Key Points

A number system is a standard, ordered sequence of symbols, names, or objects used for counting and representing quantities. Early methods included using sticks, pebbles, or tally marks in a one-to-one mapping.

Landmark numbers are specific numbers with unique symbols that act as reference points, like I, V, X, L, C, D, M in the Roman system. Numbers are formed by combining these landmark symbols in an additive way.

A base-n number system is one where landmark numbers are powers of a single number 'n', such as $n^0, n^1, n^2, n^3, \dots$. This makes arithmetic simpler than with irregular landmark numbers.

This was a base-10 system with unique symbols for powers of 10. It was additive, not positional, meaning the position of a symbol did not change its value. For example, 23 was written with two '10' symbols and three '1' symbols.

In a place value (or positional) system, the value of a digit depends on its position within the numeral. This is the most efficient method, allowing any number to be written with a finite set of symbols.

Zero is critical in a place value system as a placeholder to indicate an empty position, for example, the '0' in 502. It was also a revolutionary concept as a number in its own right, with defined arithmetic properties.

This was a base-60 (sexagesimal) place value system. It used symbols for 1 and 10 to represent numbers up to 59 in each place. Its influence is still seen in our measurement of time and angles.

Developed independently in Central America, this was a place value system written vertically. It was mostly base-20 but had an irregularity where the third place was $18 \times 20 = 360$, not $20^2=400$. It also used a symbol for zero.

This was a base-10 place value system used for calculation. It used symbols for 1-9 and alternated their orientation (vertical/horizontal) for adjacent place values to avoid confusion, which helped in the absence of a zero symbol.

This is the system we use today, also called the Indian or Hindu-Arabic system. It is a base-10 place value system that originated in India around 2000 years ago.

It uses ten unique digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). Its combination of a base-10 system, place value, and a fully functional zero makes it highly efficient for writing numbers and performing calculations.

The system was transmitted from India to the Arab world around 800 CE, popularized by mathematicians like Al-Khwārizmī. It then spread to Europe around 1100 CE, championed by figures like Fibonacci.

Europeans called them 'Arabic numerals' because they learned them from Arabs. Arabs, like Al-Khwārizmī, called them 'Hindu numerals'. The term 'Hindu-Arabic numerals' is often used today to reflect this history.

The evolution progressed through four main ideas: 1. Grouping (counting in sets). 2. Landmark Numbers (like Roman numerals). 3. The Base System (landmarks as powers of a number). 4. The Place Value System (position determines value), perfected with the use of zero.