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Complex numbers in physics

The number i, imaginary unit of the complex numbers, which contain the roots of all non-constant polynomials. The great physicist Richard Feynman said of the equation that it's the most remarkable formula in mathematics, for its single uses of the notions of addition, multiplication, exponentiation, and equality, and the single uses of the important constants 0, 1, e, i and π Definitions. Complex numbers are an extension of the real numbers. A general complex number z ∈ C z ∈ C (where C C denotes the set of all complex numbers } can be written in the form z = x+ iy x,y ∈ R. (1) (1) z = x + i y x, y ∈ R The equation (3.2.8.4) states that the complex derivative along the direction of any function can be calculated, but the result in general depends on. The derivatives for all possible angles lie on a circle, with the center and the radius A complex number is a number of the form a + bi, where a and b are real numbers, and i is an indeterminate satisfying i2 = −1. For example, 2 + 3i is a complex number. This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i2 + 1 = 0 is imposed

The importance of complex number in travelling waves. In the travelling wave, the complex number can be used to simplify the calculations by convert trigonometric functions (sin(x) and cos(x)) to exponential functions (e x) and store the phase angle into a complex amplitude. The angle velocity (ω) unit is radians per second. If you measure an angle using radian, you can simply calculate the length of the arc 1 Complex Numbers in Quantum Mechanics Complex numbers and variables can be useful in classical physics. However, they are not essential. To emphasize this, recall that forces, positions, momenta, potentials, electric and magnetic fields are all real quantities, and the equations describing them One place (out of many) where complex numbers come into play is in describing electromagnetic radiation. The radiation is described by complex functions. Whe waves are mixed, the arithmetic is that of complex numbers. Quantum theory descriptions are also using complex valued wave functions Vectors, Spinors, and Complex Numbers in Classical and Quantum Physics David Hestenes In the American Journal of Physics, Vol. 39/9, 1013{1027, September 1971. Abstract Geometric calculus is shown to unite vectors, spinors, and complex numbers into a single mathematical system with a comprehensive geometric sig-niflcance. The e-cacy of this calculus in physical applications is explicitly demon-strated complex numbers land surveying oh how I love parsing and and operators :). I am out of time right now today; however, I still by and large have the same problem in my gut so to speak. Problem being how to get a hug around the concept of complex numbers, a physical example. Even though, I saw the light when I read the example on this.

Going to physics In classical physics, complex numbers would be used as bookkeeping devices to remember the two coordinates of a two-dimensional vector; the complex numbers also knew something about the length of two-dimensional vectors. But this usage of the complex numbers was not really fundamental. In particular, the multiplication of two complex numbers never directly entered physics 11.4 Applications of complex numbers in physics Here I just give a few random examples where these concepts are useful: † Adding harmonic waves with flxed phase. Suppose you want to know what the sum of a bunch of waves is (I'll leave out the physical dimensions, but yo Complex numbers are widely used in the description of physical systems. A complex number z, z is written in the form z, equals, x, plus, i, y. z = x + i y where x, x and y, y are real numbers and i, squared, equals, minus, 1, i 2 = − 1 or i, equals, square root of, minus, 1, end square root, i = − 1

Complex Numbers Physics For Idiot

A complex number consists of a combination of a real part and an imaginary part, the former being a real number and the latter multiplying − 1, which we denote as i. (1.1.1) z = a + b i, a ≡ R e (z), b ≡ I m (z) A strictly real or imaginary number is also complex, with the imaginary or real part equal to zero, respectively COMPLEX NUMBERS AND PHYSICAL REALITY . V.V. Lyahov, V.M. Nechshadim. 1. INTRODUCTION . In XVI century Rafael Bombelly has formulated the rules of operations on complex numbers obtained as the result of solution of an algebraic equation, but has found these numbers as an useless invention. Cardano, Descartes, Newton, Leibnitz, Euler did not recognize that there was any physical or geometrical.

Introduction to Complex Numbers Physics in a Nutshel

  1. The field of complex numbers is algebraically closed, i.e. any equation of the nth power in this field has precisely n solutions. As a consequence, any linear operator in a finite-dimensional space has at least one eigenvalue. However, this is not necessarily the case if the space is infinite-dimensional
  2. PH2011 Physics 2A Maths Revision - Session 2: Complex Numbers and Vectors 1 2 Session Two - Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar to all those who have successfully passed through rst year mathematics. For our direct entrants, the School of Mathemat- ics plans to cover this stu early on in the session for your bene t.
  3. Complex numbers are used in macro physics. They are used in analysis of electrical circuits (especially when AC is involved) and in fluid dynamics. Solution of differential equations is simplified if complex numbers are used, as is Fourier analysis. Any scenario that involves periodic or cyclic functions can be modeled using complex numbers

3.2. Complex Numbers — Theoretical Physics Reference 0.5 ..

Complex numbers are broadly used in physics, normally as a calculation tool that makes things easier due to Euler's formula. In the end, it is only the real component that has physical meaning or the two parts (real and imaginary) are treated separately as real quantities. However, the situatio Main Article: Complex Plane Complex numbers are often represented on the complex plane, sometimes known as the Argand plane or Argand diagram.In the complex plane, there are a real axis and a perpendicular, imaginary axis.The complex number a + b i a+bi a + b i is graphed on this plane just as the ordered pair (a, b) (a,b) (a, b) would be graphed on the Cartesian coordinate plane Complex numbers are made up of two components, real and imaginary. They have the form a + bi, where the numbers a and b are real. The bi component is responsible for the specific features of complex numbers. The key role here is played by the imaginary number i, i.e. the square root of -1 But imaginary numbers, and the complex numbers they help define, turn out to be incredibly useful. They have a far-reaching impact in physics, engineering, number theory and geometry . And they are the first step into a world of strange number systems, some of which are being proposed as models of the mysterious relationships underlying our physical world The information these complex numbers carry isn't related to a simple physical property, like the spin of an electron. Instead, Scandolo said, it has to do with the ability to extract information..

Complex numbers enable us to find roots for any polynomial P(z) ≡ anzn+an−1zn−1 +··· +a 0. (1.14) That is, there is at least one, and perhapsas many as ncomplex numberszisuch that P(zi) = 0. Many physical problems involve such roots. In the case n= 2 you already know a general formula for the roots. There is a similar formula for the case n= 3 and historically this is important. L1.3 Necessity of complex numbers. If playback doesn't begin shortly, try restarting your device. Videos you watch may be added to the TV's watch history and influence TV recommendations. To avoid. These complex numbers, usually they're just a convenient tool, but here it turns out that they really have some physical meaning, said Tamás Vértesi, a physicist at the Institute for Nuclear Research at the Hungarian Academy of Sciences who, years ago, argued the opposite. The world is such that it really requires these complex numbers, he said

From the Pythagorean theorem, this is the length of the hypotenuse of a right triangle with legs of length a and b. Therefore the magnitude of a complex number gives its distance from the origin. An important property of complex numbers is the Euler's formula: it states that every complex number, can be rewritten in the form of re =r (cos + i sin), where e=2.71828... is the Euler's..

Complex number - Wikipedi

11.4 Applications of complex numbers in physics Here I just give a few random examples where these concepts are useful: † Adding harmonic waves with flxed phase. Suppose you want to know what the sum of a bunch of waves is (I'll leave out the physical dimensions, but yo Complex numbers were already used (see Yang and Barbour) in classical physics but merely as a convenient tool. At the end of the calculations, the real or imaginary part of the result is always taken. As Yang observes, physics was conceptually in terms of real numbers. Figure 26: Impedance measures the opposition of a circuit to a current when a voltage is applied. It is one example of. Observable physical quantities are represented by self-adjoint operators in this space. The field of complex numbers is algebraically closed, i.e. any equation of the nth power in this field has precisely n solutions. As a consequence, any linear operator in a finite-dimensional space has at least one eigenvalue. However, this is not. We use complex numbers to describe physical things that have two-dimensional properties. Consider an ideal bell that rings in a pure sine wave. One real number by itself does not describe the current state of this bell. The bell might be vibrating between -1 and +1 units from its rest state because it was struck softly, or it might be vibrating between -5 and +5 units because it was struck.

Whenever we thought of complex numbers, we first imagined a number line, then we imagined taking square-root of a negative number, and going still backwards at the number line. (/\) However, complex numbers are all about revolving around the number line. That is the reason why the numbers 1, i , -1 and -i keep repeating cyclically in complex numbers. Want an example? i^0=1 i^1=i i^2=-1 i^3=-i. Complex Numbers and Phasors Outline . Reading - Shen and Kong - Ch. 1 . Linear Systems Theory Complex Numbers Polyphase Generators and Motors Phasor Notation . 1. True / False . 1. In Lab 1 you built a motor about 5 cm in diameter. If this motor spins at 30 Hz, it is operating in the quasi-static regime. 3. This describes a 1D propagating wave: 2. The wave number k (also called the wave vector. The main reason why complex numbers show up in quantum mechanics is because they make linear algebra much easier. Let's talk about quantum mechanics for a bit. If you look at an undergraduate quantum mechanics text, you'll quickly see that the la..

The Complex numbers in real life - Science Us

Complex numbers and complex analysis show up everywhere in mathematics and physics. Algebraically, complex numbers are closed. This is a good algebraic property for a field. They've been studied in mathematics since the 17th century because of their applications to mathematics, mechanics, waves, etc. Complex numbers show up in number theory a lot. From the analytical point of view, there is a. Chapter 3: Capacitors, Inductors, and Complex Impedance - 19 - In order to perform these functions analytically, we will need to introduce a number of new concepts and some significant mathematical formalism. In this process we will also develop a number of new concepts in analyzing electronic circuits. Capacitanc Complex numbers can be de ned as pairs of real numbers (x;y) with special manipulation rules. That's how complex numbers are de ned in Fortran or C. We can map complex numbers to the plane R2 with the real part as the xaxis and the imaginary part as the y-axis. We refer to that mapping as the complex plane. This is a very useful visualization. The form x+iyis convenient with the special. In the physical sciences, the wavenumber (also wave number or repetency) is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance.Whereas temporal frequency can be thought of as the number of waves per unit time, wavenumber is the number of waves per unit distance.. In multidimensional systems, the wavenumber is the magnitude of the wave vector

1 Complex Numbers 1 De•nitions 1 Algebraic Properties 1 Polar Coordinates and Euler Formula 2 Roots of Complex Numbers 3 Regions in Complex Plane 3 2 Functions of Complex Variables 5 Functions of a Complex Variable 5 Elementary Functions 5 Mappings 7 Mappings by Elementary Functions. 8 3 Analytic Functions 11 Limits 11 Continuity 12 Derivative 12 Cauchy- Riemann Equations 13. vi Contents. Complex numbers are not as visually appealing, initially, but mathematically they operate in a similar manner to the impedance's function within a circuit. This combines two concepts that I will address separately: how does a complex impedance behave, and how a complex number represents that. While a resistance only changes the magnitude of a signal by absorbing energy, a complex impedance can. That complex number will in turn usually be represented by a single letter, such as z= x+iy. * If you think that this question is an easy one, you can read about some of the di culties that the greatest mathematicians in history had with it: \An Imaginary Tale: The Story of p 1 by Paul J. Nahin. I recommend it. James Nearing, University of Miami 1. 3|Complex Algebra 2 z 1 = x 1 +iy 1 z 2 = x.

Ferromagnetic materials like cobalt and iron produce magnetic fields whose magnitudes are measured by real numbers. Imaginary or complex fields are, however, essential in the fundamental theory that underlies the statistical physics of phase transitions, such as those associated with the onset of magnetization. Long thought to be merely mathematical constructs, a realization of these imaginary. Chapter 4: Complex numbers and functions. You first learned about complex numbers when an algebra teacher tried to convince you that made sense. It doesn't just make sense; it is a powerful way to model the real world we live in, in spite of i being a so-called imaginary number. Maple knows all about the complex plane, starting with a special variable to represent : Maple calls it ; yes. 32.1: Complex Numbers Real Numbers. Let us think of the ordinary numbers as set out on a line which goes to infinity in both positive and... Solving Quadratic Equations. The problem with this is that sometimes the expression inside the square root is negative. Polar Coordinates. Some properties of.

Complex Numbers in physics Physics Forum

  1. Imaginary and complex numbers are handicapped by the name we gave them. Imaginary has obvious and bad connotations: it implies an object made up, perhaps not useful; complex similarly seems to argue the numbers are too hard to use. As with so much in math and physics, however, the names are historical, and complexity i
  2. But the amazing thing is that pure theory of complex numbers, complex functions and complex analysis has applications almost everywhere you look, and not just within mathematics. If you have studied physics, you may already have met complex numbers and functions when looking at impedance, phase angles, and oscillating currents
  3. Complex numbers are the sum of a real and an imaginary number, represented as a + bi. Using the complex plane, we can plot complex numbers similar to how we plot a coordinate on the Cartesian.
  4. Complex numbers have become an essential part of pure and applied mathematics. It is unfortunate that such numbers are called 'imaginary' since they are no more or less real than any other type of number. It is best to regard all numbers as abstractions and not to insist on arbitrary connections with the real world. This Note describes the algebraic properties of complex numbers and their.
  5. Complex Numbers in Real Life Asked by Domenico Tatone (teacher), Mayfield Secondary School on Friday May 3, 1996: I've been stumped! 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.

Contributor; I am now going to repeat the analyses of Sections 13.1 and 13.2 using the notation of complex numbers.In the context of alternating current theory, the imaginary unit is customarily given the symbol \(j\) rather than \(i\), so that the symbol \(i\) is available, if need be, for electric currents A complex number is a number comprising area land imaginary part. It can be written in the form a+ib, where a and b are real numbers, and i is the standard imaginary unit with the property i2=-1. The complex numbers contain the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the understanding of. That's complex numbers -- they allow an extra dimension of calculation. (Yes, I know about phase shifts and Fourier transforms, but these are 8th graders, and for comprehensive testing, they're required to know a real world application of complex numbers, but not the details of how or why. I don't understand this, but that's the way it is There are a lot of applications in physics and engineering where complex numbers are useful. A common use of complex numbers is in electrical circuits, where capacitors and inductors are like an imaginary component of resistors. This only applies when using alternating current at the frequency being used. In other words, if the frequency of the current changes then the complex 'resistance.

Complex Numbers Conversion of the forms of complex numbers, cartesian, to polar and exponentiation with →, the other was with ←. The angle φ is in rad, here you can convert angle units. Calculate with cart. value transfers the cartesian number into the second calculator. a = ρ * cos(φ) b = ρ * sin(φ Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number \(a + bi\) can be identified with the point \((a, b)\) in the complex plane. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number. The complex conjugate is used in the rationalization of complex numbers and for finding the amplitude of the polar form of a complex number. One application of the complex conjugate in physics is in finding the probability in quantum mechanics. Since the wavefunction which defines the probability amplitude may be a complex function, the probability is defined in terms of the complex conjugate. Complex Numbers show up all over the place in Computer Science and Engineering as well as Scientific Computing. Examples include Fast Fourier Transforms for Signal Processing, Circuit Simulation (Complex Numbers are very common in Electrical Engineering), and Fractals which get used in Graphics and various other fields. They also show up a lot in Physics programming as Complex Numbers have.

A complex number is of the form i 2 =-1. A complex number is usually denoted by the letter 'z'. 'a' is called the real part, and 'b' is called the imaginary part of the complex number. The notion of complex numbers increased the solutions to a lot of problems. For instance, had complex numbers been not there, the equation x 2 +x+1=0. Although imaginary numbers are not commonly used in daily life, in engineering and physics they are in fact used to represent physical quantities such as impedance of RL, RC, or RLC circuit. Complex numbers are numbers that consist of two parts, one real and one imaginary. An imaginary number is the square root of a negative real number (−1. might make complex numbers a little less mysterious by thinking of z as a vector in a two dimensional vector space. The complex conjugate z* of a complex number z is defined z* = x-äy (7) so *z is the mirror image of z. Operationally if you have a complex number z you can construct the complex conjugate z* by changing the sign of the imaginary part of z. Sometimes it is convenient to write a. Complex Numbers are part of this real world solution. Let's do a quick example with actual numbers so you can see how this works. Suppose a 4-kilogram mass is attached to a spring with a stiffness measured at k = 53 and a damping of r = 8 . The quadratic equation we need to solve is. 4 x 2 + 8 x + 53 = 0

are both complex numbers, i.e. ( ; 2C) 2 Complex Numbers are Essential. Quantum mechanics is the rst physics theory that truly makes use of complex numbers. The numbers most of us use for daily life (integers, fractions, decimals) are real numbers. The set of complex numbers is denoted by C and the set of real numbers is denoted by R. Complex. A complex number has both real and imaginary parts. Any complex number z can be written in the form x+iy where x and y are real numbers. x is the real part and y is the imaginary part. For instance, 2+3i is a complex number. The complex conjugate of a complex number, represented by a star next to the number, is the same real part but the negative imaginary part. So if z=2+3i then z * =2.

Applications And Use Of Complex Numbers. A complex number is that number which comprises a real and an imaginary part. It is mainly written in the form a + bi, where a is real numbers, and i is the imaginary unit with b as also the real part of the imaginary portion with the property i2 = âˆ'1. The complex number contains. 7. Powers and Roots of Complex Numbers. by M. Bourne. Consider the following example, which follows from basic algebra: (5e 3j) 2 = 25e 6j. We can generalise this example as follows: (rejθ)n = rnejnθ. The above expression, written in polar form, leads us to DeMoivre's Theorem Definition. The complex number is basically the combination of a real number and an imaginary number. The complex number is in the form of a+ib, where a = real number and ib = imaginary number. Also, a,b belongs to real numbers and i = √-1. Hence, a complex number is a simple representation of addition of two numbers, i.e., real number and an imaginary number The Algebra of Complex Numbers. A complex number is usually expressed as this — a+ib where we say that a is the real part of the complex number and b is the imaginary part of the complex number. And we can define the following on a complex number. Let z = a +ib be a complex number then Re{z} = a and Im{z} = b thus giving us the real and the imaginary parts Chapter 1: Complex Numbers Lecture notes Math Section 1.1: Definition of Complex Numbers Definition of a complex number A complex number is a number that can be expressed in the form z = a + bi, where a and b are real numbers and i is the imaginary unit, that satisfies the equation i2 = −1. In this expression, a is the real part Re(z) and b.

The rectangular representation of a complex number is in the form z = a + bi. If you were to represent a complex number according to its Cartesian Coordinates, it would be in the form: (a, b); where a, the real part, lies along the x axis and the imaginary part, b, along the y axis. The Polar Coordinates of a a complex number is in the form (r. For a introduction in Complex numbers and the basic mathematical operations between complex numbers, read the article Complex Numbers - Introduction.. In this article we are going to explain the different ways of representation of a complex number and the methods to convert from one representation to another.. Complex numbers can be represented in several formats Argument of a Complex Number Calculator. The argument of a complex number is the direction of the number from the origin or the angle to the real axis. Sometimes this function is designated as atan2 (a,b). Please enter the two values a and b of a complex number in the form a+bi, the argument will be calculated

Physical Meaning of Complex Numbers Physics Forum

Researchers find string theory link in a class of complex numbers. by Kavli Institute for the Physics and Mathematics of the Universe. Figure 1. Extension of the concept of integral numbers. Basic complex number facts I Complex numbers are numbers of the form a + b_{, where _{2 = 1.I We add and multiply complex numbers in the obvious way. Other operations: I a + b{_ = a b{_ (conjugation). I ja + b_{j= p a2 + b2 (absolute value). Note: jzj= p z z. I We can identify a complex number a + b{_ with the point (a;b) in the plane If we represent a complex number in the Re-Im plane, its complex conjugate will be mirrored around Re axis.. Scilab has the predefined function conj(), which outputs the complex conjugate of a complex number input as argument.For example, we can find the complex conjugate of our z 1 and z 2 number with:--> conj(z1) ans = 2. - i --> conj(z2) ans = 1. - 2. Complex Numbers and Complex Functions A complex number zcan be written as z= x+ iy or z= rei˚ with r 0 where i= p 1, and x, y, r, and ˚are real numbers. Clearly, x= rcos˚and y= rsin˚ leading to a description in terms of the \complex plane. The complex conjugate of zis z = x iy or z = re i˚ The \modulus of zis jzj p zz= r= x2 + y2 and ˚is often called the \phase of z. A complex. Definition. Die komplexen Zahlen lassen sich als Zahlbereich im Sinne einer Menge von Zahlen, für die die Grundrechenarten Addition, Multiplikation, Subtraktion und Division erklärt sind, mit den folgenden Eigenschaften definieren: . Die reellen Zahlen sind in den komplexen Zahlen enthalten. Das heißt, dass jede reelle Zahl eine komplexe Zahl ist

Why complex numbers are fundamental in physics - Blogge

This is arguably the first time that complex numbers were given a physical interpretation, which led to a better understanding of elliptical and circular polarizations. This rather unknown episode of the history of physics is described in this work, and some of the pedagogical lessons that can be extracted from it are discussed. ACKNOWLEDGMENTS. The author is indebted to Jesper Lützen and. Representation of Waves via Complex Numbers. In mathematics, the symbol is conventionally used to represent the square-root of minus one: that is, the solution of (Riley 1974). A real number, (say), can take any value in a continuum of values lying between and . On the other hand, an imaginary number takes the general form , where is a real number Complex numbers (the sum of real and imaginary numbers) occur quite naturally in the study of quantum physics. They're useful for modelling periodic motions (such as water or light waves) as well. This paper aims to clarify the geometrical and physical interpretation of the Dirac theory and determine some of its implications for the interpretation of quantum theory in general. With a formalism developed in Refs. l, 2. 3 the Dirac theory can be given a completely geometrical formulation, involving neither matrices nor complex numbers. In the process, the following significant facts are. Can a physical quantity be described by a complex number? Yes, Fairly common examples involve electromagnetic waves, as well as alternating currents and corresponding voltages. For example, a voltage signal might be represented by: [math]V(t) =.

3 Complex Algebra 52 Complex Numbers Some Functions Applications of Euler's Formula Geometry Series of cosines Logarithms Mapping 4 Di erential Equations 67 Linear Constant-Coe cient Forced Oscillations Series Solutions Some General Methods Trigonometry via ODE's Green's Functions Separation of Variables Circuits Simultaneous Equations Simultaneous ODE's Legendre's Equation. Complex Numbers Paul Renteln Department of Physics California State University San Bernardino, CA 92407 May, 2009 1 The Basic Representation.

Complex numbers - Isaac Physic

Stern-Gerlach Experiments and Complex Numbers in Quantum Physics. Authors: S. Sivakumar. (Submitted on 9 Jul 2012) Abstract: It is often stated that complex numbers are essential in quantum theory. In this article, the need for complex numbers in quantum theory is motivated using the results of tandem Stern-Gerlach experiments. Subjects Analysis - Analysis - Complex analysis: In the 18th century a far-reaching generalization of analysis was discovered, centred on the so-called imaginary number i = −1. (In engineering this number is usually denoted by j.) The numbers commonly used in everyday life are known as real numbers, but in one sense this name is misleading. Numbers are abstract concepts, not objects in the physical. Complex Numbers. By M Bourne. Chapter Contents. 1. Basic Definitions of imaginary and complex numbers - and where they come from.. 2. Basic Operations - adding, subtracting, multiplying and dividing complex numbers.. 3. Graphical Representation of complex numbers.. 4. Polar Form of complex numbers . 5. Exponential Form of complex numbers . 6 The only complex number which is both real and purely imaginary is 0. Observe that, according to our definition, every real number is also a complex number. equating the real and the imaginary parts of the two sides of an equation is indeed a part of the definition of complex numbers and will play a very important role. 2 Conjugation and Absolute Value Definition 2.1 Following common. An equivalent way to represent complex numbers is provided by the polar representation. Here each corresponding vector is characterised by its length $|z|=\sqrt{z \bar{z}} \in [0,\infty)$ and the angle $\varphi \in [0,2\pi)$ between the real axis and this vector. [1] Any complex number can be illustrated as a vector in the two-dimensional.

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1.1: Complex Numbers - Physics LibreText

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COMPLEX NUMBERS AND PHYSICAL REALITY - arXi

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Why is quantum physics based on complex numbers

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