Multiplication of Points in the Plane

Below, square brackets are used to indicate polar coordinates while round brackets indicate rectangular coordinates.

Next we define using polar coordinates the product of two points in the plane. Each point or factor is located by means of angular displacement or rotation from the positive real axis, and also a nonnegative distance from the origin. The product of two points is given by a third point. Its angular displacement is the sum of the angular displacement of the factors. Its distance to the origin is the product of the distances of the factors. This is the add the angles and multiply the lengths rule. In polar coordinate notation, the multiplication rule and definition is indicated by

(r₁,q₁)·(r₂,q₂) = (r₁r₂,q₁+q₂) when 0 <  r₁r₂ 

The polar coordinates of the origin [0,0] is taken to be (0, 0 degrees). However, all polar coordinates of the form (0, ,q ) where the angle measure ,q  in degrees (etc) locate the origin.

Example. Two arrows are to be multiplied. One has length 1.3 and angle 22.62^°; the other factor has length 1.026 and angle 46.97^°; and so their product has length 1.3338 = 1.3·1.026 and angle 69.59^° = 22.62^°+46.97^°; and that is it. See the following diagram.

correction: 22.62  + 46.97 = 69.59 not 69.69s

Another Example. The product of the two points (3,80^°) and (4, 60^°) is 

(3 ^. 4, 80^°+ 60^°) = (12,140^°)

Remark:  For each point in the interior of the first quadrant with rectangular coordinates [a,b], the line segment from the origin [0,0] to [a,b] is the hypotenuse of a right triangle with length r and angle q say  at the vertex [0,0]. Thus the rectangular coordinates determine the polar coordinates (r, q).  Conversely, the polar coordinates (r,q ) where 0 < q < 90 degrees determine a line segment of length r and angle q with the positive x-axis. That line segment with the aid of the ASA triangle construction method determines horizontal and vertical legs of a triangle. The lengths of the horizontal leg gives a> 0 while the length of the vertical leg gives b > 0.  The vertex at the non-origin end of the hypotenuse has rectangular coordinates [a,b]. Here specification of polar coordinates or a point in the first quadrant determines the rectangle coordinates [a,b].  Whence specifing a point, giving its polar coordinates and giving its rectangular coordinates are equivalent operations in the first quadrant (off the coordinate axes). The foregoing can be generalized to all four quadrants as well as to points on the axes.

What Are Complex Numbers

Points in the plane with the operations of addition and multiplication just given are called the complex numbers. The plane with these two operations on its points is called the complex numbers plane, or more briefly the complex numbers.

We will now change to a more standard notation for them. We may and often will write the rectangular coordinates z = (a,b) as z = a+ib, We will further call the abscissa a, the real part of the complex number z = a+ib. We will also call the ordinate b, the imaginary part of the complex number z = a+ib.

Note: Two quantities x and y are equal modulo a third quantity c, if and only if their difference x-y = kc for some whole number or integer k.

We will say that the complex number z = a+ib is purely imaginary when its real part a = 0. The angle of a purely imaginary complex number z = a+ib = 0+ib = (0,b) is 90 degrees or 270 degrees (modulo 360 degrees), depending on the sign of the imaginary part b. When b > 0, the angle is 90 degrees (modulo 360 degrees). When b < 0, the angle is 270 degrees (modulo 360 degrees).

We will also say that z = a+ib is (purely) real when its imaginary part b is zero. The angle of a (purely) real complex number z = a+ib = a+i0 = (a,0) is 0 degrees or 180 degrees (modulo 360 degrees), depending on the sign of the real part a. If a > 0, this angle is 0 degrees (modulo 360 degrees) while if a > 0, this angle is 180 degrees (modulo 360 degrees).

Exercise: Use  b = sign(b)|b| to show that  bi = b^. i where i = [0,1]

Real Numbers as Complex Numbers

Each complex number z = a+i0 with imaginary part zero gives and is given by a real number a. We will write z = a in this situation, and say that the complex number z is also a real number.

With this practice, the real numbers can be regarded as a subset of the complex numbers; and the real number line can be identified with the horizontal axis of the plane.

Confirmation of The Law of Signs

We identify the real number line with the horizontal axis of the plane. With this identification, observe that positive numbers have angular displacement zero, modulo 360 degrees. Also observe that negative numbers have angular displacement 180 degrees, modulo 360 degrees. The magnitude of a real number is its distance to the origin.

Suppose z = a+i0 = [a, 0] and w = c+i0 = [c,0]. We want to compute the product zw with the multiply the lengths, add the angles rule. Each factor has length |a| or |c|. Each factor has angle 0 or 180 degrees (modulo 360 degrees). The relationships

• 0^° = 0^°+0^°
• 180^° = 0^°+180^° = 180^°+0^°
• 360^° = 180^°+180^° = 0^° (modulo 360^°)
  

• (+1) = (+1)(+1) as 0^° = 0^°+0^°
• (-1) = (+1)(-1) = (-1)(+1) as 180^° = 0^°+180^° = 180^°+0^°
• (-1)(-1) = (+1) as 360^° = 180^°+180^°
  

For the second example, the number -2 is identified with the point [-2,0] = (2,180^°). See the figure below.

Now multiplying the point (2,180^°) by itself leads to the product (2,180^°)² = (2²,180^°+180^°) = (4,360^°) = (4,0^°). Thus the point on the horizontal axis identified with -2 when squared gives the point identified with +4 indicated above. The 360 degrees in the diagram for the number or point 4 = [4,0] represents the doubling of the angle 180 degrees.

For an example or exercise, compute the pair-wise products of 3=3+0i, 4=4+0i, -3=-3+i0 and -4=-4+0i using the add the angles, multiply the lengths rule.

Teachers: The add the angles, multiple the lengths rule for the multiplication of complex numbers gives a rule for the multiplication of real numbers once the multiplication of nonnegative numbers with themselves is mastered. There are now three ways to introduce the law of signs. (i) give it as as part of a rule for multiplication of real numbers after students have learnt to multiply unsigned numbers;  (ii) derive it from the axioms for real numbers;  and (iii) derive it from the add the angles, multiple the lengths rule for multiplication of complex numbers, after signed numbers have been introduced as a coordinates in or along a real line and in rectangular coordinates for the plane. Approach (ii) presumes or forces a mastery of the algebraic way of reading and writing. Thus (i) and/or (iii) could be best for novices.  Both could be used to define the product of real numbers to people/students who know (a) about the addition of real numbers or coordinates and (b) about the multiplication of non-negative numbers. They would not need to have any previous knowledge of the law of signs.

More Exercises. Compute the following using the multiply the lengths, add the angles rule:

1. A = (1.5)·(2). 
2. B = (1.5)·(-2). 
3. C = (-1.5)·(-2).
4. D = (1.5)·(-2).
5. E = (10,45^°) ·(1/20,15^°).
   

Stop For A Summary. The polar coordinate definition

(r1,q1)·(r2,q2) = (r1r2,q1+q2)

of the product of two point in the plane, involves the multiplication of lengths (= distances to the origin) and the addition of angles. For points on the horizontal axis, the angles of the factors are zero or 180^° (modulo 360^°). Computing the angle of the product will involve one of the following expressions:

0°+0°	=	0°
0°+180°	=	180°
180°+0°	=	180°
180°+180°	=	360°

Since the angle 180 degrees is associated with -1, and the angles 0 and 360 degrees are both associated with the number +1, the polar coordinate definition of multiplication of points in the plane agrees with (or yields) the law of signs for the multiplication of positive and negative numbers.

In rectangular notation, the foregoing implies  the product [a,0]·[d,0] = [ad,0] holds for all real numbers a and c.  We call the latter the real-real product formula.

1. Multiplication of points [a,0] by the number i = (1, 90 degrees) in the case that a > 0 equal (a , 0 degrees) · (1, 90 degrees) = (a, 90 degrees) = [0, a]
2. Multiplication of points [a,0] by the number i = (1, 90 degrees) in the case that a < 0 equal (|a| , 180 degrees) · (1, 90 degrees) = (|a|, 270 degrees) = [0, a] as well. 

Items 1 and 2  imply  i·[a,0] = [0, a] =  [a,0]·i for all real numbers a. We call this the Basic imaginary-real product formula: 

Now [a,0]·[0,d] =  [a,0]·([d,0]·i) = ( [a,0]·[d,0]) i = [ad,0] i  = [0, ad] since multiplication in the plane is associative. Hence the product

 [a,0]·[0,d]  = [0, ad] = [0, da] = [d,0]·[0,a] 

We call the latter the real-imaginary or imaginary-real product formulas. 

Now the imaginary-imaginary product

[0,a]·[0,d] = (+1) [0,a]·[0,d]  

= (-i²) [0,a]·[0,d]) since -i² = +1
= -1 i[0,a] ·i[0,d] since multiplication is commutative & associative
= -[a,0]·[d,0] due to the basic imaginary-real product formula
= -[ad,0] due to the real-real product calculation formula
= [-ad, 0] due to the definition of polar multiplication.

Our conclusion is that   [0, a]·[0,d] =   [-ad, 0].  We call this the imaginary-imaginary product calculation formula. 

Summary of Key Rectangular Coordinate, Product Calculation Formulas:

• [a,0]·[d,0] = [ad,0]   - the real-real case product formula
• [a,0]·[0,d]  = [0, ad] - the real-imaginary product formula
• [d,0]·[0,a] = [0, da]  - the imaginary-real product formula
• [0, a]·[0,d] =   [-ad, 0] - the imaginary-imaginary product formula

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