Wikipedia Draft:Comparison of number bases

Main article: Radix

This article is a comparison of different numeral systems, based on their different properties.

Binary
Main article: Binary number

Binary is the base of modern computing. This is due to the fact that binary is easy to implement in computers. The presence of current is read as 1, and the lack of current, 0. However, computer programmers have found it very inconvenient for human use and thus, use hexadecimal.

Ternary
Main article: Ternary numeral system

Ternary is the most efficient integer base when measured by radix economy. This makes it, in an ideal world, the best number base. However, it's a prime base, making division and divisibility tests difficult. It also does the first two primes with easy to remember inverses, 1/2=0. 1, 1/3=0.1.

Quaternary
Main article: Quaternary numeral system

Quaternary, being the first square base (apart from 1) doesn't have any maximal representations, and thus, lack cyclic numbers. Having factors of 2, being one after three, and one before 5 means it does the first three primes with easy to remember inverses, 1/2=0.2, 1/3=0. 1, 1/5=0. 03.

Quinary
Main article: quinary

Senary
Main article: Senary

Senary is occasionally suggested as an alternative to decimal. It's proponents argue that duodecimal quarters being single digits doesn't matter as it can be derived on the fly by halving a half, which is easy for even bases, while easy-to-remember inverse primes are more important as they can only be derived from long division, and that senary has easy to remember inverses for the first four primes: 1/2=0.3, 1/3=0.2, 1/5=0. 1, 1/7=0. 05 . They also argue that senary numbers getting bigger more quickly doesn't matter as hexatrigesimal can be used as a compression system for senary and that 6 has the lowest possible result for the Euler totient function, 2.

Octal
Main article: Octal

Octal was once used in computer science to compress binary.

Decimal
Main article: Decimal

Decimal is the standard base used to represent numbers, as of 20th February 2022. Many ancient cultures have used decimal as a base, and it is hypothesized this happened because hands have ten fingers/digits. Proponents of the idea of keeping this number base argue that the transition will cause confusion and that this outweighs any potential benefits.

Undecimal
Main article: Undecimal

Undecimal is a system that uses eleven as its base. While there are no known cultures that have used the system, two cultures are speculated to have used it: the Māori, and the Pañgwa. Undecimal was briefly considered as a base to be used during the French Revolution.

Duodecimal
Main article: Duodecimal

Duodecimal is frequently suggested as an alternative to decimal. Its proponents argue that halves, thirds, quarters and sixths are single digits in duodecimal, while quarters take up two digits, and thirds and sixths are repeating digits. They also argue that senary numbers are harder to remember, as the numbers get bigger more quickly.

Hexadecimal
Main article: Hexadecimal

Hexadecimal is frequently used by computer scientists to compress binary. The proponents of hexadecimal argue that ease of arithmetic doesn't matter due to electronic devices being everywhere, and that as said devices use binary, computers would have an easier time dealing with hexadecimal inputs.

Vigesimal
Main article: Vigesimal

Hexatrigesimal
Hexatrigesimal is frequently suggested as a number base to supplant senary to remember numbers more efficiently.

Sexagesimal
Main article: Sexagesimal

Sexagesimal is a number base with a funny name. It was used in Babylon, and the way we measure time came from them. There aren't many proponents of this base, but most argue that in base 60, halves, thirds, quarters, fifths and sixths have single digit representations. However, others argue that base 60 isn't good because there are 3600 multiplications to learn, and most of them are difficult because the Euler totient function for 60 is quite high.

Other numeral systems
Imaginary bases (can represent all complex numbers in one string of numbers)

Negative bases (can represent all real numbers in one string of numbers)