www.whyslopes.com > Volume 1B,  Mathematics Curriculum Notes,  1996  >   11 Complex Numbers     [ Back ]

Chapter 11: Complex Numbers

By means of measurement coordinates locate points on line and in the plane with both rectangular and polar coordinates. Navigational examples can be employed to introduce, illustrate and motivate vector and displacement addition. The earlier described operation on vectors of multiplying lengths and adding angles can be used to define multiplication in the plane and extend the concept of multiplication on real line from pairs of positive numbers, to any pair of real numbers – points on a horizontal axis. The foregoing defines the complex numbers (minus a representation of products in terms of real and imaginary parts). The polar coordinate definition of multiplication multiply lengths and add angles applied to real numbers provides or agrees with the law of multiplication and law of signs.

This development is pre-algebraic, but rule-based. It is in accordance with the expositional principle of putting the material easiest to understand first. Before this development only a familiarity with addition, subtraction and multiplication of positive numbers (fractions or decimal representation) is assumed. Negative numbers need only be employed as coordinates. Operations with them need not be defined before the exposition of complex numbers. An elaboration of these ideas follow. This development represents an innovation for elementary mathematics instruction, and it has some consequence for intermediate mathematics instruction.

This new perspective introduces operations on real numbers (signed decimals, signed fractions or points on a horizontal axis of the complex plane) without relying on algebra or algebraically described properties of real numbers. It is computational and pre-algebraic.

Coordinates On the Line

Before the introduction of negative numbers, the notion of positive numbers is not emphasized. Students may have a knowledge of unsigned (positive) numbers. These numbers can be employed as coordinates on an infinite half-line to locate points. After this, signed numbers, positive and negative can be employed to locate points, that is to serve as coordinates on the bi-infinite real line. This allows students to graphically comprehend the role of positive and negative numbers, and zero too, as coordinates or marks on a coordinate line. No arithmetic is required. Examples of coordinate lines are provided perhaps by temperature scales, by water levels (the signed height of tides, reservoirs or river waters above or below a zero mark) and by bank account balances. Accountants today employ parentheses to avoid writing negative signs. Prior to the 15th century, negative numbers were thought to be imaginary – figments of the imagination.

Coordinates in the Plane

Ordered pairs of positive and then arbitrary real numbers can be introduced as rectangular coordinate for the plane after the selection of an orthogonal pair of axes. Following Descarte, ordered pairs of positive or unsigned number locate points in the first quadrant. Following Newton (or others before Newton), signed coordinates can be employed to locate points in all four quadrants. This role of signed coordinates offers another motivation for having and employing positive and negative numbers.

Displacement and Vector Addition

Points in the plane can be identified with vectors (issuing from the origin). The transport of these vectors and the head-to-tail addition of vectors can be described graphically, and then in terms of rectangular coordinates. The rules for this can be drawn from examples in an inductive fashion. Motivation can be provided by the problem of planar navigation, and moving from point to point on a map. Here the addition of vectors representing displacement on a map can be introduced. The resultant of two successive displacement can be declared to be the linear displacement between the initial point of the first displacement and the terminal point, following the second displacement. Students will find from examples and exercises that the addition methods appear to be repeatable and reproducible, and thus verifiable in a pre-algebraic and pre-deductive fashion.

Restriction to An Axis

Addition of vectors or points on a coordinate axis or coordinate line can then be viewed as a special or restricted case of the more easily visualized situation in the plane, an application of the head to tail vector addition method to pairs of points, alias vectors, on the coordinate line. This will lead via examples to easily visualized rules for addition of positive and negative numbers, that is points on the horizontal axis with positive and negative coordinates. Rules for the addition and subtractions of numbers, vectors or displacements in the horizontal coordinate line, can now be extracted from the planar case: regarded as the special or limiting case of motion restricted to a single line in the plane. This provides another means to visualize mathematics.

Multiplication of Planar Points

Both polar and rectangular coordinates with respect to a pair of axes can be determined (measured) for points. Given or measured values of polar or rectangular coordinates can also be used to locate points. The foregoing geometrically suggests that polar and rectangular coordinates are interchangeable. It provides a method, geometric measurement, for obtaining polar coordinates from rectangular, and vice-versa. This approach is hands-on, physically dependent and while not deductive, it is repeatable, reproducible, and thus secure. Angles in polar coordinates can be computed, modulo 360 degrees.

Given a pair of nonzero vectors issuing from the origin, that is two points in the plane, their angles can be measured, and their lengths measured and represented by an unsigned decimal number – a unit-free length. Adding the angles together, modulo 360 degrees, and multiplying the unit-free lengths together yield the angle and unit-free length of third vector, their product. This defines via polar coordinates, the multiplication of points or vectors in the plane.

Multiplication of Real Numbers

Following the identification of the horizontal axis with the real number line, a polar-coordinate representation or visualization of the product of real numbers follows. This may define for students such products. This definition implies the law of signs for the product of real numbers. Moreover, the identification also provides a context and location for the definition of square roots of negative numbers. These square roots can be found on the vertical, alias imaginary, axis.

Consequences for Intermediate Instruction

The foregoing offers in elementary instruction a computational and visual comprehension of arithmetic with real and complex numbers. In intermediate level instruction, there is choice of how expressions for the real and imaginary parts of the product of two complex numbers are to be obtained. Assumption of the distributive law of multiplication over addition in the complex plane immediately implies expressions for the real and imaginary parts of the product in terms of those of the factors. Ease of exposition may justify the assumption: Intermediate instruction need only offer strands of reasoning. Threading them together in a purely deductive fashion may be left to advanced courses. But the distributive law can be seen or justified via geometric arguments:

The distributive law itself can be geometrically implied or suggested by viewing multiplication by a nonzero complex number as the consequence of multiplying by a positive length and following up by a rotation. The operations commute. Both are distributive over addition. Reasons for the latter follow.

For just a rotation, one can physically show the distribution law by considering the rotation of a parallelogram.

For just a positive stretch factor, one can show this in the special case of small whole numbers, and then proceed inductively to the case of rational numbers. The case of irrationals now follows by an assumption of and intuitive appeal to continuity.

The geometric argument has the appeal that it applies to real multiplication as well.

As a third alternative, the distributive law can be assumed for real numbers only and then later on in a trigonometry course, the distributive law for complex numbers can be obtained from the angle sum formulas. The rotate-a-triangle proof of these formulas may be less of a surprise and more accessible to students who have seen the add the angles, multiple the lengths polar coordinate method for complex multiplication. Against this third approach, I suspect that many students on learning the distributive law for real numbers will apply it to complex numbers without a second thought, and with little patience for the notion that it should be derived. They may be correct.

For ease of exposition, and to provide a greater command of mathematics, the distributive law for complex numbers can be assumed, and from it the distributive law for real numbers obtained as a special. In the derivation of mathematics from set theoretic foundations for arithmetic (axiomatic set theory), both distributive laws, the one for reals and the one for complex numbers, are almost equidistant from the axioms in terms of the work required for their respective derivation. The first exposition of complex numbers like that of trigonometry and calculus may mixed algebraic and geometric arguments which illustrate the deductive aspect of mathematics.

The geometric argument, the first alternative outline above, avoids the semantic problem of which distributive law to assume first. A second chapter on complex numbers in the companion book Why Slopes and More Math explores these possibilities in more detail. A fourth alternative is to present the distribution law as a theorem and leave its proof as an intellectual IOU.

Footnotes:

1. This educational writer has no first hand experience of the elementary school classroom as teacher. The image here is based on home-based observations of the children of others, how they learn, and this author’s memory as a student in the classroom. This author as a child was an adult in waiting – mentally alert and observant, if not informed, attending the future and a reason for being.
2. Numbers may be just adjectives that become objects when discussed separately from the description of other objects.
3. Another context for subtraction is provided by the coordinate line. Addition of a number n corresponds to n steps or a movement in a forward or positive direction. Subtraction corresponds to  step in another direction.
4. flimsy evidence perhaps
5. Technical detail: if a measurement is known or seen to lie between a lower limit L and an upper limit U, the measurement can be recorded as equal
6. Technical Note: Arithmetic operations with the decimal expansions of whole numbers is a modification of the polynomial multiplication process which takes into account the carrying and borrowings which avoids coefficients with values  >9 the base -1.
7. A Caution: Isolation in its use should be avoided.

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