Complex numbers are often represented on the complex plane, sometimes known as the Argand plane or Argand diagram. In the complex plane, there are a real axis and a perpendicular, imaginary axis. The complex number \(a+bi\) is graphed on this plane just as the ordered pair \((a,b)\) would be graphed on the Cartesian coordinate plane.

One importance of conjugation comes from the fact the product of a complex number with its conjugate, is a real number!! (See the operation c) above.) This can come in handy when simplifying complex expressions. It is like rationalizing a rational expression. Let's look at an example to see what we mean.

Complex exponentiation extends the notion of exponents to the complex plane.That is, we would like to consider functions of the form \(e^z\) where \(z = x + iy\) is a complex number.

In plane geometry, complex numbers can be used to represent points, and thus other geometric objects as well such as lines, circles, and polygons. They are somewhat similar to Cartesian coordinates in the sense that they are used to algebraically prove geometric results, but they are especially useful in proving results involving circles and/or …

The complex plane (also called the Argand plane or Gauss plane) is a way to represent complex numbers geometrically. It is basically a modified Cartesian plane , with the real part of a complex number represented by a displacement along the \(x\)-axis, and the imaginary part by a displacement along the \(y\)-axis.

Contour integration is a method of evaluating integrals of functions along oriented curves in the complex plane. It is an extension of the usual integral of a function along an interval in the real number line. Contour integrals may be evaluated using direct calculations, the Cauchy integral formula, or the residue theorem. A curve in the complex plane …

In complex analysis, Euler's formula provides a fundamental bridge between the exponential function and the trigonometric functions. For complex numbers \( x \), Euler's formula says that \[ e^{ix} = \cos{x} + i \sin{x}. \] In addition to its role as a fundamental mathematical result, Euler's formula has numerous applications in physics and ...

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A complex fraction is a fraction with a numerator or denominator that also contains a fraction. As an example, the following is a complex fraction: ...

5 days ago · The primitive \(n^\text{th}\) roots of unity are the complex numbers \[ e^{2\pi i k/n} : 1\le k \le n, \text{gcd}(k,n)=1.