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In mathematics, a dual system, dual pair or a duality over a field is a triple (,,) consisting of two vector spaces, and , over and a non-degenerate bilinear map:. In mathematics , duality is the study of dual systems and is important in functional analysis .
In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form a + bε, where a and b are real numbers, and ε is a symbol taken to satisfy = with .
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"A base is a natural number B whose powers (B multiplied by itself some number of times) are specially designated within a numerical system." [1]: 38 The term is not equivalent to radix, as it applies to all numerical notation systems (not just positional ones with a radix) and most systems of spoken numbers. [1]
The base-2 numeral system is a positional notation with a radix of 2.Each digit is referred to as a bit, or binary digit.Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because ...
A self-dual code is one which is its own dual. This implies that n is even and dim C = n/2.If a self-dual code is such that each codeword's weight is a multiple of some constant >, then it is of one of the following four types: [1]
The first true written positional numeral system is considered to be the Hindu–Arabic numeral system.This system was established by the 7th century in India, [1] but was not yet in its modern form because the use of the digit zero had not yet been widely accepted.
According to George Dantzig, the duality theorem for linear optimization was conjectured by John von Neumann immediately after Dantzig presented the linear programming problem. Von Neumann noted that he was using information from his game theory , and conjectured that two person zero sum matrix game was equivalent to linear programming.