A complex number is an extension of a real number whose properties can be used to solve previous unsolvable problems and that model 2D geometry. Complex numbers appear in many engineering and math equations such as the Fundamental Theorem of Algebra, Euler’s Formula and the Mandelbrot Set.

There are two assumptions, shown below, that form the definition of a complex number. These two assumptions and the properties they exhibit when combined with arithmetic and other math operators form the basis of a complex number.

- Assume there is some number so that .
- Give a home in the complex plane.

The notation for a complex number is shown below, where and are real numbers and . When talking about the set of complex numbers, the symbol is used.

The first assumption is that there exists some number that satisfies the equation . This by itself can be used in combination with the rules of algebra to solve previously unsolvable problems. However, a lot of the usefulness of a complex number comes from its application to two dimensional geometry in the coordinate plane. So first, we will give the notion of a complex number a home in the complex plane.

The second assumption is that a complex number has a “home” in the complex plane and can be drawn as an arrow from the origin to some point in the plane. The vertical axis marks the complex part of the number and the horizontal axis marks the real part of the number. For example, the complex number is drawn as an arrow starting from the origin and going units to the right and units up as shown in the figure below.

Now that we have established the foundation of a complex number, we will look at some of the basic operations that can be perfomed from both a algebraic and geometric perspective.

The properties of addition for two complex numbers follow the rules of algebra. The real parts of the number are added together and the complex parts of the number are added together. Visually, this is represented by drawing the two arrows corresponding to each of the complex numbers “tail-to-tail” as shown below. The resulting number is the arrow drawn from the origin to the tip of the second arrow.

The properties of complex number multiplication demonstrate why complex numbers are able to elegantly express an area of mathematics. Conceptually, multiplying by a complex number corresponds to stretching and rotating a notation of the complex plane.

Multiplying any complex number by results in . Visually, this correspond with stretching and rotating the number so that it lies on top of the complex number in the complex plane. Conceptually, you can imagine stretching and rotating all of the horizontal and vertical axis lines in the first plane.

The second property of complex number mutliplication is multiplying a number by is equivalent to rotating by . For example, the complex number multiplied by is equal to the rotation shown below:

This is where the usefulness of complex numbers starts to appear; the multiplication of two complex numbers can be geometrically related to rotation. Let’s verify the result we saw above using algebra and look at another example.

The last property of multiplication hopefully allows us to visualize what is happening when any two complex numbers are multiplied together. For example, to multiply the complex numbers and together, first we can visualize the stretch and rotation of the coordinate system from the first number , then we can draw the complex number arrow using this new stretched grid to find the product.

We can verify this result using algebra.

The fact that multiplying two complex numbers together results in a stretch and rotation in the complex plane suggestively hints that complex numbers may be useful when applied to trigonometry. Recall, a point on the unit circle which corresponds to the angle rotation along the perimeter of the circle.

Perhaps the most notorious formula in math and physics is Euler’s Formula as shown above.

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The trigonometric identities are a set of equations that are used to transform and manipulate math expressions. Assuming that you have had exposure to trigonometry and these identities, this is one of the areas where the usefulness of complex numbers shines. The complex plane represents an elegant way to think about and transform trigonometric expressions.

The sum of two angles identities can be derived using the properties of the complex plane by defining a complex number in terms of the sine and cosine of its angle.

This example demonstrates who to derive the difference of two angle identities.

This example derives the trig. half angle identities in the complex plane.

The trigonometric double angle identites can be derived using the properties of the complex plane by defining a complex number in terms of the sine and cosine of its angle.

Euler's Formula returns the point on the unit circle in the complex plane when given an angle.

The set of complex numbers contains all possible complex numbers. Each complex number has a real part and an complex part.

The unit circle is a circle of radius one placed at the origin of the coordinate system. This article discusses how the unit circle represents the output of the trigonometric functions for all real numbers.