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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.
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.}
In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit j satisfying =, where . A split-complex number has two real number components x and y , and is written z = x + y j . {\displaystyle z=x+yj.}
Even though contains copies of the complex numbers, it is not an associative algebra over the complex numbers. Because it is possible to divide quaternions, they form a division algebra. This is a structure similar to a field except for the non-commutativity of multiplication.
The key to the argument is the following Claim. The set V of all elements a of D such that a 2 ≤ 0 is a vector subspace of D of dimension n − 1. Moreover D = R ⊕ V as R-vector spaces, which implies that V generates D as an algebra. Proof of Claim: Pick a in D with characteristic polynomial p(x). By the fundamental theorem of algebra, we ...
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, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory , the theory of group characters , and the discrete Fourier transform .
The Gaussian integers are the set [1] [] = {+,}, =In other words, a Gaussian integer is a complex number such that its real and imaginary parts are both integers.Since the Gaussian integers are closed under addition and multiplication, they form a commutative ring, which is a subring of the field of complex numbers.