[superluserdo]

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.

[jprete]

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.

[andsoitis]

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

[alexb_]

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.

[jvanderbot]

Straight and true.

The other view is fine too: Base-9 would mean "max number is 9".

[balnaphone]

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]

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