Intro to Imaginary Numbers

In our study of mathematics, you may have noticed that some quadratic equations do not have any real number solutions.

For example, try as you may, you will never be able to find a real number solution to the equation x2 = -1. This is because it is impossible to square a real number and get a negative number!

However, a solution to the equation x2 = -1 does exist in a new number system called the complex number system.

The imaginary unit

The backbone of this new number system is the imaginary unit, or the number i. The following is true of the number i:

  • i = √ -1
  • i2 = -1

The second property shows us that the number i is indeed a solution to the equation x2 = -1. The previously unsolvable equation is now solvable with the addition of the imaginary unit!

Why do we need them?

The answer is simple. The imaginary unit i allows us to find solutions to many equations that do not have real number solutions. This may seem weird, but it is actually very common for equations to be unsolvable in one number system but solvable in another, more general number system.

Here are some examples with which you might be more familiar:

  • With only the counting numbers, we can’t solve x + 8 = 1; we need the integers for this!
  • With only the integers, we can’t solve 3x – 1 = 0; we need the rational numbers for this!
  • With only the rational numbers, we can’t solve x2 = 2. Enter the irrational numbers and the real number system!

And so, with only the real numbers, we can’t solve x2 = -1. We need the imaginary numbers for this!

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