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In mathematics, change of base can mean any of several things: Changing numeral bases , such as converting from base 2 ( binary ) to base 10 ( decimal ). This is known as base conversion .
Base √ 2 behaves in a very similar way to base 2 as all one has to do to convert a number from binary into base √ 2 is put a zero digit in between every binary digit; for example, 1911 10 = 11101110111 2 becomes 101010001010100010101 √ 2 and 5118 10 = 1001111111110 2 becomes 1000001010101010101010100 √ 2.
Horner's method can be used to convert between different positional numeral systems – in which case x is the base of the number system, and the a i coefficients are the digits of the base-x representation of a given number – and can also be used if x is a matrix, in which case the gain in computational efficiency is even greater.
Mixed-radix numbers of the same base can be manipulated using a generalization of manual arithmetic algorithms. Conversion of values from one mixed base to another is easily accomplished by first converting the place values of the one system into the other, and then applying the digits from the one system against these.
The factorial number system is a mixed radix numeral system: the i-th digit from the right has base i, which means that the digit must be strictly less than i, and that (taking into account the bases of the less significant digits) its value is to be multiplied by (i − 1)!
Addition and subtraction are particularly simple in the unary system, as they involve little more than string concatenation. [9] The Hamming weight or population count operation that counts the number of nonzero bits in a sequence of binary values may also be interpreted as a conversion from unary to binary numbers. [10]
Negative numerical bases were first considered by Vittorio Grünwald in an 1885 monograph published in Giornale di Matematiche di Battaglini. [3] Grünwald gave algorithms for performing addition, subtraction, multiplication, division, root extraction, divisibility tests, and radix conversion.
For real numbers the quater-imaginary representation is the same as negative quaternary (base −4). A complex number x+iy can be converted to quater-imaginary by converting x and y/2 separately to negative quaternary. If both x and y are finite binary fractions we can use the following algorithm using repeated Euclidean division: