There is a natural way to "add" or "multiply" two points in the Euclidean plane. By "natural" I mean that the definitions have turned out to be useful for many applications, and that the definitions are fairly simple. Unfortunately, the definitions take their simplest forms if we use different coordinate systems for the addition and multiplication operations.
The "addition" of points is described most simply as vector
addition. A vector can be represented by a directed line-segment; two
vectors are considered equal if they point in the same direction and have the
same length. (See diagram.) We can change the representation of a vector by
moving it (i.e., "translating" it) to a new position parallel to the original
position.
To add two
vectors V1 and V2, represent them with directed
line-segments so that the initial end of V2 is located at the
terminal end of V1. Thus the arrows in the diagram form a path:
start at the initial end of V1, proceed to its terminal end, then
turn a corner and follow V2 from its initial end to its terminal
end. The sum, or resultant, V1+V2, is the journey
going from the initial end of V1 to the terminal end of
V2. That sum is represented by a single directed line-segment, the
dashed third side of the triangle.
To
represent vectors with the Cartesian coordinate system, draw a vector V so
that its initial end is at the origin (0,0). Then the coordinates of the
location of its terminal end are used as the coordinates of the vector. (See
diagram.)
If we use that coordinate system, then the formula for vector addition is very simple: The first coordinate of V1+V2 is the sum of the first coordinates of V1 and V2, and the second coordinate of V1+V2 is the sum of the second coordinates of V1 and V2. That is,
| (a,b) + (c,d) = (a+c, b+d) |
The "multiplication" that we want to use can also be described in Cartesian coordinates: (a,b) ´ (c,d) = (ac-bd, ad+bc). But that's a bit complicated and nonintuitive; it looks somewhat arbitrary and contrived. We get a much simpler, more geometrically appealing definition if we switch to polar coordinates. Let a point be represented by <r,q> if it has radius r and angle q -- i.e., if it is located r units away from the origin, and on a ray that is q radians counterclockwise from the ray that points toward the right. That point has Cartesian coordinates (r cosq, r sinq). If you substitute those values into our Cartesian formula for multiplication, and then simplify using some trigonometric identities, you'll end up with this much simpler definition of multiplication:
| If P1 has polar coordinates <r1,q1> and P2 has polar
coordinates <r2,q2>, then the product P1P2 is defined to be the point with polar coordinates <r1r2, q1+q2>. |
In other words, multiply the radii and add the angles. The effect of multiplying points in the plane by P2 is to rotate the plane through an angle of q2 and stretch (or shrink) the plane by a magnification factor of r2. This concept is very simple, and it's quite useful in engineering, which is often concerned with describing rotations (e.g., of engines).
When addition and multiplication are defined as above, then the points in the plane are called complex numbers, for reasons that will be discussed a few paragraphs from now.Since (a,0)+(c,0)=(a+c,0) and (a,0)´(c,0)=(ac,0), the points along the horizontal axis have an arithmetic just like "ordinary" numbers; we will write (a,0) more briefly as a. For instance, (5,0) will be written as 5. The points along the vertical axis also have a shorter notation: the point (0,b) will be written more briefly as bi; for instance, (0,5) will be written as 5i. The i stands for "imaginary", for reasons explained below.
Using either the formula (a,b) ´ (c,d) = (ac-bd, ad+bc) or the definition in terms of polar coordinates, the beginner should now verify that i2 = -1. That will be important in the discussion below.
Information provided by Eric Schechter