individuals is nine.
This idea applies to Chana, who requested from Hashem to fulfill her personal desire in a manner that was above the way of nature. She was not fit to give birth and requested that Hashem provide a miracle to help her as an individual, which is very much along the lines of individualistic Divine Providence. That is the reason that she mentioned Hashem’s names nine times in her prayers to show the great extent of individual Divine Providence that Hashem employs.
On Rosh Hashana, we must particularly concentrate on picturing in our mind’s eye the matter of Divine Providence for the individual. This can help us improve our individual actions and increase our evoking of mercy, each one according to his own needs. This is the way to build the success of the collective, which requires the success of its constituent parts. For this reason, the prayers of Rosh Hashana were built upon the number nine. The Needs of the Individual and of the Collective
(based on Berachot 2:43) Gemara:
Master of the universe, it is clear and known to you that we desire to do Your desire, but who stands in the way? The yeast in the dough and the subjugation of the kingdoms. May it be Your will to subdue them from before us and after us, and we shall return to the laws of Your desire with a full heart. Ein Ayah:
A person should deal with his shleimut (completeness) in regard both to his personal matters and the shleimut of the collective by joining together with K’lal Yisrael in a manner that will bring spiritual and physical benefit to the nation by his actions. The main causes of lacking in an individual’s actions are a bad nature and its faults, which are referred to as the yeast in the dough. Regarding communal needs, which are in a low point without the strengthening of Hashem’s Torah and all good goals and attributes, it is the subjugation of the kingdoms. In all matters, we can look forward (the main part) or backward (the minor part). Really, a person is complete only when he is attached to the collective. Therefore, the front part is the communal, and the back is the individual. We ask that we should not be held back in either realm.
The matter of returning to the laws of Your desires refers to the laws of the Torah. The laws complete a person in a true way; some of them are designed to complete the collective. Therefore, only when one is prepared to improve himself, others, and the collective can he do the laws that Hashem desires with a full heart, for then he will feel the internal pleasantness of the holy Torah and mitzvot. Then Hashem’s pleasantness will be upon him to set the path of his actions.