Cartesian Coordinate System

A straight line with an associated direction, a selected point and a unit length is known as the number line, especially when the numbers of interest are integers. Otherwise, it may be called a number or real axis. The selected point is called the origin. Points on the real axis relate to real numbers such that the origin is associated with 0 and the point at the unit distance from it in the chosen direction with 1. All other points are assigned real numbers which are their distances to the origin measured with the given unit length and taken with the sign plus or minus depending whether they are on the same side from the origin as the number 1.

(It is assumed that any point on a line divides the line into two rays so that the division point separates the points on the rays. In case of a real axis and its origin, the rays are known as the positive and negative half-axes).

The number associated with a point is called its coordinate.

Complex numbers are points in the plane endowed with additional structure. The y-unit is then denoted i and the points on the y-axis are written as yi. The points on the x-axis are denoted by the single real number x, as if it was the only number axis. Instead of (x, y), we then write x + yi and call the expression the complex coordinate of a point. Thus

The sign "+" really signifies an operation of addition defined for complex numbers. In fact, yi actually means the product of y and i. The two claims imply yi = 0 + yi and x + 0i = x.

For a point (x, y) the distance to the origin (0, 0) is determined from the Pythagorean theorem,

(1)

.

More generally, the distance between two points (x₁, y₁) and (x₂, y₂) is defined as

(2)

.

In complex notations, we write instead

(1')

.

and call |x + yi| the modulus, or absolute value, of the complex number x + yi.

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. Remarks on the History of Complex Numbers
   • First Geometric Interpretation of Negative and Complex Numbers
9. Complex Numbers: an Interactive Gizmo
10. Cartesian Coordinate System
11. Fundamental Theorem of Algebra
12. Complex Number To a Complex Power May Be Real
13. One can't compare two complex numbers
14. Riemann Sphere and Möbius Transformation
15. Problems
    • Product of Diagonals in Regular N-gon
    • Sum of the Nth Roots of Unity
    • Napoleon's Relatives
    • Distance between the Orthocenter and Circumcenter

|Contact| |Front page| |Contents| |Algebra| |Store|

Copyright © 1996-2012 Alexander Bogomolny