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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.
The formula is still valid if x is a complex number, and is also called Euler's formula in this more general case. [ 1 ] Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering.
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.}
The same formula applies to octonions, with a zero real part and a norm equal to 1. These formulas are a direct generalization of Euler's identity, since and are the only complex numbers with a zero real part and a norm (absolute value) equal to 1.
x is the argument of the complex number (angle between line to point and x-axis in polar form). The notation is less commonly used in mathematics than Euler's formula, e ix, which offers an even shorter notation for cos x + i sin x, but cis(x) is widely used as a name for this function in software libraries.
Figure 1. This Argand diagram represents the complex number lying on a plane.For each point on the plane, arg is the function which returns the angle . In mathematics (particularly in complex analysis), the argument of a complex number z, denoted arg(z), is the angle between the positive real axis and the line joining the origin and z, represented as a point in the complex plane, shown as in ...
Alternatively, the same system of complex numbers may be defined as the complex numbers whose real and imaginary parts are both constructible real numbers. [13] For instance, the complex number i {\displaystyle i} has the formulas − 1 {\displaystyle {\sqrt {-1}}} or 0 − 1 {\displaystyle {\sqrt {0-1}}} , and its real and imaginary parts are ...
The analogue of Euler's formula for the split-complex numbers is exp ( j θ ) = cosh ( θ ) + j sinh ( θ ) . {\displaystyle \exp(j\theta )=\cosh(\theta )+j\sinh(\theta ).} This formula can be derived from a power series expansion using the fact that cosh has only even powers while that for sinh has odd powers. [ 2 ]