Complex Numbers and Quadratic Equations
The square root of −1 is represented by the symbol i. It is read as iota.
i = or i2 = −1
Any number of the form a + ib, where a and b are real numbers, is known as a complex number. A complex number is denoted by z.
z = a + ib
For the complex number z = a + ib, a is the real part and b is the imaginary part. The real and imaginary parts of a complex number are denoted by Re z and Im z respectively.
For complex number z = a + ib, Re z = a and Im z = b
A complex number is said to be purely real if its imaginary part is equal to zero, while a complex number is said to be purely imaginary if its real part is equal to zero.
For e.g., 2 is a purely real number and 3i is a purely imaginary number.
Two complex numbers are equal if their corresponding real and imaginary parts are equal.
Complex numbers z1 = a + ib and z2 = c + id are equal if a = c and b = d.
- Let's now try and solve the following puzzle to check whether we have understood this concept.
Verify that each of the following numbers is a complex number.
can be written as, which is of the form a + ib. Thus, is a complex number.
is not of the form a + ib. But it is known that every real number is a complex number.
Thus, is a complex number.
1 − 5i is of the form a + ib. Thus, 1 − 5i is a c…
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Complex Numbers in Real LifeAsked by Domenico Tatone (teacher), Mayfield Secondary School on Friday May 3, 1996:
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".
Followup question by Greg Castle, Dickson College (Australia) on October 21, 1996:
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
V = I Zwhere these are all complex numbers and the multiplication is complex multiplication.
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.
Asked by Melissa Bellin and Stephanie Carlson, students, Maple Grove Senior High on January 14, 1997:
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.
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