# 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":

(x_{1}, y_{1}) + (x_{2}, y_{2}) = (x_{1} + x_{2}, y_{1} + y_{2}),

but are multiplied in a more intricate way:

(x_{1}, y_{1})(x_{2}, y_{2}) = (x_{1}x_{2} - y_{1}y_{2}, x_{1}y_{2} + x_{2}y_{1}).

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 z_{k},

z_{1}·z_{2} = (z_{1}·Conj(z_{2}) + z_{1})·z_{2})/2

and the complex product as

z_{1}×z_{2} = (z_{1}·Conj(z_{2}) - Conj(z_{1})·z_{2})/2.

What if applet does not run? |

### Worthy Observations

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

Conj(Conj(z) = z. Like vectors, complex numbers are added according the parallelogram rule.

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.

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

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.

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

^{o}counterclockwise.Multiplication by -1 reflects a number in the origin.

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.

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

- Algebraic Structure of Complex Numbers
- Division of Complex Numbers
- Useful Identities Among Complex Numbers
- Useful Inequalities Among Complex Numbers
- Trigonometric Form of Complex Numbers
- Real and Complex Products of Complex Numbers
- Complex Numbers and Geometry
- Plane Isometries As Complex Functions
- Remarks on the History of Complex Numbers
- Complex Numbers: an Interactive Gizmo
- Cartesian Coordinate System
- Fundamental Theorem of Algebra
- Complex Number To a Complex Power May Be Real
- One can't compare two complex numbers
- Riemann Sphere and Möbius Transformation
- Problems

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Copyright © 1996-2017 Alexander Bogomolny

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