A REALLY Brief Intro to Complex Numbers

Venn Diagram of Number Sets from Keith’s Think Zone

Imaginary numbers and complex numbers interest me a lot.  Aside from the properties of infinity (\infty ), complex numbers \mathbb{C} are, well… complex! They’re very strange and yet they open a whole new world of numbers beyond the Real numbers \mathbb{R} .

Let’s find out what real numbers and complex numbers are.

  • The Real numbers \mathbb{R} are the set made up of all the Rational numbers \mathbb{Q} and Irrational numbers together.
  • To make our transition into the complex numbers, we need to define what \sqrt{-1} is since it is not a real number.  That is to say, there is no real number i so that when we square it, it is equal to -1, (i^2=-1).  So we make up a new number to make it work.  We define the Imaginary unit, i to equal \sqrt{-1}, (i= \sqrt{-1}.
  • An Imaginary number is a specific kind of complex number written in the form bi where b is a real number and i is the imaginary unit.
  • The Complex numbers \mathbb{C}  are the set that involves a combination of real and imaginary numbers.  They are written in the form a+bi, where a and b are real numbers.

 

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