The Meaning of Complex Numbers

You may still be wondeing: Why study complex numbers and complex arithmetic?

We've mentioned how complex numbers provide us with a new perspective on the real numbers. From an imaginary vantage point, the real line opens into the complex plane, and our well-beaten path along the reals is seen for what it is: A one-dimensional slice of a much bigger picture.

The benefits of this bigger picture, however, may be unclear at the moment. Indeed, many of the advantages of working with complex numbers can only be appreciated after one has some familiarity with the elements of calculus. Parts of calculus appear in greatly simplified form when viewed from a complex perspective. In precalculus, however, we find ourselves before our journey into calculus, and talk of complex numbers may seem premature.

Not so. Complex numbers allow us to gain a deeper understanding of the subject at the center of both precalculus and calculus: families of functions.

Here are just a few examples:

• Polynomial Functions

  Recall the Fundamental Theorem of Algebra: It tells us that a polynomial of degree  n  has at most  n  real roots. If we are willing to consider complex roots, it tells us that there are exactly  n  roots. By observing the behavior of the entire set of roots, both real and complex, we can see more clearly when roots will be real and where they will be located.

    Explore complex roots of polynomials
  
  
  

• Exponential & Trigonometric Functions

  In 1748, Euler published a remarkable formula that connected the family of trigonometric functions to the family of exponential functions through a shared heritage in the complex plane:

  eiq = cos(q) + i sin(q)

  Working with exponentials is often much simpler than working with trigonometric functions. Explorations of Euler's formula and a few of its many applications follow.

    Explore Euler's formula

    Explore Euler's formula & complex arithmetic

    Explore Euler's formula & trig identities

    Explore Euler's formula & periodic phenomena
  
  
  

• Inverses & Multi-Valued Functions

  By definition, functions must always pass the vertical line test for single-valuedness — each input must correspond to a unique output value. They needn't, however, always pass the horizontal line test — certain outputs may correspond to multiple inputs. This is what prevents some functions from having a global inverse function.

  In 1851, is his doctoral dissertation, Riemann proposed a clever way to describe the multiple values of an inverse as "branches" of a single-valued complex function. By doing so, he forever changed the way we think of functions and their inverses.

  The following exploration of Riemann's idea is slightly challenging, but it provides a good review of many precalculus topics.

    Explore the branches of the complex square root

There is much more to be said about the complex numbers. If you're curious, and would like to find out more right away, you might look here.