Base N Convert

The base is the number of distinct digits used to represent numbers (in a positional number system).

A number $ N_{(b)} $ written in base $ b $ is composed of $ n $ digits $ \{ c_{n-1}, c_{n-2}, \cdots, c_1, c_0 \} $, it can be written as a polynomial where the digits are the coefficients and the base $ b $ is the variable:

$$ N_{(b)} = \{ c_{n-1}, \cdots, c_1, c_0 \}_{(b)} = c_{n-1} \times b^{n-1} + \cdots + c_1 \times b^1 + c_0 \times b^0 $$

To make a base change (from one base $ b_1 $ to another $ b_2 $), it will most often be necessary to first convert the number from base $ b_1 $ to base 10, then convert from base 10 to base $ b_2 $.

It is sometimes possible to convert directly from one base to another without going through base 10, if base $ b_2 $ is a power of base $ b_1 $, but this does not work for arbitrary bases.

To convert a number $ N $ written in base 10 to base $ b $, use the successive division method:

1 - Divide $ N $ by $ b $

2 - Record the remainder $ r $ and the quotient $ q $

3 - Repeat steps 1 and 2, replacing $ N $ with the quotient $ q $ until you get a quotient of zero.

4 - The number converted to base $ b $ is obtained by reading the remainders from the last division (highest digit) to the first division (units digit).

To convert a number $ N_{(b)} $ from base $ b $ to base $ 10 $, the most direct method is to use the formula $$ N_{(10)} = c_{n-1} \times b^{n-1} + \cdots + c_1 \times b^1 + c_0 \times b^0 $$

Another method uses polynomial accumulation:

1 - Initialize the conversion by setting $ R = 0 $

2 - Scan the digits $ c_i $ from left to right and for each digit, perform: $ R = R \times b + c_i $

3 - The final result $ R $ corresponds to the value in base 10, $ N_{(10)} = R $

A number in base 10 (or lower bases) is written with the digits 0123456789. For higher bases, it is customary to use letters, and more specifically the following characters: 0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ to write numbers up to base 62.

Other bases allow for the simplification of certain calculations or representations depending on the context.

— base 2 (binary system - base2) in informatics

— base 3 (trinary or ternary system - base3)

— base 8 (octal system - base8)

— base 9 (nonary system - base9)

— base 10 (decimal system - base10)

— base 12 (duodecimal system - base12), for month or hours

— base 16 (hexadecimal system - base16) in informatics for bytes

— base 20 (vigesimal system - base20) used by Mayan numeral system (and Aztecs)

— base 26 (alphabetic system - base26)

— base 27 (alphabetic system + special character - base27)

— base 36 (alphanumeric system - base36)

— base 37 (alphabetic system + special character - base37)

— base 60 (sexagesimal system - base60) for minutes, seconds by Sumerians and Babylonians.

— base 62 (full alphanumeric system - base62)

All the basics can be used for computer coding or any other math problem.