|This article/section deals with mathematical concepts appropriate for a student in mid to late high school.|
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The complex numbers are a mathematical construction extending the real numbers (which are, in turn, a construction extending the rational numbers, which extend the integers, which extend the natural numbers.)
|“|| God made the integers [natural numbers], and all the rest is the work of man.
The complex numbers are truly one of the miracles of mathematics. Their rise from the simple question "What are the roots of x2 + 4x + 8 = 0" to their widespread application in many subjects, both theoretical and applied, is an amazing success story.
- 1 Historical Digression
- 2 Construction of the complex numbers
- 3 Analysis
- 4 Are there other extensions one could make? How about using more than 2 components?
- 5 Applications
- 6 Polar notation
- 7 Complex Numbers as Matrices
- 8 In popular culture
The history of mathematics is a history of abstract constructions. The "natural numbers" are just the way people count things. They are properly recognized as only being positive. With just the natural numbers, subtraction isn't always defined—one can't subtract 7 from 5.
The "integers" are the extension to negative numbers, solving the subtraction problem. (Mathematicians haven't always made this distinction so precisely, as in Kronecker's quote above.) But division, as the inverse of multiplication, isn't always defined.
The "rational numbers" are the extension to fractions, thus solving the division problem. With rational numbers, division is always possible, unless the denominator is zero. But things like the square root of 2 aren't defined.
The "real numbers" solve the square root problem, for positive numbers. The construction of the reals from the rationals is actually a complicated theoretical problem to solve rigorously. It is generally solved either by Dedekind cuts or Cauchy sequences. But on a practical scale, people don't usually notice the difference. When one measures something (length, temperature, area, voltage) one is usually satisfied with either kind of number, because they are both good enough. (The theoretical term here is that both sets are "dense".)
But the ancient Greeks studied the theory underlying their mathematics very deeply, and were troubled by such things as the fact that couldn't be represented by their mathematics. This led to the "mathematical problems of antiquity", including "squaring the circle" (now recognized as the problem of the existence of pi), "duplicating the cube" (now recognized as the problem of the existence of the cube root of 2), and "trisecting an angle" (now recognized as the problem of solving cubic equations.) The mathematics of the ancient Greeks were very limited, because they did everything by geometric constructions. Modern mathematics has gone far beyond those limitations, and has no problem, using the real and complex numbers, representing the things that troubled the Greeks. Furthermore, modern mathematicians have a deep understanding of just why those problems couldn't have been solved geometrically.
But there was a remaining problem with the reals—polynomials couldn't always be factored into monomials, or, equivalently, polynomials equations couldn't always be solved. Over the period from about 300BC to 1600AD, the necessary theory (that is, "algebra") was worked out. The problem is now recognized as the failure of the real numbers to be algebraically closed. The construction of the complex numbers solves that. The complex numbers are an algrebraically closed field, leading to the fundamental theorem of algebra, which states that all polynomials (even polynomials whose coefficents are themselves complex numbers—that's what "closed" means) have all of their roots if the roots are allowed to be complex numbers. Equivalently, every Nth degree polynomial can be factored into N 1st degree polynomials.
Construction of the complex numbers
A complex number is constructed as a pair of real numbers. They are called the "real part" and the "imaginary part". The term "imaginary" is unfortunate—it suggests some defectiveness in the existence of this part. The mathematicians who had been frustrated by this problem for so long may be forgiven for using this word.
Now, when one measures some ordinary quantity, like the length of a table, one gets a plain real number, and it is understandable that one might treat the "imaginary part" of that quantity as not existing. But, in the deeper study of mathematics, complex numbers play an enormously important role, including many real-world applications, as will be noted below.
Complex numbers have all the normal arithmetic operations defined on them—addition, subtraction, multiplication, and division (but not by zero). See complex arithmetic.
Being a pair of real numbers, complex numbers may be plotted on a 2-dimensional plane, with the real part on the x-axis and the imaginary part on the y-axis. Such a plane is often called an "Argand diagram" or just "complex plane".
When they are plotted this way, the first miracle comes into view: Addition of complex numbers has a geometrical interpretation on the Argand diagram—it is just "vector addition" using the usual "parallelogram rule". Multiplication is even more remarkable. If a complex number's distance from the origin of the diagram is called the "modulus" or "magnitude" (it's the same as the "absolute value" for real numbers) and its angle from the x-axis is the "phase" (sometimes called the "argument") the product of two complex numbers has the product of their moduli and the sum of their phases! This is related to the addition formulae in trigonometry, and leads to DeMoivre's theorem.
Just from this much, we can already see where all the missing roots went. There are 5 5th roots of 1. They are all on the circle of radius 1 on the Argand diagram, evenly spaced around that circle. They are on the x-axis, at 72 degrees from the x-axis, 144 degrees, 216 degrees, and 288 degrees. Only the one exactly on the x-axis has an imaginary part of zero, so it is the only one that was visible to the ancients. But they are all present in the complex plane. So the polynomial x5 = 1 has 5 roots. A little more algebra shows that, for example, x2 + 4x + 8 = 0 has two roots (the way quadratics are supposed to): at (-2, 2) and (-2, -2).
The algebraic properties of the complex numbers mean that one can also represent them as ordinary algebraic expressions, involving a quantity "i", which is the unit on the y-axis. The complex number with coordinates (3,5) on the Argand diagram can be represented as "3+5i". When done this way, all the arithmetic operations described in Complex arithmetic can be thought of as ordinary operations, with the understanding that
- i2 = -1
The quantity "i" is called the "imaginary unit", or "the square root of -1". (But note that, like all nonzero numbers, -1 has two roots: i and -i. To consider i to have been defined as would make it ambiguous.)
Complex numbers drive an enormously rich branch of mathematics known as complex analysis. In addition to the obvious role in algebra, complex analysis involves such subfields as calculus, analytic function theory, functional analysis, integration and "residue analysis", convergence theory, etc. etc.
Analytic function theory is one of the miracles of complex numbers. It even has applications to real-valued functions. It tells us why, for example, the power series for the arctangent function converges out to x=1, but no farther. See User:SamHB/Complex analytic function.
Are there other extensions one could make? How about using more than 2 components?
There have been a few other similar extensions, somewhat along the lines of complex numbers, that have been made, such as quaternions and octonions. Quaternions have some applicability to the representation of rotations in 3-dimensional space. But, on the whole, none of these extensions come close to the simple elegance and wide-ranging significance of complex numbers.
Far beyond being an abstract mathematical concept, complex numbers have a large number of far-reaching applications in science, technology, and society.
The image on the computer screen that you are looking at was probably processed by computer algorithms developed with the complex Discrete Fourier Transform, commonly called the "Fast Fourier Transform", or "FFT", and perhaps the most important signal-processing computer algorithm ever invented.
More generally, the Fourier transform, in its many guises, lies at the heart of many branches of mathematics, physics, and engineering. For example, the well-known trade-off between aperture and angular resolution in optics, radio astronomy, and so on, are related to a Fourier transform.
Complex numbers are the cornerstone of the study of electrical engineering. The science underlying radio, telephone, and television communication has developed, over the last 140 years or so, around complex numbers. In electrical engineering (as in related areas of mechanical engineering, for example) a variant of the Fourier transform known as the Laplace Transform is often used. This is sort of a Fourier transform placed sideways in the complex plane. It allows for the convenient analysis of "complex frequencies", which includes both damped (exponential) and oscillatory phenomena. Various types of signal filters, used, for example, in audio processing (Butterworth, Chebyshev, Bessel...) are characterized by the location of the "poles" (the complex roots of the denominator polynomial) of their "system function" in the plane resulting from a Laplace transform. When the poles and zeros are plotted this way, one can easily visualize such things as passband sharpness, passband ripple, and phase distortion. For example, the Butterworth filter has no passband ripple. This follows from the location of its poles on an ellipse in the complex plane.
Complex numbers are widely used in number theory, that is, the study of ordinary integers. For example, from 1896 to 1948, the only known proof of the prime number theorem used complex numbers.
Many theorems of number theory involve the complex Riemann zeta function. One of the great unsolved mysteries of mathematics is the Riemann hypothesis, which says that all of the nontrivial zeros of the Riemann zeta function lie on the line in the complex plane with real part equal to 1/2. It has remained unsolved since 1859.
"Modular forms" are a mathematical construction involving complex analysis. They have wide applications to number theory and algebraic topology. The celebrated proof of Fermat's last theorem, by Andrew Wiles and Richard Taylor in 1993-1994, uses modular forms.
As indicated above, the modulus (often denoted ) and phase (often denoted ) play a crucial role in the characterization of a complex number. If a line is drawn on the Argand diagram from the origin to a given complex number , the length of that line is and the angle it makes to the x-axis is . We have:
- (with a small amount of ambiguity)
The representation in terms of and is called polar notation.
This is related to Euler's remarkable formula:
Complex Numbers as Matrices
The field F on complex numbers is isomorphic to the field F' of 2x2 matrices of the form
- [a -b]
- [b a],
with mapping as a function f to the above matrix. We can see that F and F' are isomorphic because: The function f is clearly 1-to-1 and onto, , and .
In popular culture
In Yevgeny Zamyatin's satirical novel We, the narrator's psychological distress at contemplating the concept of complex numbers becomes a metaphor for the limitations of totalitarian systems of thought.