Help for Understanding and Using Complex Numbers

Complex Numbers in Several Ways

The best way to master this site area is to begin with this Complex No. Intro - a page outside of this site area due to site history. Look for the easy and helpful consequences of teaching complex numbers early in high school - as soon as there an explicit or implicit discussion of polar coordinates for points in the plane.

An earlier development with connections to vectors and trig. Items B2 to B10 are still recommended.

New (August 3, 2001, Revised January 2006): Site pages on Complex Numbers (this introduction not in this site area) and on further page in this site area offer a short way to reach and explain trigonometry, the Pythagorean theorem, trig formulas for dot- and cross-products, the cosine law and a converse to the Pythagorean Theorem. See the easy consequence in left margin.

Remark: The development of complex numbers here offers a leaner route for senior high school mathematics and a justification of the complex number based perspective of trig often given to engineering and physics students without explanation. There-in lies a paradigm shift for mathematics education.

A1 Add Points	B5 Rt Triangle Similarity	D1 Arrow Navigation
A2 Polar Coords	B6 Trig Functions	D2 Sum of Motions
A3 Polar Multiply A4 Complex Numbers	B7 Dot & Cross Products	D3 Addition Method I
A5 Real Numbers	B8 Cosine Law	D4 Addition Method II
A6 Law of Signs	B9 Exponential & cis fns	D5 Addition Method III
B1 Cartesian Multiply	B10 Easy Trig Identities	D6 Coordinate Addition
B2 Complex Conjugates	B11 Set Viewpoint	D7 1st Distributive Law
B3 Pythagoras	C1 Unsigned Coords	D8 2nd Distributive Law
B4 Distance	C2 Signed Coords	D9 3rd Distributive Law

This Complex Number (java) applet , online earlier, illustrates the addition and multiplication of points, arrows and complex numbers in the plane.

See B2 to B10 for the easy consequences of the key arithmetic properties of complex numbers, normally algebraically described include the Pythagorean theorem, trig formulas for dot- and cross-products, the cosine law and a converse to the Pythagorean Theorem.

The sequence of lessons A1 to A6, B1 to B11, C1, C2 and D1 to D9 represents an older development of mateiral which can be replaced by Analytic Geometry lessons. In the older sequence the two webpages Complex Numbers & Trig for Today's Students and Distributive Law for Complex Numbers, could be read first and followed by easy consequences B2 to B10.

The two webpages Complex Numbers & Trig for Today's Students and Distributive Law for Complex Numbers, one or both should be read first, and followed by the easy consequences B2 to B10. The idea of introducing complex numbers geometrically stems from a 1976 lectures of the late Richard Feynman, one of three public lectures given in fall 1976. In one or two, he described physic as the addition and multiplication of arrows in the plane, with addition given using the parallelogram law and multiplication being given with polar coordinate rule, add the angles, multiply the lengths. All was presented without mentioning complex numbers. ]

Site Ways to understand and explain Complex Numbers

Several listed in reverse chronological order follow. The sixth or fifth way is recommended on first reading.

The seventh way (September 2005) in Complex No. Intro was first indicated in the Number Theory site areas. It provides a derivation of the field properties of real numbers based on counting principles and the properties of Fractions. A key notion in the derivation is the concept that a change of units should not affect the sum of two vectors - whence multiplication distributes over addition of real (or complex) numbers.
The sixth way appears in the site Analytic Geometry (Summer 2005). It combines the field properties of real numbers with some Euclidean Geometry (hand-waving geometry before coordinates) to obtain a coordinate geometry approach to complex numbers.
The fifth and apparently easiest way, posted online August 3, 2001, is described in the two webpages Complex Numbers & Trig for Todays Students and Distributive Law for Complex Numbers with some help from pages B2 to B10 in this site area.
A fourth way is given by the (offline) Euclidean Geometry webvideos. See the Mathematics How-TOs and Leading Questions at this site for a written description or variation of this fourth way.
The third way (given below) and posted online in 2000 is for enriched studies and after the second, derives key properties of complex numbers from geometric assumptions instead of assuming them directly See lessons A1 to A6 and B1 to B12 on the addition and multiplication of point or vectors in the plane, and on the consequences of the key properties. The key properties are justified in lessons D1 to D9 below. These lessons could be read before B1 to B12 if you wish. See too the first twelve web-video lessons and then the further web video lessons on complex numbers and trigonometry.
The second way (also given below) and posted online in 1999 assumes key properties of complex numbers instead of deriving them from geometric assumptions. See lesson A1 to A6 and B1 to B12 below on the addition and multiplication of point or vectors in the plane, and on the consequences of the key properties. See too the web-video lessons.
A first way appears in Volume 3 (1996) in the Calculus and Beyond Why Slopes and More Math site area.. It is for students who have studied trigonometry and not yet complex numbers. It includes the rotate-a-triangle in the unit circle proof of angles sum formula for cos(A+B) and some other thoughts on how to prove the distributive law.

All these ways stem directly or indirectly from the work of Wessel, Gauss and (?) others in providing a geometric representation of complex numbers and the square root of negative numbers.

Steps and Substeps for 2nd & 3rd ways

A. Add and Multiply

How to Add Points in the Plane with rectangular coordinates.

Polar Coordinates - how to locate points in the plane, and switch between polar and rectangular coordinates for them. (Optional if you know about polar coordinates)

How to Multiply I. The polar coordinate way to compute products of points or vectors (arrows) in the plane, adds their angles and multiplies distances to get the polar coordinates of the product. Later you will see how to compute products using rectangular coordinates. The equality of two different ways to compute products in the plane provides a key to understanding and explaining complex numbers, trigonometric functions, the Pythagorean theorem, the cosine law, the trigonometric interpretations of dot and cross products of points (or vectors) in the plane, the Converse to Pythagoras theorem, and complex number based short cuts for trigonometry favored in college if not always in high school.

What are Complex Numbers? Answer: Points in the plane with the rectangular coordinate method of addition and the polar coordinate way for multiplication provide a geometric viewpoint and understanding of complex numbers with their real and imaginary parts of complex numbers are identified. The word imaginary has a strange history in mathematics. From the twelfth to fifteenth Centuries, negative numbers were considered imaginary or figments of the imagination, useful in calculations, but not to be presented in public. But in the fifteenth Century, familiarity with them aloud them to use in public. It was no longer necessary amongst the learned to rewrite calculation involving negative numbers so only positive or unsigned numbers appeared. Then the question of how to compute square roots of negative appeared, possibly due to algebraic manipulations of formulas.. These square roots could be used algebraically, but there was no geometric representation of them. So the square root of -1 was considered to be imaginary. The name has remain. but this lesson, the previous and the next, show how you may see complex numbers and real numbers as points in the plane which may be added and multiplied.

Real Numbers as Complex Numbers and the square roots of -1. Here real numbers are identified with points on the horizontal axis. (The discussion of square roots of other points in the plane may be too cryptic - skip it on first reading.)

The Law of Signs for real numbers follows immediately from the polar coordinate way of multiplying points in the plane. The previous five lessons introducing complex numbers and the real numbers as points in the plane could be understood if you did not know how to compute products with negative numbers in them. Then this lessons could be an introduction to the law of signs for products of real numbers. I am wondering if there is a route for introducing signed numbers as coordinates along a line or in the plane, which uses vector viewpoint of addition to define the addition and subtraction of signed numbers, but does not say how to multiply them. And this route, the properties of addition and multiplication of real and complex numbers would follow from geometric principles: segment arithmetic. Here could be geometric alternative to the set theoretic viewpoint of arithmetic with real and complex numbers. The challenge here is to provide a path which many can follow easily. For more details, see the discussion of segment arithmetic in chapter III, section 15, of the work Foundations of Geometry, by David Hilbert. This an exercise for advance students.

Key Properties. This lesson observes how the algebraic properties of complex numbers follow from the corresponding properties of real numbers due to polar and rectangular definitions of sums and products, and from the assumption of the distributive law. The latter provides the basis for this quick description of complex numbers. Including the lessons below on how the distributive law follows from geometric assumptions would lengthen this description or introduction, and make it harder to follow. Those geometric assumptions in turn may be justified starting with a more detailed discussion of geometry. So the quickest way to understand and explain complex numbers and enjoy the consequences is to assume the distributive law. Chains of reason leading to it may be studied later.

B. Consequences of the Key Properties

The Rectangular Way to Multiply. The distributive law in its left and right forms, assisted by associative laws for addition and multiplication of points in the plane (see the previous lesson) implies a second way to compute products using the rectangular coordinates, or real and imaginary parts, of the factors. This is first consequence of the key properties. Many further consequences follow from it.

Complex Conjugates and Reciprocals f complex numbers are illustrated here. The Key Properties lessons covered the existence of reciprocals or multiplicative inverses of complex numbers without the diagrams.

Pythagorean Theorem. Here is the complex number based proof, another consequence of assuming the distributive law and obtaining two different ways to compute the product of a complex number with its complex conjugate. Multiplication of an point by its reflection across the horizontal axis can be done in two different ways with the aid of polar coordinates and with the aid of real and imaginary. The equality of the results implies a new proof of the Pythagorean Identity in which there is no mention of areas.

Pythagorean Distance Formula. The distance between two points in the plane may be computed with the aid of their rectangular coordinates and a formula due to the Pythagorean Theorem.

Right Triangle Similarity. The geometric unit circle definition of trigonometric functions can be explained in the first instance without a discussion of the consequences of similarity for right triangles. The right triangle based, geometric computation of these functions and the independence from the choice of unit of length in the initial definition, depends on right triangle similarity principles or assumptions. (A mix of assumptions about geometric and about arithmetic is required to introduce mathematics. The further set theoretic codification of arithmetic and geometry may show a small minority of pupils and teachers how to avoid dependence on geometric sketches for definitions of objects and proofs of their properties.)

Trigonometric Functions. This lesson show how the unit circle definition could be put before the right triangle computation of trig functions. With the foregoing development of complex numbers and their properties, the unit circle may be used to define trig functions for angles q (positive, negative or zero) and to derive many of their algebraic properties from the equality of different ways to multiply complex numbers, or the complex-valued expression exp(i q) = cos (q) + i sin(q). The latter algebraic method is often shown to science students without explanation. Then similarity properties of triangles implies the classical right triangle methods involving the ratios of adjacent side, opposite side and/or hypotenuse.

Dot and Cross Products. The complex number viewpoint of their definitions leads to a quick trigonometric interpretation of these products in the plane. The challenge for advance for students is to show how the trigonometric interpretation given here can be extended to the points or position vectors in three dimensions. Hint: Show the dot product is invariant under rotation matrices. Extend the foregoing argument to a pair of non-collinear vectors A & B in three dimensions by starting with an orthonormal basis for plane spanned by A and B, and then showing that a rotation preserves inner products and cross products. The rotation may be constructed from the orthonormal basis plus a unit vector normal to them, whose existence may have to be assumed in synthetic geometric, that is, if coordinates are not used.

The Cosine Law. The cosine law, and its consequence, a converse to the Pythagorean theorem, follow immediately from the trigonometric interpretation of dot products in the plane.

Trig Short Cuts I. The cis or exponential functions. Statistics, Engineering and Science courses in college which employ trigonometric functions may introduce the complex-valued cis or exponential function to simplify and accelerate computations: turn them into simpler algebraic problems.

Trig Shortcuts II. More Identities Here are few identities that follow from use of the exponential function.

Complex No.Axioms Here is a summary of the set theoretic viewpoint or codifiction of complex numbers.

C. Optional Reading (for Later)

D. Proof of the Distributive Law

The above introduction to complex number and the properties which link it to trigonometry depend on the assumption of the distributive law for multiplication over addition. The next lesson offer a proof.

Vectors in Navigation. On a map, a sequence of straight line motions may be used to precisely or approximately represent the path of an object (ship, plane or person) over land or sea. These motions and their directions may be represented by arrows with tail at the starting point of a motion and head at the other end or last point in that motion. Here is Motivation and a context for the use of arrows, or vectors, in navigation.

Resultant of A Sequence of Movements. A straight line arrow from one point to another may summarize the movement of an object. The object itself may follow a curved path between the tail or initial point of the arrow and the head or terminal point. Similarly when a sequence of straight line motions is followed, one after another, the arrow joining the initial point of the first motion to the terminal point of the last motion summarizes or gives the sum or resultant of the intermediate motions. Here is a context and motivation for the head-to-tail addition of a sequence of arrows or vectors in navigation. This addition is associative.

First Addition Method, head-to-tail addition. This method for vector or arrow addition is suggested by the navigational use of arrows or vectors to represent or summarize a sequence of movements. This method is associative.

Second Addition Method with parallelograms. When two arrows or vectors (representing motions if you wish) have a tail at the same place, they may be added together by moving the tail of one to the head of the other with the aid of a parallelogram, and then using the head to tail method for addition. This gives the parallelogram method for adding a pair of arrows or vector addition. The resultant arrow does not depend on which arrow, the first or second, is moved. This addition method is commutative. More generally, parallelograms can be used to displace or move arrows from one location to another without changing their lengths or directions. But that is another story, or chain of reason, not illustrated here. Assumption: Two sides of a parallelogram uniquely determine the set of points its sides occupy..

Third Addition Method . In a plane, the intersection of two perpendicular lines, one horizontal and the other vertical, defines a reference point or origin for the plane. Each arrow in the plane is equal to the sum or resultant of a horizontal and vertical arrows, its so-called horizontal and vertical components. This representation or decomposition of an arrow as the sum of horizontal and vertical components leads to a third method for arrow addition given by the addition of components. The horizontal components of an arrow sum is given the arrow sum of the horizontal components. Likewise, the vertical components of an arrow sum is given the arrow sum of the vertical components. Here is a technical observation with little motivation except for consequences that will follow. Optional Exercise: Think about components of vectors with respect to intersecting lines, not mutually perpendicular.)

Fourth Addition Method. In a plane, the intersection of two perpendicular lines, one horizontal and the other vertical, defines a reference point or origin for the plane. The head of each arrow in the plane has coordinates. Ordered pairs of vertical and horizontal coordinates, ordinates and abscissa, can be employed to add arrows together or find the position of the head of their sum, when the tails of the vectors in the sum are both located at the origin. This gives a fourth method for arrow addition given by the addition of coordinates. This method is very similar to the third method for addition with components. Here is another technical observation with little motivation except for consequences that will follow.

Scalar Multiplication and a First Distributive Law for it. The repeated addition of an arrow to itself, n-1 additions, leads to the notion of a scalar multiple: n times the arrow. Drawing parallelograms, tessellating the plane with them, implies or suggests that multiplication of vectors by whole numbers and then fractions distributes over the sum of two different vectors. Here is motivation if not a Euclidean proof, for the distributive of scalar multiplication over vector addition.

Rotation distributes over addition. Here is a second distributive law. It follows from the assumption that a parallelogram is a rigid body.

Polar Multiplication distributes over addition. Multiplication by a point with polar coordinates (r, q) in the plane consists of two operations (i) multiplication by a length r, which corresponds to a scalar multiplication where the scalar r > 0; and (ii) addition of an angle q which corresponds with a rotation through that angle. The two operation (i) and (ii) of length multiplication and angle addition may be done simultaneously, or one after another in either order. This distributive law follows from the previous two distributive laws, one for scalar multiplication by a positive factor and the second for rotation