Introduction to Complex Numbers

Up to this point, we have avoided taking the square root of negative numbers, such as √(−1). We’ve simply dismissed the notion by saying an output does not exist, and rightfully so given the real number system we’ve been brought up with. In this section, we deal with them by introducing a new number system known as imaginary numbers.

Out of curiosity, can you find a real number, when multiplied in itself gets you  “-4”?

… probably not because there are no real numbers that can get you that.

Here’s a historical link as to why we need a new number system.

When math was first invented, early mathematicians tried solving equations like x − 5 = 0, so they asked, what is x? The naturals solved them (easily).

  • Then we asked, “what about x + 5 = 0?” So, negative numbers were invented.
  • Then we asked “what about 2x = 1?” So, rational numbers were invented.
  • Then we asked “what about x² = 2?” So, irrational  numbers were invented.
  • Finally, we asked, “what about x² = −1?”

This is the only question that was left, so mathematicians decided to invent the “imaginary” numbers to solve it. All the other numbers, at some point, didn’t exist and didn’t seem “real”, but now they’re fine.

Now that we have imaginary numbers, we can ‘solve’ every polynomial, so it makes sense that that’s the last place to stop.

A second interpretation

Another way to look at it is to ask yourself, do negative numbers really exist.

  • Of course they do not. You can’t have a negative number of apples.

Yet, the beauty of negative numbers is that when we define them, then all of a sudden we can use them to solve problems we were never ever able to solve before, or we can solve them in a much simpler way. Negative numbers were invented as a tool to help us solve real life problems.

Therefore, the idea that imaginary numbers exist or not has no bearing on whether they are actually useful in solving the problems of every day life, or making them many times more easy to solve.


Definition of Imaginary Numbers

Recall that in the real number system, the equation


Has no solution because there was no real number such that its square is –1.

Now, if you run into this situation, we can use a letter i to denote the answer  to this. Therefore, we call i an imaginary unit that represents:


Definition of Complex Numbers

A complex number is any number, real and/or imaginary, that can be written in the form:


  • where a and b are real numbers, and i = √(–1) is the imaginary unit. A complex number written this way is said to be in rectangular form. You’ll find out later how to convert complex numbers written in rectangular form into polar form just like we did with polar equations.
It’s somewhat unusual to think of an ordinary integer as a “complex number”, but by definition, all numbers belong to this family.

Some examples of complex numbers are shown below. Notice that if a real number exists, it’s written as the first term, followed by the imaginary unit.

  • 4+2i
  • −7+8i
  • 5.92−2.93i
  • 27i
  • 83

Combining Complex Numbers

Part of this unit includes knowing how to combine complex numbers via addition and subtraction. Luckily, they’re treated no different than how you combine like-terms in algebra. The following examples are combined in the video below:

(a) 3i + 5i(b) 2i + (6 – 5i)(c) (2 – 5i) + (–4 + 3i)(d) (–6 + 2i) – (4 – i)

Raising i to a Power

To understand how to calculate i when raised to any power (i.e. iⁿ), you need to know what happens specifically to i2, i3, and i4. Knowing these will help you understand how to reduce i raised to larger powers. The video below thorough explains how.

Summary:   If we represent n as the exponent of iⁿ,

  • When n is 1i = √(–1)
  • When n is 2, i² = –1
  • When n is 3, i³ = –√(–1) or simply –i
  • When n is 4, i4n = 1 (where n > 1, and n is a whole number)

Graphing Complex Numbers

To graph a complex number, you use axes similar to the rectangular coordinate system. The only difference is that the horizontal axis is the real axis, and the vertical axis is the imaginary axis. Such a coordinate system defines what is called the complex plane. Real numbers are graphed as points on the horizontal axis; pure imaginary numbers are graphed as points on the vertical axis.

Such a plot is called an Argand diagram, named for Jean Robert Argand (1768–1822).

To plot a complex number a + bi in the complex plane, simply locate a point with a horizontal coordinate of a and a vertical coordinate of b.

In case you’re asked to graph the conjugate of a complex number in the form a + bi, the conjugate of a complex number is obtained by changing the sign of the imaginary term (↓).