on numbers

numbers were the first writing system to exist. it started with the same symbol being used for all numbers: IIIIIIIIIIIII

• earliest improvement: group symbols together.
  • IIIIIIIIIIIII → IIII IIII III
• allowed people to go from tallying "another one, another one..." to counting "forty-one, forty-two..."

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a bigger improvement came along with the idea to have different symbols for different numbers.

the best example of this new kind of writing systems are Roman numerals. they used 7 symbols (I, V, X, L, C, D, M) and 3 main rules:

1. symbols must be ordered by decreasing value, from left to right.
   • VI+VII → VIVII → VVIII
2. having symbols side-by-side is the same as adding them together.
   • VVIII = 5+5+1+1+1
3. a symbol cannot be repeated if it can be combined.
   • VVIII → VV=X → XIII

• these simple rules made Roman numerals faster and more accurate for mathematics than tally symbols, which is why they stayed in use for ~2400 years, from around -900 until the 1500s.

the biggest leap Roman numerals made was allowing people to manipulate numbers without needing to count each symbol individually: their design implicitly teaches people how to do math.

• the system, however, relied on as as the symbol with the highest value (1000).
• this meant that performing operations on large numbers could only be done by carefully counting each instance of the same symbol (=counting tally marks).
  • MMMMMMCDII + MMMMMMMMCCDIII
• simple operations on large numbers became very difficult and error-prone → new system needed

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1500s, writing numbers as letters

Hindu-Arabic numerals were invented in 9th century India but only became commonly used until the 1500s.

• they use a set of 10 symbols (0-9) and only one single rule: the value of a digit depends on its position in the number.
  • 002=2 ⠀ 020=20 ⠀ 200=200
• this rule allows the system to be universal and work like an alphabet: every number in existence can be represented in a unique, distinct and understandable way (just as no word exists that can be said but can't be written).
• their design also simplified complex operations and significantly reduced chances of errors: an avalanche of new mathematical progress followed shortly after their adoption (Galileo, Newton, the Enlightenment, birth of modern Physics...)

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the leap in progress from Hindu-Arabic numerals is a consequence of deep insights on how math, numbers and design work.

every numeral system incorporates certain knowledge about which relationships of numbers are the most interesting to use.

• tally symbols → Roman numerals: new knowledge about addition and rules about arbitrarily defined symbols.
• Roman numerals → Hindu-Arabic numerals: complex mathematical ideas (commutativity, associativity, distributivity) expressed through profound design ideas (abstract representation, parsimony, spatiality).

∴ design and mathematical insights work together: math and design —science and art— are entangled.

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the main advantage of improving the design of a numeral system is making it useful for unpredictable innovations and progress.

Roman numerals allowed new ideas and uses of numbers like salaries, taxes and interest rates to spread, but their design also deeply limited scientific progress.

⤷ no major abstract mathematical discovery was made during the ~2400 years they were in use.

• when thinking about how our numeral system could be limiting further unknown progress, the question becomes: how does one make the leap to a smarter way of writing numbers?