Proof We know that, for a triangle with the circumcenter at the origin, the sum of the vertices coincides with the orthocenter . (3.5) Thus argz is the angle that the line joining the origin to z on the Argand diagram makes with the positive x-axis. If x, y, p, q are real and x + iy = p + iq then x = p and y = q. Proof of the Rule for Dividing Complex Numbers in Polar Form. Let us see some example problems to understand how to find the modulus and argument of a complex number.
Proof: Since x + iy = p + iq Hence x − p = -i(y − q) ⇒ (x − p) 2 = i 2 (y − q) 2 Some Useful Properties of Complex Numbers Complex numbers take the general form z= x+iywhere i= p 1 and where xand yare both real numbers.
The modulus of a complex number The product of a complex number with its complex conjugate is a real, positive number: zz = (x+ iy)(x iy) = x2+ y2(3) and is often written zz = jzj2= x + y2(4) where jzj= p x2+ y2(5) is known as the modulus of z. 1 Algebra of Complex Numbers We define the algebra of complex numbers C to be the set of formal symbols x+ıy, x,y ∈ Geometrically, in the complex plane, as the 2D polar angle φ from the positive real axis to the vector representing z.The numeric value is given by the angle in radians and is positive if measured counterclockwise. We give a short proof that the limsup of the p th root of the modulus of the p th moment of a sequence of complex numbers is equal to the modulus of the maximum of the sequence. The trigonometric form of a complex number provides a relatively quick and easy way to compute products of complex numbers.
The principal argument of z is denoted The argument of a complex number is defined as the angle inclined from the real axis in the direction of the complex number represented on the complex plane. => z is a purely real number => z = z – (ii) arg(z) = π/2 , -π/2 => z is a purely imaginary number => z = – z – Note that the property of argument is the same as the property of logarithm. These are respectively called the real part and imaginary part of z. The angle \(\theta\) is called the argument of the argument of the complex number \(z\) and the real number \(r\) is the modulus or norm of \(z\). An argument of the complex number z = x + iy, denoted arg(z), is defined in two equivalent ways: .
We write: