Modulus-Argument Form Complex Numbers Modulus-Argument Form

Modulus-Argument Form The complex number z is marked on the Argand diagram. Re Im Here is the real and imaginary part of the complex number. y Cartesian coordinates are not the only way to specify a position on a plane. The angle (argument) and distance from the origin (modulus) could also be used. This is called the modulus-argument form of the complex number. x Using simple trigonometry allows us to find x and y in terms of r and θ.

Multiplication Im Re

Argand diagram – the argument Im Re

The modulus

The argument

Summary When two complex numbers u and v are multiplied together, the modulus of the product uv is equal to the modulus of u multiplied by the modulus of v. The argument of uv is equal to the sum of the arguments of u and v.

Division - modulus

Division - argument

Summary When one complex number u is divided by another v, the modulus of u/v is equal to the modulus of u divided by the modulus of v. The argument of u/v is equal to the arguments of u minus the argument of v.

Using modulus-argument form If complex numbers are written in modulus-argument form, it is easy to find the modulus and argument of any product or quotient of the numbers and hence the actual product and quotient. for hence

Using modulus-argument form hence