6 ms·

Speaking of mathematical missteps relating to bases, I've always been baffled by why we refer to a base system by the number above the highest representable sin

Speaking of mathematical missteps relating to bases, I've always been baffled by why we refer to a base system by the number above the highest representable single digit. Every base is "base 10" in that case! Why is binary referred to as "base 2", when the number 2 doesn't even appear? Wouldn't it make infinitely more sense to refer to our conventional number system as "base 9", binary as "base 1", unary as "base 0", and hexadecimal as "base F"? Or we could have used a more sensical word like "ceiling" or "roof" in that case, to convey that it's referring to the highest single-digit value in the system.

andsoitis 2y ago
Because it refers to the number of digits in the system. Base 10 comprises ten digits, viz. count(0, 1, 2, 3, … 9) = 10z Base 2 is count(0, 1) = 2 Base 16 (hexadecimal) is count(0, 1, 2, 3, … 9, a, b, … f) = 16
- jvanderbot 2y ago
  Straight and true. The other view is fine too: Base-9 would mean "max number is 9".
  - davrosthedalek 2y ago
    The base b is actually the multiplier for the value in each digit: the first digit is b^0, the next to the left is b^1, the next left is b^2. Similar for right of the dot: b^-1, b^-2....
- alexb_ 2y ago
  But your counts are all using base 10! If we were to count the number of digits in binary using binary, we would get 10 digits. If we were to count the number of digits in hexadecimal in hexadecimal, we would still get 10 digits. This is true for all bases.
  - maxerickson 2y ago
    Changing the representation doesn't change the number. Ten, two, sixteen...
  - andsoitis 2y ago
    Try this: Base 10 is count(0, 1, 2, 3… 9) = |||||||||| Base 16 is count(0, 1, 2,… a, b, …f) = ||||||||||||||||
jprete 2y ago
It's the base, as in "base and exponent", of the value of a digit position. If the lowest digit is marked digit zero, then each digit of a number contributes (digit x base^position) to the total value. E.g. 1567(base 10) = 7x10^0 + 6x10^1 + 5x10^2 + 1x10^3.
balnaphone 2y ago
Steve Mann resolved this notational dilemma by using the term "crown", as in binary = crown 1, octal = crown 7, decimal = crown 9, hex = crown F, and so on.
xigoi 2y ago
What would you call base -2? Or base φ? Or base i-1?

andsoitis 2y ago

Because it refers to the number of digits in the system. Base 10 comprises ten digits, viz. count(0, 1, 2, 3, … 9) = 10z Base 2 is count(0, 1) = 2 Base 16 (hexadecimal) is count(0, 1, 2, 3, … 9, a, b, … f) = 16

jvanderbot 2y ago
Straight and true. The other view is fine too: Base-9 would mean "max number is 9".
- davrosthedalek 2y ago
  The base b is actually the multiplier for the value in each digit: the first digit is b^0, the next to the left is b^1, the next left is b^2. Similar for right of the dot: b^-1, b^-2....
alexb_ 2y ago
But your counts are all using base 10! If we were to count the number of digits in binary using binary, we would get 10 digits. If we were to count the number of digits in hexadecimal in hexadecimal, we would still get 10 digits. This is true for all bases.
- maxerickson 2y ago
  Changing the representation doesn't change the number. Ten, two, sixteen...
- andsoitis 2y ago
  Try this: Base 10 is count(0, 1, 2, 3… 9) = |||||||||| Base 16 is count(0, 1, 2,… a, b, …f) = ||||||||||||||||

jprete 2y ago

It's the base, as in "base and exponent", of the value of a digit position. If the lowest digit is marked digit zero, then each digit of a number contributes (digit x base^position) to the total value. E.g. 1567(base 10) = 7x10^0 + 6x10^1 + 5x10^2 + 1x10^3.

balnaphone 2y ago

Steve Mann resolved this notational dilemma by using the term "crown", as in binary = crown 1, octal = crown 7, decimal = crown 9, hex = crown F, and so on.

xigoi 2y ago

What would you call base -2? Or base φ? Or base i-1?