applet-magic.com
Thayer Watkins
Silicon Valley
& Tornado Alley
USA

The Nature of Complex Numbers

θ

Complex Numbers

Probably most students of mathematics are somewhat suspicious of complex numbers when first introduced to them. They seem to be predicated upon the existence of a logical contradiction; i.e., the square root of minus one. Any positive number times itself is a positive number. Any negative number times itself is likewise a positive number. Therefore a number times itself that is negative seems impossible. And so it is impossible for any real number. To hypothesize the existence of a square root of minus one and call it imaginary does not asuage the skepticism of the student. But there is a simpler, more concrete way to define the complex numbers.

Complex numbers are ordered pairs of real numbers for which multiplication is defined in a special way. Let (a,b) and (c,d) to ordered pairs of real numbers. The product of (a,b) with (c,d) is defined as:


(a.b)*(c,d) = (ac-bd,ad+bc).
 

For any two pairs such that the second components are zero:


(a,0)*(c,0) = (ab,0)
 

Thus such ordered pairs with second component zero behave exactly like real numbers.

Consider now the product of the complex number (0,1) with itself. The result is:


(0,1)*(0,1) = (0-1,0+0) = (-1,0).
 

So (0,1) is a number such that its square is equal to (-1,0) which is equivalent to -1. Thus the square root of -1 is not fictitious or imaginary in the ordinary sense but is simply (0,1).

The negative numbers can be considered to have arisen as solutions of algebraic equations of the form x+a=0. The rational numbers can be considered to have arisen as solutions of algebraic equations of the form ax+b=0 for a and b integral. The irrational numbers can be considered to have arisen as solutions of algebraic equations of the form x²-2=0. Likewise complex numbers can be considered to have arisen as solutions of algebraic equations of the form x²+1=0. The question occurs as to whether new types of numbers arise in the solution of equations of the form ax²+bx+c=0 if the coefficients a, b and c are complex. The answer is no. Complex numbers are as far as we have to go. There is a theorem, called the Fundamental Theorem of Algebra, that establishes that any polynomial equation with complex coefficients will have all its roots as complex numbers.

The addition of complex numbers is very simple; the sum of two complex numbers is the complex numbers whose components are the sums of the components of the two numbers. That is to say,


(a,b) + (c,d) = ((a+b),(c+d))
 

The representation of complex numbers given above is the rectangular representation. There is another equivalently representation of complex numbers that is more convenient for some purposes. It is the polar representation of complex numbers. A complex number as a point in a two dimensional plane can be completely characterized by a radius (the distance of the point from the origin) and an angle. The radius of a complex number is also called its modulus. The relationship between the rectangular representation (x,y) or x+iy and the polar representation (r,θ) is:


r = (x2+y2)1/2
θ = tan-1(y/x)
 

One of the virtues of the polar representation is that multiplication of two complex numbers is simple; i.e.,


(r,θ)*(s,φ) = (rs, θ+φ)
 

Thus finding the power of a complex number in polar form is especially easy; i.e.,


(r,θ)n = (rn ,nθ)
 

For a web calculator of a complex number to a complex power click here.


HOME PAGE OF applet-magic.com
HOME PAGE OF Thayer Watkins