Did the discovery of zero affect our calendar and how we count days? Are we off by a tenth?

As long as you use positional digit system you can calculate properly even without the zero numeral. The ancient Mesapotamians did their math in sexagesimal number system without use of zero symbols; the ancient Chinese did not use zero numerals when doing computations with counting rods or abacus, but originally use blank placeholders or none value settings.

Although later use of zero numerals undeniably facilitated the number system and expressions.

You are conflating the quantity or number zero, denoting none, with the numeral or symbol explicitly denoting the quantity of none, and represented in our Arabic Numeral system by '0' but also named zero. The significance of the symbol '0' is its utility as a placeholder when performing complex calculations on paper.

Note that numeral systems prior to the introduction of a symbol Zero, such as Roman Numerals, had no difficulty maintaining a Base 10 representation system, but simply used different symbols at each power of ten:

• I and V to represent one and five units respectively
• X and L to represent one and five tens respectively
• C and D to represent one and five hundreds respectively
• M to represent one thousands

Note that the Romans did not require a Zero placeholder for calculation as they performed calculations on an abacus rather than on paper. The inspiration for an explicit placeholder probably originates, and coincides, with additional representations for the development of Algebra.

The Ancient Babylonians also had no difficulty maintaining extensive accounting records without a placeholder for Zero by using a Base 60 numeral system.

The concept of nothingness, the number zero, is as old as numbers themselves. However the explicit representation of numerical zero, and specifically its use as a placeholder to simplify the representation of large numbers, is more recent. The precise origins are debated, but it certainly appears in India and Middle East early in the Christian Era, as well as independently in the Mayan records about the same time.

The base-10 Arabic number system you are using in your question also didn't exist back then. It couldn't have because, as you have noticed, the entire concept relies on having the concept of "0". The system was invented in India around 700 AD (not so coincidentally only about a generation after someone there discovered the concept of 0). A bit more than a century later it was being used in the Arab world, and a bit over a century after that in Europe.

In their previous written systems, of course there was a number between their 9 and their 11. For example, for the Romans and people in their domains the sequence 9, 10, 11 would be written "IX, X, XI".

The only lasting effect this had on our calendar system is that there is no year 0. The Gregorian Calendar we use goes straight from 1 BC to 1 AD with no year in between them.

Most folks are used to positional numbers meaning they have a position to indicate the number of ones, the number of tens, the number of hundreds, and so on. The common Hindu-Arabic numerals are positional. "102" is one hundred plus zero tens plus two ones.

You can see why a zero is necessary. Otherwise "12" could be "one hundred and two" or "ten and two (twelve)" or "one thousand and two tens".

Many prior number systems don't work that way. For example, Roman numerals do the opposite. They use a different glyph for each magnitude, and repetition to indicate how many. "CII" is one hundred plus two ones.

There's no need for the numeral "zero". There's no ambiguity.

Similarly, Greek numbers have specific numerals for 1 to 10, but also 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, and so on. 102 in ancient Greek 102 is ΡΒ or rho beta. 100 + 2.

Still others, such as Chinese counting rods, were positional but had clear positions laid out on a grid or mat. 102 would be written as | || with the space indicating an empty place. (You might be surprised to know that spaces to divide words are also a fairly recent invention. Reading was hard back then.) Later various glyphs were used such as the Zetian 〇, | 〇 ||. These weren't "zero" but rather to clarify a vacant position.

Tracing the parallel evolution of the numeral zero is pretty well known and was a pragmatic thing.

The concept of "zero" as its own stand alone number has puzzled mathematicians and philosophers for a long time. To many, numbers were for counting and adding real things. Five sheep. Ten gold pieces. Two hundred acres. Numbers represented real things. These are the natural numbers. 1, 2, 3, and so on.

Saying "zero sheep" seemed weird. If you have no sheep just don't write down sheep on your list. How can nothing be something? How can you "have" no sheep? How can there even be nothing? In the classical view there was always something. There was always a medium to exist in, be it air, water, earth, or aether.

This is similar to a modern person wondering about infinity. Is infinity a real thing? Or just something mathematicians faff about with? But unlike infinity which is not a number it's a concept, zero has come around to being accepted as a number. Infinity is not a number. You can't add, subtract, multiply, or divide infinities (not without defining whole new number systems with new axioms).

While weird things happen around zero, like dividing by zero (which is not infinity), you can add, divide, multiply, and subtract zero. Including zero in your mathematical system makes many things easier.

The acceptance of zero as its own number is a bit more muddled. If you want to get into that, you might be interested in Zero: The Biography of a Dangerous Idea and Finding Zero: A Mathematician's Odyssey to Uncover the Origins of Numbers.