SIMILARITY OF RECTANGULAR, COMPLEX, TRIG, VECTOR, AND POLAR FORMS OF A SINGLE POINT

During fourth quarter, the academic brain may begin to feel so full that not even one more concept could be stuffed in.  This is the time to ORGANIZE existing knowledge, making comparisons and identifying similarities and differences. 

And so it is with the introduction of POLARS to the Precalculus or Advanced Algebra curriculum.   There’s the rectangular way of noting a coordinate, (x,y), that we’ve known about since Algebra I.  Then came the complex numbers, i, that appeared when the quadratic formula gave us square roots of negative numbers.  After that came Trigonometry and the unit circle.   Vectors may have been included in some courses.  Now we have polar coordinates and graphs.

At this point we have so many ways to identify a single point that similarities among the terms can create a perplexing cacophony. 

        (x, y)...................................RECTANGULAR
        (x + iy)................................COMPLEX
        (cos, sin)............................TRIG
        (r cos theta, r sin theta)......VECTOR
        (r, theta).............................POLAR I
        r (cos theta + i sin theta)....POLAR II

Equations for converting from one configuration to another are standard instruments in the math toolkit, but visual representations can highlight the relationship between the forms and possibly reduce confusion.

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(x,y) RECTANGULAR coordinates are ubiquitous and should be well understood by now.  They describe things like sales, speed, population, distance, and many more common real life situations.  The x axis is what we control, the independent variable like time; the y axis reports results, outcomes.


i = √-1  Imaginary numbers were a convenience when we ran into the square root of a negative number, a frequent result from the quadratic formula.  Mathematicians are uncomfortable with answers like ‘cannot be determine,’ so in the 1700s, Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855) popularized the use of i, √-1, elevating imaginary numbers from a derogatory classification to a useful tool.

(3 + 4i) Combining a real number with an imaginary one creates a COMPLEX NUMBER that can be graphed on the ‘imaginary plane’ in the same way that (x,y) appear on the coordinate graph.  Complex forms frequently appear in electronics and in descriptions of electromagnetic fields, but higher level computations with them can be tedious. 

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(COS, SIN)  The Unit Circle in TRIGONOMETRY is our first foray into what will become a template for the polar graphing system.  But first, extend the trig information to express…

VECTORS in (r, theta) form indicate magnitude and direction and (r cos theta, r sin theta) describe horizontal and vertical positions on the standard coordinate plane. 

Trig class examples are made practical by asking about navigation, aviation, and cartography.

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(r, theta)  POLAR coordinates and r(cos theta + i sin theta), the POLAR form of a complex number, combine what we know about Imaginary numbers, Complex numbers, and the Circle.  Real life uses include navigation, robotics, and electronics.

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The visual aide graphs demonstrate that each of the forms uses the same related information.

RECTANGLE         (5, 12)

COMPLEX             (5 + 12i)

TRIG, VECTOR     (5, 12) 

                              (13 cos 67.4˚, 13 sin 67.4˚)

                                      or

                              (13 cos 1.18 radians, 13 sin 1.18 radians)

POLAR                  (13, 1.18 radians)

                               13 (cos 1.18 + i sin 1.18)

Any two pieces of data provide the basis for calculating the others and constructing the required form of a point.