Complex Numbers

1. Complex Numbers Modulus-Argument Form

2. Im y Re x Modulus-Argument Form The complex number z is marked on the Argand diagram. Here is the real and imaginary part of the complex number. 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. Using simple trigonometry allows us to find x and y in terms of r and θ.

3. Multiplication Im Re

4. Argand diagram – the argument Im Re

5. The modulus

6. The argument

7. 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.

8. Division - modulus

9. Division - argument

10. 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.

11. 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

12. Using modulus-argument form for hence