Place-value notation

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Place-value notation is a system of numeration where the value of a numeral symbol (a digit) depends on its position within the numeral string: a digit in a higher position is a multiple of a higher power of the base, while a digit in a lower position is a multiple of a lower power of a base; these multiples are then added up to give the value of the number. Given an integer base $b>1$, the numeral system uses $b$ digits: 0, and 1 to $b-1$. Position 0 is the 1's place, since $b^{0}=1$,, position 1 corresponds to multiples of the base, position 2 corresponds to multiples of the square of the base, etc.; these describe the integer part of the number. Going in the other direction we have the fractional part of the number, with position –1 corresponding to multiples of ${\frac {1}{b}}$, position –2 to multiples of ${\frac {1}{b^{2}}}$, etc. For example, the numeral string 12021.1 would mean (in almost any base except binary) $b^{4}+2b^{3}+2b^{2}+1+{\frac {1}{b}}$. Now a couple of examples in specific bases: 525 in octal is $5\times 8^{2}+2\times 8+5=341$; 301 in base 24 is $3\times 24^{2}+1=1729$.

Quinary, duodecimal, vigesimal and even sexagesimal were all at one time serious contenders for choice of numeral system for humans, but the choice is now undeniably the decimal numeral system. Almost all computers use the binary numeral system for their internal calculations, usually accepting input and displaying output in decimal (computer programmers often use octal and hexadecimal as convenient shorthands for the lengthy binary strings).

The foregoing should not be taken to mean that the base of a numeral system using place-value must be a positive integer. Indeed negative integers may be used (negabinary uses –2 for a base), irrational numbers like the golden ratio and even imaginary numbers like $2i$ (see quater-imaginary base). At the theoretical level, there are only three numbers that can be dismissed out of hand as the base of a numeral system:

• 1, as any number other than 0 can have an infinite number of representations by inserting any arbitrary amount of 0s into the numeral string;
• 0, as it is incapable of representing any number other than 0; and
• –1, as there are even more ways of creating ambiguous number representations (e.g., 10101 and 1111000010101 both mean 3).

At the practical level, many other considerations enter into play that make some bases more suitable for certain purposes but not intrinsically better than others. HERE OUGHT TO GO A DISCUSSION ABOUT NON-TERMINATING REPRESENTATIONS OF RATIONAL AND IRRATIONAL NUMBERS.

Also note that the direction of the powers is entirely a matter of convention. Other than tradition, nothing stops us from writing 24 / 61 = 526.2 where we would normally write 42 / 16 = 5.625.

With these various trade-offs, place-value numeral systems are still far superior to most of the ancient numeral systems. What is occasionally gained in conciseness (e.g., 1000 as M in Roman numerals uses three fewer symbols) is lost in a need to create more and more symbols for larger numbers. Even a moderately "small" Mersenne prime like 170141183460469231731687303715884105727 presents problems for Roman numerals, and an ancient Roman versed in the concept of primality would still be unlikely to at least guess that II^CCXXVII – I could be a prime number. But of course truly large numbers are problematic for all numeral systems.