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Hexadecimal (also known as base-16 or simply hex) is a positional numeral system that represents numbers using a radix (base) of sixteen. Unlike the decimal system representing numbers using ten symbols, hexadecimal uses sixteen distinct symbols, most often the symbols "0"–"9" to represent values 0 to 9 and "A"–"F" to represent values from ten to fifteen.
Of particular interest are the quater-imaginary base (base 2i) and the base −1 ± i systems discussed below, both of which can be used to finitely represent the Gaussian integers without sign. Base −1 ± i , using digits 0 and 1 , was proposed by S. Khmelnik in 1964 [ 3 ] and Walter F. Penney in 1965.
The Natural Area Code, this is the smallest base such that all of 1 / 2 to 1 / 6 terminate, a number n is a regular number if and only if 1 / n terminates in base 30. 32: Duotrigesimal: Found in the Ngiti language. 33: Use of letters (except I, O, Q) with digits in vehicle registration plates of Hong Kong. 34
By using a dot to divide the digits into two groups, one can also write fractions in the positional system. For example, the base 2 numeral 10.11 denotes 1×2 1 + 0×2 0 + 1×2 −1 + 1×2 −2 = 2.75. In general, numbers in the base b system are of the form:
The space K(ℓ 2) of compact operators on the Hilbert space ℓ 2 has a Schauder basis. For every x, y in ℓ 2, let x ⊗ y denote the rank one operator v ∈ ℓ 2 → <v, x > y. If {e n} n ≥ 1 is the standard orthonormal basis of ℓ 2, a basis for K(ℓ 2) is given by the sequence [17]
Thus we have a characterization of d + 1 mutually unbiased bases as those for which the uncertainty relations are strongest. Although the case for two bases, and for d + 1 bases is well studied, very little is known about uncertainty relations for mutually unbiased bases in other circumstances. [27] [28]
In a positional numeral system, the radix (pl.: radices) or base is the number of unique digits, including the digit zero, used to represent numbers.For example, for the decimal system (the most common system in use today) the radix is ten, because it uses the ten digits from 0 through 9.
An ideal does not have any zero (the system of equations is inconsistent) if and only if 1 belongs to the ideal (this is Hilbert's Nullstellensatz), or, equivalently, if its Gröbner basis (for any monomial ordering) contains 1, or, also, if the corresponding reduced Gröbner basis is [1]. Given the Gröbner basis G of an ideal I, it has only a ...