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A Real Exploration of Euler's Imaginary I: Isomorphism And

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A Real Exploration of Euler's Imaginary I: Isomorphism And Session ETD 525 A Real Exploration of Euler’s Imaginary i: Isomorphism and Applications to AC Circuit Theory Andrew Grossfield, Ph.D., P.E. Vaughn College of Technology Abstract In elementary schools the students are taught that negative numbers do not have square roots. The appearance of a square root of a negative number in the course of a computation indicates that either the problem has no solution or an error has occurred. Subsequently students are told that negative numbers have “imaginary” square roots which can be constructed using the symbol i which represents the square root of –1. However, this reasoning appears logically inconsistent. There is nothing imaginary about the symbol i or its use. This paper treats the following interesting topics in the theory of functions of a complex variable: 1) sensible introductions to Euler’s i that conform to the way engineers and technicians use the symbol in analyzing alternating current circuits and mechanically vibrating systems; 2) the derivation of the algebraic and topological features of the complex plane and a comparison of these features to the properties of “real” numbers; 3) the description of the isomorphism between phasors and combinations of same-frequency sinusoidal oscillations that underlies the theory of alternating current analysis promoted and successfully used by C. P. Steinmetz for the distribution of electrical power throughout the United States; 4) the derivation of the linear 2-dimensional rotational mappings represented by a system of 2×2 matrices with real number entries. These mappings can represent complex numbers and serve as an alternative definition of the meaning of the symbol i. This paper should provide a reasonable introduction to theories of alternating currents and vibrations and encourage further study of the theory of complex variables. Background The history of the development of the “real” number system began with the positive whole numbers to which zero and the positive rational fractions were added. These numbers were represented by points to the right of the origin on the “real” number line. The points to the left of the origin on the line were added with the inclusion of the minus ( – ) symbol. Multiplying any number on the horizontal positive axis by –1 reversed its direction. The new system together with the irrational numbers included all the points from – ∞ to + ∞ on the 1-dimensional “real” number line. Proceedings of the 2019 Conference for Industry and Education Collaboration Copyright ©2019, American Society for Engineering Education Session ETD 525 To create a 2-dimensional complex plane a symbol is needed which on multiplication would lift points off the horizontal “real” axis. This conventional symbol is i which has the property that it rotates vectors by 90º counterclockwise. In this system the Cartesian form, a + bi of a complex number can be constructed by adding the horizontal value a to the value b rotated by 90º counterclockwise. On the real number line this is the same construction that is capable of producing negative numbers by adding the positive value a to the positive value b rotated by 180º counterclockwise, that is, a + b × (–1). The value i times i has the effect of adding two 90º counterclockwise rotations producing a rotation of 180º which is the same as multiplication by –1; that is, it produces a reversal in direction. Taking this as a starting point, all the properties of complex numbers can be derived which will clarify the conceptualization and the computation of complex numbers. There is nothing imaginary here. See figures 1, 2, and 3. 0 + i 0 + i i × i = -1 1.0 1.0 Figure 1 1 × i =90° counterclockwise rotation Figure 2 i × i =180° rotation y 3 + 2i 3 + 2 3 + 2i*i = 3 - 2 x Figure 3 Locating the point 3 + 2i Note: While complex numbers add like 2-dimensional vectors they do not multiply like vectors. Vectors can be n-dimensional while the complex numbers are inherently 2-dimensional. Dot products of vectors are scalars; (that is, ordinary non-dimensional numbers) while cross products of vectors lie in a space perpendicular to the plane of the multiplying vectors. Products and quotients of complex numbers lie in the same 2-dimensional plane as the complex factors. With this in mind it is better not to call the points in the complex plane “vectors.” Electrical engineers more appropriately use the words “phasors” or “impedances.” In this paper I will try to use the words “phasors” or “complex numbers” instead of vectors and additionally I will try to use the Proceedings of the 2019 Conference for Industry and Education Collaboration Copyright ©2019, American Society for Engineering Education Session ETD 525 words “horizontal” and “vertical” component instead of the words “real” and “imaginary.” However, setting convention aside is not easy. The 2-dimensional algebraic plane of complex numbers The algebraic properties of the points (complex numbers, phasors) in the complex plane have a lot in common with our familiar ordinary numbers. Their algebraic properties permit the basic operations of addition, subtraction, multiplication and division. As in the case of the ordinary numbers, the commutative, associative and distributive laws apply to the operations of addition and multiplication of complex numbers. Because these laws apply we need not worry when doing complex additions or multiplications about the order or grouping of the terms or factors. There are situations where it is advantageous to complete the complex plane by appending an additional single point called “infinity” which results from division by zero. This point is constructed by identifying both ends of any line as this single point, thereby forming the line into a ring. This single point lies at both ends of every line in every direction beyond any bound. The addition of this single point allows the 2-dimensional flat plane to be mapped in a one-to-one correspondence onto a sphere. We must note that this complex number system differs from our conventional number system where division by zero is not permitted. At this point I strongly recommend that the reader view the marvelous video “Mobius Transformations Revealed.” 1 I should also note that while in the real number system negative numbers do not have square roots, in the complex number system every number except zero and infinity has two distinct square roots. In fact, every nth degree polynomial has n roots, not necessarily distinct. The Cartesian and polar forms of complex numbers There are two common forms for describing 2-dimensional vectors. The Cartesian form describes the vector in terms of its horizontal and vertical coordinates. The polar form describes the vector in terms of its distance from the origin or magnitude and the angle the vector makes with the horizontal axis. Modern scientific calculators provide the capability of converting between the two forms. Because the conversion requires two values for the computation and yields two values, the calculator manual should be consulted to see how a particular calculator model handles the separation of the arguments and the results. The conversion equations as seen in Figure 4 are: Polar to Cartesian: (r, θ) → (x, y) Cartesian to Polar: (x, y) → (r, θ) y x = r cos(θ), y = r sin(θ) r = √x2 + y2, θ = arctan( ) x Proceedings of the 2019 Conference for Industry and Education Collaboration Copyright ©2019, American Society for Engineering Education Session ETD 525 2 2 r = √ (x + y ) α = arctan(y/x) x = r cos(α) . Figure 4 Polar ⇔ Cartesian conversion In the Cartesian form addition and subtraction of vectors is easy. Addition is simply performed by adding the horizontal components of the summands and then adding the vertical components. Similarly subtraction is performed by successively subtracting the co-ordinate components. An application of addition of 2-dimensional vectors is found in mechanics in the computation of the resultant of given forces. The vector forces are given as magnitudes and angles, usually provided in polar form. To compute the resultant, convert the forces to Cartesian form and then add their horizontal and vertical components. Convert the sum back to polar form to obtain the magnitude and direction of the resultant force. Addition and subtraction of complex numbers in polar form are not very easy, but multiplication and division are. The distributive law of numbers applies to the multiplication of complex numbers in Cartesian form: (a + bi)*(c + di) = ac + bci + adi + bd i*i . Since multiplication by i is a rotation of 90º, the product i*i of two rotations is a rotation of 180º or a multiplication by –1 so (a + bi)*(c + di) = ac – bd + (bc + ad)i y i*(4 + 3i) = –3 + 4i 4 + 3i 3 4 x − − − − Figure 5 Multiplying the phasor 4 + 3i by i induces a 90° counterclockwise rotation to –3 + 4i. Proceedings of the 2019 Conference for Industry and Education Collaboration Copyright ©2019, American Society for Engineering Education Session ETD 525 We can now derive the rule for multiplication of complex numbers in polar form. iα iβ (x1 + iy1)*(x2 + iy2) = r1 e * r2 e r1 r2 {cos α + i sin α}{ cos β + i sin β} = r1 r2 {(cos α cos β – sin α sin β) + i (sin α cos β + sin β cos α)} i(α +β) = r1 r2 {cos (α + β) + i sin (α + β)} = r1 r2 e . We see that in polar form to multiply complex numbers simply multiply the magnitudes and add the angles. It can similarly be proven that division of complex numbers in polar form can be performed by dividing their magnitudes and subtracting their angles; 푖α x1 + iy1 r1e r1 i(α – β) ( ) = 푖β = e x2 + iy2 r2e r2 Now we should note that while complex numbers are inherently 2-dimensional objects, they can be manipulated to form functions almost identically to the way the ordinary “real” numbers are manipulated.
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