Electrical engineering in particular has many uses with complex numbers. Imaginary numbers are used to measure amplitude and phase of an electrical oscillation such as audio signals, radio waves, or even waves used in transmitting telephone calls. Processing signals is a key component of complex numbers such as those used in large computer systems, telecommunications, radar for airplane navigation, or even in biology to analyze some of the firing events from neurons in the brain. These numbers are also used in measuring the electrical voltage and current that power different types of electrical appliances. One electrical engineer stated that "there are often more imaginary numbers than real numbers in electrical problems".
The problems that electrical engineers face use complex numbers primarily dealing with electric circuits. For example, some circuits have no real current which means that the measure of the current flow would be zero. Therefore, imaginary numbers are a necessity in these types of circuits to produce energy flow through these types of circuits. Regarding this idea, an engineer stated: "think of it like this: a coil is just a wire, so if you run electricity through it there is no real voltage drop because there is no real resistance. A capacitor is just two pieces of metal that do not touch so if you put a voltage through it no real current can flow.". Some of these examples or applications may not be intuitive for those who are not familiar with electric circuits or other subjects within the electrical engineering field. However, the following example will provide a more simple explanation regarding the subject or electricity or voltage. (Jeremiah, 2015) (Douglas, 2015)
Electrical engineers are able to measure voltage output from all different power sources. Sometimes the voltage, which is a combination of current and impedance, alternates in direction and amplitude and thus possess other dimensions such as phase shift and frequency. For this reason, engineers needed to represent multidimensional quantities in these alternating current (AC) circuits, and complex numbers provided this convenience. Thus, quantities of voltage, current, and impedance are represented in the form (a+jb). When using imaginary numbers in electricity or electronics, j is substituted for i as the imaginary number since I is a symbol for current in electronics.
Now, given the equation E=I*Z where E is the voltage, I is the current, and Z is the impedance, electrical engineers are able to find the voltage, in any AC circuit. For example, consider a generator like one below that is powering a house. Suppose this electric circuit had a current of 5+4j amps and an impedance of 3-j ohms, then one can multiply the current and impedance (5+4j)*(3-j) to find the voltage, (19+7j) volts.(Roberts)
The video below gives a more extensive introduction as to specifically how complex numbers help when working with AC circuits.
While electrical engineering has many uses for complex numbers, other fields contain applications as well. Certain engineers frequently use complex numbers in modeling fluid flow in and around objects or pipelines. Complex numbers also help to study stresses on beams and structures; follow the link below to see one simple example of how imaginary numbers are introduced in the construction of bridges. When analyzing these structures for resilience, matrices are used that contain eigenvalues and eigenvectors that often come from complex domain. (Jerry, 2015) The field of Physics also has several applications of complex numbers. In quantum mechanics for example, one can calculate a probability distribution in order to comprehend position in space; this can only be done using complex variables.(Tom, 2015)
The following podcast gives a review of the history of imaginary numbers as well as how they can be used. With regards to applications, Melvyn Bragg and his guest discuss some of the applications that complex numbers can be used for. This audio recording is very helpful to understand why complex numbers are vital in society, specifically for engineers.
For your amusement, follow the link below and read through a cartoon that illustrates some of the mathematical extensions that come from complex numbers.
As mentioned earlier, some of the shared applications and examples may seem complicated for those who are unfamiliar with these engineering topics. However, I find it fascinating that complex numbers are present in so many of the conveniences that we live with. Imagine for a moment living without the luxury of electricity, listening to the radio, or even talking on a cell phone. None of these would be possible without complex numbers. Because of their necessity and usefulness, complex numbers are everywhere. It is true that complex numbers can be difficult to comprehend; nonetheless, I don't think I will ever cross a bridge over a canyon or river again without being happy that there are engineers that do understand how to analyze and use complex numbers.