The Shulchan Aruch dedicates a whole chapter for one short halacha, teaching that even a person who eats pat akum (kosher bread made by a non-Jew) throughout the year, should refrain from doing so on the ten days of teshuva. Since Hashem is not to be fooled by temporary measures which are not intended to be long lasting, the idea of accepting something upon one’s self for merely a few days needs clarification. Being stricter during the ten days of teshuva alone could be interpreted as hypocrisy. However, it could have a deeper meaning. By managing to be stricter for ten full days, one is indicating a message – this is not a coincidence but rather a direct outcome of an entire year of Avodat Hashem, serving Hashem. i.e. ten is a complete number. This is translated in endless ways; the Ten Commandments, the Ten Sephirot in Kabbala and in several more ways listed in the fifth chapter of Pirkei Avot (e.g. the number of the Divine utterances which the world was created by, Avraham’s tests, miracles in Egypt, plagues). Therefore when something is achieved for a few days, it may be a fluke, but when achieved for ten days it is genuine.
Avraham’s main argument to Hashem is that if there are ten righteous people in the city, it shows that the city is not that wicked, since the righteous people are produced by the city. However, if there are less than ten righteous people in the city, it is but a coincidence and they are in fact righteous in their own right regardless of the city, leaving Avraham with no valid claim.
The number ten is not merely a complete number; it has the ability to create a new reality. The most frequent example is that of a minyan. Only when in a minyan may key parts of the tefillah be said. Furthermore, Rabbi Akiva, though capable of davening a much loftier tefillah when davening alone preferred the tefillah in a minyan. Also, if one is forced to daven alone, it is halachically recommended to try and daven the same time as the local minyan. The minyan creates a new reality, so that when davening alone, one’s tefillah is accepted according to who they are and what merits they have individually. However, when davening in a minyan, neither one’s merits nor the entire minyan’s merits are a precondition and the prayer is welcomed in Heaven.