**Question Corner and Discussion Area**

I've been stumped!I'm going to give you my answer; if anybody else out there has some other particularly effective examples, please share them with us using the form below.After teaching complex numbers, my students have asked me the obvious question: Where is this math used in real life!

Your assistance would be greatly appreciated.

There are two distinct areas that I would want to address when discussing complex numbers in real life:

- Real-life quantities that are naturally described by complex numbers rather than real numbers;
- Real-life quantities which, though they're described by real numbers, are nevertheless best understood through the mathematics of complex numbers.

The problem is that most people are looking for examples of the first kind, which are fairly rare, whereas examples of the second kind occur all the time.

Here are some examples of the first kind that spring to mind. In
electronics, the state of a circuit element is described by two real
numbers (the voltage *V* across it and the current *I* flowing through
it). A circuit element also may possess a capacitance *C* and an
inductance *L* that (in simplistic terms) describe its tendency to
resist changes in voltage and current respectively.

These are much better described by complex numbers. Rather than the
circuit element's state having to be described by two different real
numbers *V* and *I*, it can be described by a single complex number
*z *= *V *+ *i I*. Similarly, inductance and capacitance can be thought of as the
real and imaginary parts of another single complex number *w *= *C *+ *i L*.
The laws of electricity can be expressed using complex addition and
multiplication.

Another example is electromagnetism. Rather than trying to describe an electromagnetic field by two real quantities (electric field strength and magnetic field strength), it is best described as a single complex number, of which the electric and magnetic components are simply the real and imaginary parts.

What's a little bit lacking in these examples so far is why it is complex numbers (rather than just two-dimensional vectors) that are appropriate; i.e., what physical applications complex multiplication has. I'm not sure of the best way to do this without getting too far into the physics, but you could talk about a beam of light passing through a medium which both reduces the intensity and shifts the phase, and how that is simply multiplication by a single complex number.

Much more important is the second kind of application of complex numbers, and this is much harder to get across. I'm inclined to do this by analogy. Think of measuring two populations: Population A, 236 people, 48 of them children. Population B, 1234 people, 123 of them children. You might say that the fraction of children in population A is 48/236 while the fraction of children in population B is 123/1234, and that 48/236 (approx. 0.2) is much less than 123/1234 (approx. 0.1), so population A is a much younger population on the whole.

Now point out that you have used fractions, non-integer numbers, in a problem where they have no physical relevance. You can't measure populations in fractions; you can't have "half a person", for example. The kind of numbers that have direct relevance to measuring numbers of people are the natural numbers; fractions are just as alien to this context as the complex numbers are alien to most real-world measurements. And yet, despite this, allowing ourselves to move from the natural numbers to the larger set of rational numbers enabled us to deduce something about the real world situation, even though measurements in that particular real world situation only involve natural numbers.

In the same way, being willing to think about what happens in the
larger set of complex numbers allows us to draw conclusions about real
world situations even when actual measurements in that particular real
world situation only involve the real numbers. You can point out that
this happens all the time in engineering applications. If your
students have seen some calculus, you can talk about trying to solve
equations like *a y*" + *b y*' + *c y *= 0 (*) for the unknown
function *y*. State that there's a way to get the solutions provided one
can solve the quadratic equation *a r*^2 + *b r *+ *c *= 0 for the variable
*r*. In the real numbers, there may not be any solutions. However, in
the complex numbers there are, so one can find all complex-valued
solutions to the equation (*), and then finally restrict oneself to
those that are purely real-valued. The starting and ending points of
the argument involve only real numbers, but one can't get from the
start to the end without going through the complex numbers. Since
equations like (*) need to be solved all the time in real-life
applications such as engineering, complex numbers are needed.

Those are some thoughts on how I would try to answer the question "where are complex numbers used in real life".

I am doing an assignment on complex numbers and their applications in the real world. I have found some fields where they are used, in engineering for example, but I really require formulas. Any formulas involving complex numbers that are used in the real world would be appreciated.It's a little difficult to answer you're question without knowing what kind of formulas you're asking for (it's sort of like asking somebody for a sentence; you could write a sentence about just about anything!)

You can have formulas for simple laws; for example, the basic law relating
current to voltage in a DC circuit, *V *= *IR* where *V* = voltage, *I* = current,
and *R* = resistance, generalizes through the use of complex numbers
to an AC signal of frequency passing through a circuit with resitance,
capacitance, and/or inductance, in the following way:

A sinusoidal voltage of frequency can be thought of as the real-valued part of a complex-valued exponential function

Similarly, the corresponding current can be thought of as the
real-valued part of a complex-valued function *I*(*t*). These complex-valued
functions are examples of the *second* kind of application of
complex numbers I described above: they don't have direct physical
relevance
(only their real parts do), but they provide a better context in which
to understand the physically relevant parts.

When such a voltage is passed through a circuit of resistance *R*,
capacitance *C*, and inductance *L*, the circuit impedes the signal.
The amount by which it impedes the signal is called the *impedance*
and this is an example of the *first* kind of application
of complex numbers I described above: a quantity with direct physical
relevance that is described by a complex number. It is given by

and the circuit law becomes

where these are all complex numbers and the multiplication is complex multiplication.V=I Z

So there's one example of a simple formula used in circuit analysis, generalizing the resistance-only case to the case of inductance, resistance, and capacitance in a single-frequency AC circuit.

Other formulas using complex numbers arise in doing calculations even
in cases where everything involved is a real number. For example,
there's an easy direct way to solve a first order linear differential
equation of the form *y*'(*t*) + *a y*(*t*) = *h*(*t*). But in applications,
such as any kind of vibration analysis or wave motion analysis, one
typically has a second order equation to solve.

Consider, for instance,
the equation *y*"(*t*) + *y*(*t*) = 1. For a direct solution, one would like
to "factor out" the differentiation and write the equation as
( (*d*/*dt*) + *r *) ( (*d*/*dt*) + *s *) (*y*(*t*)) = 1. Then you can let *g*(*t*)
denote ( (*d*/*dt*) + *s *) (*y*(*t*)), and we have the first-order equation
*g*'(*t*) + *r g*(*t*) = 1 which can be solved for *g*(*t*) using the method
for first-order equations. Finally, you then use the fact that
*y*'(*t*) + *s y*(*t*) = *g*(*t*) to solve for *y*(*t*) using first-order methods.

However, in order for ( (*d*/*dt*) + *r *) ( (*d*/*dt*) + *s *) (*y*(*t*))
to be the same as *y*"(*t*) + *y*(*t*) (so that the method will work),
it turns out that *r* and *s* have to be roots of the polynomial
, so we need *r*=*i*, *s*=-*i*. Therefore, passing through
complex numbers gives a direct method of solving a differential
equation, even though the equation itself and the final solution
are all real-valued.

I hope the formulas in this and the previous example are of some use to you.

For my algebra class I need to find out how and why a specific job uses the square roots of negative numbers. Thanks for your help!Some examples that come to mind are electrical engineers, electronic circuit designers, and also anyone in a profession where differential equations need to be solved. Besides, of course, mathematicians and physicists!

For more information, you might want to look at the answers given previously in this question.

This part of the site maintained by (No Current Maintainers)

Last updated: April 19, 1999

Original Web Site Creator / Mathematical Content Developer: Philip Spencer

Current Network Coordinator and Contact Person: Joel Chan - mathnet@math.toronto.edu

Go backward to The Origin of Complex Numbers

Go up to Question Corner Index

Go forward to More Complex Number Questions

Switch to text-only version (no graphics)

Access printed version in PostScript format (requires PostScript printer)

Go to University of Toronto Mathematics Network
Home Page