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In arithmetic, a complex-base system is a positional numeral system whose radix is an imaginary (proposed by Donald Knuth in 1955 [1] [2]) or complex number (proposed by S. Khmelnik in 1964 [3] and Walter F. Penney in 1965 [4] [5] [6]).
A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. Re is the real axis, Im is the imaginary axis, and i is the "imaginary unit", that satisfies i 2 = −1.
He led the development of the Complex Number Calculator (CNC), completed in November 1939 and put into operation in 1940. Employing electromagnetic relay binary circuits for its operations, rather than counting wheels or gears, the machine executed calculations on complex numbers. [9]
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The multiplication of two complex numbers can be expressed more easily in polar coordinates: the magnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argument of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a ...
A split-complex number is an ordered pair of real numbers, written in the form z = x + j y {\displaystyle z=x+jy} where x and y are real numbers and the hyperbolic unit [ 1 ] j satisfies
It is very similar in construction to the Fuller instruments but its pointers have multiple indices so additional trigonometrical functions can be used. It cost slightly less than the Fuller-Bakewell and a 1919 example is held by the Science Museum, London. [14] [15] [16] In 1962 the Whythe-Fuller complex number calculator was introduced.
Every complex number (every number of the form a+bi) has a quater-imaginary representation. Most numbers have a unique quater-imaginary representation, but just as 1 has the two representations 1 = 0. 9 in decimal notation, so, because of 0. 0001 2 i = 1 / 15 , the number 1 / 5 has the two quater-imaginary representations 0 ...