Complex Numbers

1. Algebraic Structure of Complex Numbers
2. Division of Complex Numbers
3. Useful Identities Among Complex Numbers
4. Useful Inequalities Among Complex Numbers
5. Trigonometric Form of Complex Numbers
6. Real and Complex Products of Complex Numbers
7. Complex Numbers and Geometry
8. Remarks on the History of Complex Numbers
   • First Geometric Interpretation of Negative and Complex Numbers
9. Complex Numbers: A Dynamic Tool
10. Cartesian Coordinate System
11. Fundamental Theorem of Algebra
12. Complex Number To a Complex Power May Be Real
13. One Can't Compare Two Complex Numbers
14. Problems
    • Product of Diagonals in Regular N-gon

Trigonometric Form of Complex Numbers

Except for 0, any complex number can be represented in the trigonometric form or in polar coordinates:

where aArg(z). r, the modulus, or the absolute value of z, is easy to find:

.

But how do we find a? As we know, a is not unique, but is found modulo 2p. The main value, arg(z) belongs to the interval [0, 2p). Assume, z = x + yi. Then a is the angle formed with the x-axis by the radius vector of the point (x, y) or the point where the latter intersects the unit circle, viz.,

Apparently, a can be found from

Similar consideration must be brought to bear in the general case (1), where x ≠ 0. Since tan() is periodic with period p and the base interval (-p/2, p/2), we'll have to account for the signs of both x and y.

In the quadrants I and III, tan(t) is positive. In the quadrants II and IV, tan(t) < 0. In quadrant I, arg(z) = arctan(y/x). But, for quadrant III, the positive arctan(y/x) must be upgraded with the addition of p. Similarly, for z in quadrant II, arctan(y/x) is negative. To bring it to the second quadrant we have to add p. For z in the fourth quadrant, arctan(y/x) is again negative, but its location is correct. To keep it there we now have to add 2p, so that the value falls into the interval [0, 2p). To sum up, denote a₀ = tan(y/x). Then

or in a shorter form,

The trigonometric form is intimately related to the operation of multiplication. Let z = r(cosa + i·sina) and w = s(cosb + i·sinb). Then

The latter follows from the definition of the multiplication of complex numbers and the addition formulas for sine and cosine.

(2) tells us that the modulus of the product of two numbers is the product of their moduli, something we already knew. But it also tells us something we did not: the argument of the product is the sum of the arguments of the multiplicands. (The latter of course needs to be taken modulo 2p.)

Colloquially, to multiply two complex numbers one has to multiply their absolute values and add their arguments.

It is easy to see that arg(z') = -arg(z) (mod 2p). From which we conclude that, for w ≠ 0,

By induction, (3+) and (3-) imply de Moivre's identity: for any integer n ≠ 0,

For n = 0 and z ≠ 0, we define z⁰ = 1 and then de Moivre's formula holds for all integer n, with no exception. In general, (2) leads to

(As a matter of fact, see [Kline, p. 408], de Moivre has stated the eponymous formula only implicitly. Its standard form is due to Euler and was generalized by him to any real n.)

Next, let's think of what it takes to evaluate roots of complex numbers. It is clear form (4), that, for a given w, choosing |z| = |w|^1/n and arg(z) = arg(w)/n will solve the problem, w = zⁿ:

The important thing to note is that, save for the trivial case n = 1, we can get additional roots. For example, we might have chosen arg(z) = (arg(w) + 2p)/n with exactly the same effect. Indeed, since both cosine and sine have period 2p, we would have

Even more generally, for an integer k and arg(z) = (arg(w) + 2kp)/n, we have

so that k and m define exactly the same complex number. Well, we see that in (5) not all numbers are different. But how many are? The answer is n! Which means that, for any complex numbers w, the equation zⁿ = w has n different roots! This is all because the argument of a complex number is only defined modulo 2p. For example, if n = 4, then in the sequence

the fifth term (i.e. the one with the coefficient 8) equals modulo 2p the first term and, starting with that term, the sequence repeats itself having modulo 2p only four distinct terms.

This is not so surprising. After all, we were solving zⁿ = w which is a polynomial equation of order n. According to the Fundamental Theorem of Algebra, any polynomial equation with real (in fact, complex) coefficients, has exactly n roots.

We may also draw on our real numbers experience. For any positive real number a, there are two square roots, a and -a. As complex numbers, the first (i.e. positive) one has argument 0, the second (negative) has argument p. I.e., for a positive real number a,

arg(a) = 0, arg(-a) = p.

Example

Let's find all the third roots of w = 8 - 8i. In the trigonometric form, w = 8(cos(3p/4) + i·sin(3p/4)). Thus we get three distinct roots:

Recollecting that p = 180^o, we get the three roots in a little different form:

all evenly spaced around the origin.

References

1. T. Andreescu, D. Andrica, Complex Numbers From A to ... Z, Birkhäuser, 2006
2. C. W. Dodge, Euclidean Geometry and Transformations, Dover, 2004 (reprint of 1972 edition)
3. Liang-shin Hahn, Complex Numbers & Geometry, MAA, 1994
4. M. Kline, Mathematical Thought From Ancient to Modern Times, v. 2, Oxford University Press, 1972
5. E. Landau, Foundations of Analisys, Chelsea Publ, 3^rd edition, 1966

Copyright © 1996-2008 Alexander Bogomolny