Complex Numbers Formula

1. Euler's formula

Euler's formula is a fundamental relationship in mathematics that links complex numbers to trigonometry. It states that for any real number x, e^(ix) = cos(x) + isin(x), where e is the base of the natural logarithm, i is the imaginary unit, cos(x) is the cosine function, and sin(x) is the sine function.

This formula was discovered by the Swiss mathematician Leonhard Euler.

e^{\left(ix\right)}=\cos x+i\sin x

2. De Moivre's theorem

De Moivre's theorem is a formula that relates complex numbers to trigonometry. It states that for any complex number z = r(cos(theta) + isin(theta)), where r is the modulus of z and theta is the argument of z, z^n = r^n(cos(n*theta) + isin(n*theta)).

This formula was discovered by the French mathematician Abraham de Moivre.

z^n=r^n(\cos(n\theta)+i\sin(n\theta))

3. Polar form formula

The polar form formula represents a complex number in terms of its magnitude (which is the distance from the origin in the complex plane) and argument (which is the angle from the positive real axis). It is given by z = r(cos(theta) + isin(theta)), where r is the modulus of z and theta is the argument of z.

z=r(\cos(\theta)+i\sin(\theta))

4. Argand diagram formula

The Argand diagram formula is a graphical representation of complex numbers in the complex plane. It consists of a horizontal real axis and a vertical imaginary axis, with each complex number z = x + iy corresponding to a point (x, y) in the plane.

This formula was developed by the French mathematician Jean-Robert Argand.

z = x + iy

5. Modulus-Argument form formula

The modulus-argument form formula represents a complex number in terms of its modulus and argument. It is given by z = r(cos(theta) + isin(theta)), where r is the modulus of z and theta is the argument of z.

z = r(\cos(\theta)+ i\sin(\theta))

6. Conjugate formula

The conjugate formula involves changing the sign of the imaginary part of a complex number. If z = x + iy, then the conjugate of z is denoted by z* and is given by z* = x - iy.

7. Cartesian form formula

The Cartesian form formula represents a complex number in terms of its real and imaginary parts. It is given by z = x + iy, where x is the real part and y is the imaginary part.