Complex Numbers
A Dynamic Tool

This is just a short refresher of what is a complex number and a tool that helps investigate some of their properties and the operations

Complex number is an ordered pair (x, y) of real numbers. It can also be written in the algebraic form as

(x, y) = x + yi.

Complex numbers are added "componentwise":

(x1, y1) + (x2, y2) = (x1 + x2, y1 + y2),

but are multiplied in a more intricate way:

(x1, y1)(x2, y2) = (x1x2 - y1y2, x1y2 + x2y1).

This is the common operation of multiplication. There are two more that can be defined. One is the real product, the product of two complex numbers that is always real, the other is complex product, the product of complex numbers that is alway complex. For two complex numbers zk, k = 1, 2, the real product is defined as

z1·z2 = (z1·Conj(z2) + z1)·z2)/2

and the complex product as

z1×z2 = (z1·Conj(z2) - Conj(z1)·z2)/2.

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Complex Numbers, a Dynamic Tool

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Worthy Observations

  1. The conjugate Conj(z) is a reflection of z in the real axis. This explains why Conj(Conj(z) = z.

  2. Like vectors, complex numbers are added according the parallelogram rule.

  3. When one of two complex numbers is fixed and the other traces a straight line their sum also traces a straight line. The two lines are parallel.

  4. When two complex numbers have fixed modules, so does their (common) product.

  5. Multiplying a complex number by a real number (i.e. dragging one of the dots along the x-axis) does not change the argument of the number but only its module.

  6. Multiplication by a purely complex number (i.e., the one lying on the y-axis) rotates a number by 90o counterclockwise.

  7. Multiplication by -1 reflects a number in the origin.

  8. The real product of two numbers is always real: it always lies on the x-axis. It is zero when the radius-vectors of the two numbers are perpendicular.

  9. The complex product of two numbers is always complex: it always lies on the y-axis. It is zero when the two numbers are collinear with the origin, i.e., when their quotient is real.

Complex Numbers

  1. Algebraic Structure of Complex Numbers
  2. Division of Complex Numbers
  3. Useful Identities Among Complex Numbers
  4. Useful Inequalities Among Complex Numbers
  5. Trigonometric Form of Complex Numbers
  6. Real and Complex Products of Complex Numbers
  7. Complex Numbers and Geometry
  8. Plane Isometries As Complex Functions
  9. Remarks on the History of Complex Numbers
  10. Complex Numbers: an Interactive Gizmo
  11. Cartesian Coordinate System
  12. Fundamental Theorem of Algebra
  13. Complex Number To a Complex Power May Be Real
  14. One can't compare two complex numbers
  15. Riemann Sphere and Möbius Transformation
  16. Problems

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