The Algebra of Complex Numbers

When we wrote:

we were making some assumptions about the new, "complex" system in which real and imaginary numbers co-mingle as if they were old friends. Certain operations — factoring out the  –1  from the  –4 , pulling the  2  outside of the square root, cancelling the  2 s  in both the numerator and the denominator — are standard procedures in the algebra of real numbers. Are these procedures still valid once we've added "imaginary" numbers? We know that at least one algebraic property has changed in the new system: Negative numbers can now have square roots!

In fact, adding the imaginary numbers with their "unusual" negative squares can be accomodated into an integrated system that does not present any further surprises. In the complex number system, real numbers continue about their business as usual. We "imagine" only new roots.

This is good news: When working with complex numbers, we will not have to sweep away all of the algebra and arithmetic we have learned in an effort to please our new "guests".

Here's how the mutual accomodation works.

First, we will have to acknowledge the presence of the imaginary number  i , with its exotic personal habits:

After that, nothing else needs to be said. Real numbers behave as before. "Complex" numbers combine the two world-views only when necessary. Here are some examples:

• Addition

  
  
  

• Subtraction

  
  
  

• Multiplication

  
  
  

• Division

  
  
  
  

  Explore the geometry of complex arithmetic