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This formula distinguishes the complex number i from any real number, since the square of any (negative or positive) real number is always a non-negative real number. With this definition of multiplication and addition, familiar rules for the arithmetic of rational or real numbers continue to hold for complex numbers.
A split-complex number is an ordered pair of real numbers, written in the form = + where x and y are real numbers and the hyperbolic unit [1] j satisfies = + In the field of complex numbers the imaginary unit i satisfies =
The division with remainder or Euclidean division of two natural numbers provides an integer quotient, which is the number of times the second number is completely contained in the first number, and a remainder, which is the part of the first number that remains, when in the course of computing the quotient, no further full chunk of the size of ...
In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers. According to the theorem, every such algebra is isomorphic to one of the following: R (the real numbers) C (the complex numbers) H ...
This equivalence can be extended to a metrical equivalence by mapping each complex number to a point on the sphere via inverse stereographic projection, with the resulting spherical distance applied as a new definition of distance between complex numbers; and in general the geometry of the sphere can be studied using complex arithmetic, and ...
In the complex numbers, , there are exactly two numbers, i and −i, that give −1 when squared. In H {\displaystyle \mathbb {H} } there are infinitely many square roots of minus one: the quaternion solution for the square root of −1 is the unit sphere in R 3 . {\displaystyle \mathbb {R} ^{3}.}
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a {\displaystyle a} and b {\displaystyle b} are real numbers, then the complex conjugate of a + b i {\displaystyle a+bi} is a − b i . {\displaystyle a-bi.}
Such complex logarithm functions are analogous to the real logarithm function: >, which is the inverse of the real exponential function and hence satisfies e ln x = x for all positive real numbers x. Complex logarithm functions can be constructed by explicit formulas involving real-valued functions, by integration of /, or by the process of ...