127-741: In physics , a plane wave is a special case of a wave or field : a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position x → {\displaystyle {\vec {x}}} in space and any time t {\displaystyle t} , the value of such a field can be written as F ( x → , t ) = G ( x → ⋅ n → , t ) , {\displaystyle F({\vec {x}},t)=G({\vec {x}}\cdot {\vec {n}},t),} where n → {\displaystyle {\vec {n}}}
254-401: A b = a b {\displaystyle {\sqrt {a}}{\sqrt {b}}={\sqrt {ab}}} , which is valid for non-negative real numbers a and b , and which was also used in complex number calculations with one of a , b positive and the other negative. The incorrect use of this identity in the case when both a and b are negative, and the related identity 1
381-583: A = 1 a {\textstyle {\frac {1}{\sqrt {a}}}={\sqrt {\frac {1}{a}}}} , even bedeviled Leonhard Euler . This difficulty eventually led to the convention of using the special symbol i in place of − 1 {\displaystyle {\sqrt {-1}}} to guard against this mistake. Even so, Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, Elements of Algebra , he introduces these numbers almost at once and then uses them in
508-526: A 0 , ..., a n , the equation a n z n + ⋯ + a 1 z + a 0 = 0 {\displaystyle a_{n}z^{n}+\dotsb +a_{1}z+a_{0}=0} has at least one complex solution z , provided that at least one of the higher coefficients a 1 , ..., a n is nonzero. This property does not hold for the field of rational numbers Q {\displaystyle \mathbb {Q} } (the polynomial x − 2 does not have
635-608: A traveling plane wave , whose evolution in time can be described as simple translation of the field at a constant wave speed c {\displaystyle c} along the direction perpendicular to the wavefronts. Such a field can be written as F ( x → , t ) = G ( x → ⋅ n → − c t ) {\displaystyle F({\vec {x}},t)=G\left({\vec {x}}\cdot {\vec {n}}-ct\right)\,} where G ( u ) {\displaystyle G(u)}
762-499: A Platonist by Stephen Hawking , a view Penrose discusses in his book, The Road to Reality . Hawking referred to himself as an "unashamed reductionist" and took issue with Penrose's views. Mathematics provides a compact and exact language used to describe the order in nature. This was noted and advocated by Pythagoras , Plato , Galileo, and Newton. Some theorists, like Hilary Putnam and Penelope Maddy , hold that logical truths, and therefore mathematical reasoning, depend on
889-426: A complex exponential plane wave . When the values of F {\displaystyle F} are vectors, the wave is said to be a longitudinal wave if the vectors are always collinear with the vector n → {\displaystyle {\vec {n}}} , and a transverse wave if they are always orthogonal (perpendicular) to it. Often the term "plane wave" refers specifically to
1016-454: A complex number is an element of a number system that extends the real numbers with a specific element denoted i , called the imaginary unit and satisfying the equation i 2 = − 1 {\displaystyle i^{2}=-1} ; every complex number can be expressed in the form a + b i {\displaystyle a+bi} , where a and b are real numbers. Because no real number satisfies
1143-488: A frame of reference that is in motion with respect to an observer; the special theory of relativity is concerned with motion in the absence of gravitational fields and the general theory of relativity with motion and its connection with gravitation . Both quantum theory and the theory of relativity find applications in many areas of modern physics. While physics itself aims to discover universal laws, its theories lie in explicit domains of applicability. Loosely speaking,
1270-407: A mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations , even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has
1397-486: A standard basis . This standard basis makes the complex numbers a Cartesian plane , called the complex plane . This allows a geometric interpretation of the complex numbers and their operations, and conversely some geometric objects and operations can be expressed in terms of complex numbers. For example, the real numbers form the real line , which is pictured as the horizontal axis of the complex plane, while real multiples of i {\displaystyle i} are
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#17328764841431524-452: A basic awareness of the motions of the Sun, Moon, and stars. The stars and planets, believed to represent gods, were often worshipped. While the explanations for the observed positions of the stars were often unscientific and lacking in evidence, these early observations laid the foundation for later astronomy, as the stars were found to traverse great circles across the sky, which could not explain
1651-748: A complex number z is denoted Re( z ) , R e ( z ) {\displaystyle {\mathcal {Re}}(z)} , or R ( z ) {\displaystyle {\mathfrak {R}}(z)} ; the imaginary part is Im( z ) , I m ( z ) {\displaystyle {\mathcal {Im}}(z)} , or I ( z ) {\displaystyle {\mathfrak {I}}(z)} : for example, Re ( 2 + 3 i ) = 2 {\textstyle \operatorname {Re} (2+3i)=2} , Im ( 2 + 3 i ) = 3 {\displaystyle \operatorname {Im} (2+3i)=3} . A complex number z can be identified with
1778-837: A complex number and as a real number are equal. Using the conjugate, the reciprocal of a nonzero complex number z = x + y i {\displaystyle z=x+yi} can be computed to be 1 z = z ¯ z z ¯ = z ¯ | z | 2 = x − y i x 2 + y 2 = x x 2 + y 2 − y x 2 + y 2 i . {\displaystyle {\frac {1}{z}}={\frac {\bar {z}}{z{\bar {z}}}}={\frac {\bar {z}}{|z|^{2}}}={\frac {x-yi}{x^{2}+y^{2}}}={\frac {x}{x^{2}+y^{2}}}-{\frac {y}{x^{2}+y^{2}}}i.} More generally,
1905-591: A complex number as a point in the complex plane ( above ) was first described by Danish – Norwegian mathematician Caspar Wessel in 1799, although it had been anticipated as early as 1685 in Wallis's A Treatise of Algebra . Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806 Jean-Robert Argand independently issued a pamphlet on complex numbers and provided
2032-450: A complex number. (This is in contrast to the roots of a positive real number x , which has a unique positive real n -th root, which is therefore commonly referred to as the n -th root of x .) One refers to this situation by saying that the n th root is a n -valued function of z . The fundamental theorem of algebra , of Carl Friedrich Gauss and Jean le Rond d'Alembert , states that for any complex numbers (called coefficients )
2159-407: A consequence of the trigonometric identities for the sine and cosine function.) In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. The picture at the right illustrates the multiplication of ( 2 + i ) ( 3 + i ) = 5 + 5 i . {\displaystyle (2+i)(3+i)=5+5i.} Because
2286-404: A false point of view and therefore found a mysterious darkness, this is in large part attributable to clumsy terminology. Had one not called +1, −1, − 1 {\displaystyle {\sqrt {-1}}} positive, negative, or imaginary (or even impossible) units, but instead, say, direct, inverse, or lateral units, then there could scarcely have been talk of such darkness. In
2413-436: A fixed complex number is a similarity centered at the origin (dilating by the absolute value, and rotating by the argument). The operation of complex conjugation is the reflection symmetry with respect to the real axis. The complex numbers form a rich structure that is simultaneously an algebraically closed field , a commutative algebra over the reals, and a Euclidean vector space of dimension two. A complex number
2540-617: A function of one scalar parameter (the displacement d = x → ⋅ n → {\displaystyle d={\vec {x}}\cdot {\vec {n}}} ) with scalar or vector values, and S {\displaystyle S} is a scalar function of time. This representation is not unique, since the same field values are obtained if S {\displaystyle S} and G {\displaystyle G} are scaled by reciprocal factors. If | S ( t ) | {\displaystyle \left|S(t)\right|}
2667-420: A hard-to-find physical meaning. The final mathematical solution has an easier-to-find meaning, because it is what the solver is looking for. Physics is a branch of fundamental science (also called basic science). Physics is also called " the fundamental science" because all branches of natural science including chemistry, astronomy, geology, and biology are constrained by laws of physics. Similarly, chemistry
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#17328764841432794-411: A large homogeneous region of space can be well approximated by plane waves when viewed over any part of that region that is sufficiently small compared to its distance from the source. That is the case, for example, of the light waves from a distant star that arrive at a telescope. A standing wave is a field whose value can be expressed as the product of two functions, one depending only on position,
2921-433: A natural way throughout. In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 Abraham de Moivre noted that the identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be re-expressed by
3048-393: A quelquefois aucune quantité qui corresponde à celle qu'on imagine. ] A further source of confusion was that the equation − 1 2 = − 1 − 1 = − 1 {\displaystyle {\sqrt {-1}}^{2}={\sqrt {-1}}{\sqrt {-1}}=-1} seemed to be capriciously inconsistent with the algebraic identity
3175-453: A rational root, because √2 is not a rational number) nor the real numbers R {\displaystyle \mathbb {R} } (the polynomial x + 4 does not have a real root, because the square of x is positive for any real number x ). Because of this fact, C {\displaystyle \mathbb {C} } is called an algebraically closed field . It is a cornerstone of various applications of complex numbers, as
3302-487: A rigorous proof of the fundamental theorem of algebra . Carl Friedrich Gauss had earlier published an essentially topological proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of −1". It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane, largely establishing modern notation and terminology: If one formerly contemplated this subject from
3429-476: A scalar or a vector, is called the amplitude of the wave; the scalar coefficient f {\displaystyle f} is its "spatial frequency"; and the scalar φ {\displaystyle \varphi } is its " phase shift ". A true plane wave cannot physically exist, because it would have to fill all space. Nevertheless, the plane wave model is important and widely used in physics. The waves emitted by any source with finite extent into
3556-560: A solution which is a complex number. For example, the equation ( x + 1 ) 2 = − 9 {\displaystyle (x+1)^{2}=-9} has no real solution, because the square of a real number cannot be negative, but has the two nonreal complex solutions − 1 + 3 i {\displaystyle -1+3i} and − 1 − 3 i {\displaystyle -1-3i} . Addition, subtraction and multiplication of complex numbers can be naturally defined by using
3683-409: A special case the fundamental formula This formula distinguishes the complex number i from any real number, since the square of any (negative or positive) real number is always a non-negative real number. With this definition of multiplication and addition, familiar rules for the arithmetic of rational or real numbers continue to hold for complex numbers. More precisely, the distributive property ,
3810-465: A specific practical application as a goal, other than the deeper insight into the phenomema themselves. Applied physics is a general term for physics research and development that is intended for a particular use. An applied physics curriculum usually contains a few classes in an applied discipline, like geology or electrical engineering. It usually differs from engineering in that an applied physicist may not be designing something in particular, but rather
3937-426: A speed much less than the speed of light. These theories continue to be areas of active research today. Chaos theory , an aspect of classical mechanics, was discovered in the 20th century, three centuries after the original formulation of classical mechanics by Newton (1642–1727). These central theories are important tools for research into more specialized topics, and any physicist, regardless of their specialization,
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4064-399: A subfield of mechanics , is used in the building of bridges and other static structures. The understanding and use of acoustics results in sound control and better concert halls; similarly, the use of optics creates better optical devices. An understanding of physics makes for more realistic flight simulators , video games, and movies, and is often critical in forensic investigations. With
4191-461: A substantial treatise on " Physics " – in the 4th century BC. Aristotelian physics was influential for about two millennia. His approach mixed some limited observation with logical deductive arguments, but did not rely on experimental verification of deduced statements. Aristotle's foundational work in Physics, though very imperfect, formed a framework against which later thinkers further developed
4318-429: A topic in itself first arose in the 16th century when algebraic solutions for the roots of cubic and quartic polynomials were discovered by Italian mathematicians ( Niccolò Fontana Tartaglia and Gerolamo Cardano ). It was soon realized (but proved much later) that these formulas, even if one were interested only in real solutions, sometimes required the manipulation of square roots of negative numbers. In fact, it
4445-463: A transverse planar wave satisfies ∇ ⋅ F → = 0 {\displaystyle \nabla \cdot {\vec {F}}=0} for all x → {\displaystyle {\vec {x}}} and t {\displaystyle t} . Physics Physics is the scientific study of matter , its fundamental constituents , its motion and behavior through space and time , and
4572-935: A wave in a one-dimensional medium. Any local operator , linear or not, applied to a plane wave yields a plane wave. Any linear combination of plane waves with the same normal vector n → {\displaystyle {\vec {n}}} is also a plane wave. For a scalar plane wave in two or three dimensions, the gradient of the field is always collinear with the direction n → {\displaystyle {\vec {n}}} ; specifically, ∇ F ( x → , t ) = n → ∂ 1 G ( x → ⋅ n → , t ) {\displaystyle \nabla F({\vec {x}},t)={\vec {n}}\partial _{1}G({\vec {x}}\cdot {\vec {n}},t)} , where ∂ 1 G {\displaystyle \partial _{1}G}
4699-440: Is a unit-length vector , and G ( d , t ) {\displaystyle G(d,t)} is a function that gives the field's value as dependent on only two real parameters: the time t {\displaystyle t} , and the scalar-valued displacement d = x → ⋅ n → {\displaystyle d={\vec {x}}\cdot {\vec {n}}} of
4826-458: Is also denoted by some authors by z ∗ {\displaystyle z^{*}} . Geometrically, z is the "reflection" of z about the real axis. Conjugating twice gives the original complex number: z ¯ ¯ = z . {\displaystyle {\overline {\overline {z}}}=z.} A complex number is real if and only if it equals its own conjugate. The unary operation of taking
4953-425: Is an expression of the form a + bi , where a and b are real numbers , and i is an abstract symbol, the so-called imaginary unit , whose meaning will be explained further below. For example, 2 + 3 i is a complex number. For a complex number a + bi , the real number a is called its real part , and the real number b (not the complex number bi ) is its imaginary part . The real part of
5080-535: Is bounded in the time interval of interest (which is usually the case in physical contexts), S {\displaystyle S} and G {\displaystyle G} can be scaled so that the maximum value of | S ( t ) | {\displaystyle \left|S(t)\right|} is 1. Then | G ( x → ⋅ n → ) | {\displaystyle \left|G({\vec {x}}\cdot {\vec {n}})\right|} will be
5207-546: Is called the direction of propagation . For each displacement d {\displaystyle d} , the moving plane perpendicular to n → {\displaystyle {\vec {n}}} at distance d + c t {\displaystyle d+ct} from the origin is called a " wavefront ". This plane travels along the direction of propagation n → {\displaystyle {\vec {n}}} with velocity c {\displaystyle c} ; and
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5334-413: Is clear-cut, but not always obvious. For example, mathematical physics is the application of mathematics in physics. Its methods are mathematical, but its subject is physical. The problems in this field start with a " mathematical model of a physical situation " (system) and a "mathematical description of a physical law" that will be applied to that system. Every mathematical statement used for solving has
5461-461: Is computed as follows: For example, ( 3 + 2 i ) ( 4 − i ) = 3 ⋅ 4 − ( 2 ⋅ ( − 1 ) ) + ( 3 ⋅ ( − 1 ) + 2 ⋅ 4 ) i = 14 + 5 i . {\displaystyle (3+2i)(4-i)=3\cdot 4-(2\cdot (-1))+(3\cdot (-1)+2\cdot 4)i=14+5i.} In particular, this includes as
5588-419: Is concerned with bodies acted on by forces and bodies in motion and may be divided into statics (study of the forces on a body or bodies not subject to an acceleration), kinematics (study of motion without regard to its causes), and dynamics (study of motion and the forces that affect it); mechanics may also be divided into solid mechanics and fluid mechanics (known together as continuum mechanics ),
5715-400: Is concerned with the most basic units of matter; this branch of physics is also known as high-energy physics because of the extremely high energies necessary to produce many types of particles in particle accelerators . On this scale, ordinary, commonsensical notions of space, time, matter, and energy are no longer valid. The two chief theories of modern physics present a different picture of
5842-483: Is defined only up to adding integer multiples of 2 π {\displaystyle 2\pi } , since a rotation by 2 π {\displaystyle 2\pi } (or 360°) around the origin leaves all points in the complex plane unchanged. One possible choice to uniquely specify the argument is to require it to be within the interval ( − π , π ] {\displaystyle (-\pi ,\pi ]} , which
5969-590: Is denoted by C {\displaystyle \mathbb {C} } ( blackboard bold ) or C (upright bold). In some disciplines such as electromagnetism and electrical engineering , j is used instead of i , as i frequently represents electric current , and complex numbers are written as a + bj or a + jb . Two complex numbers a = x + y i {\displaystyle a=x+yi} and b = u + v i {\displaystyle b=u+vi} are added by separately adding their real and imaginary parts. That
6096-449: Is detailed further below. There are various proofs of this theorem, by either analytic methods such as Liouville's theorem , or topological ones such as the winding number , or a proof combining Galois theory and the fact that any real polynomial of odd degree has at least one real root. The solution in radicals (without trigonometric functions ) of a general cubic equation , when all three of its roots are real numbers, contains
6223-425: Is expected to be literate in them. These include classical mechanics, quantum mechanics, thermodynamics and statistical mechanics , electromagnetism , and special relativity. Classical physics includes the traditional branches and topics that were recognized and well-developed before the beginning of the 20th century—classical mechanics, acoustics , optics , thermodynamics, and electromagnetism. Classical mechanics
6350-429: Is generally concerned with matter and energy on the normal scale of observation, while much of modern physics is concerned with the behavior of matter and energy under extreme conditions or on a very large or very small scale. For example, atomic and nuclear physics study matter on the smallest scale at which chemical elements can be identified. The physics of elementary particles is on an even smaller scale since it
6477-529: Is now a function of a single real parameter u = d − c t {\displaystyle u=d-ct} , that describes the "profile" of the wave, namely the value of the field at time t = 0 {\displaystyle t=0} , for each displacement d = x → ⋅ n → {\displaystyle d={\vec {x}}\cdot {\vec {n}}} . In that case, n → {\displaystyle {\vec {n}}}
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#17328764841436604-593: Is often called the central science because of its role in linking the physical sciences. For example, chemistry studies properties, structures, and reactions of matter (chemistry's focus on the molecular and atomic scale distinguishes it from physics ). Structures are formed because particles exert electrical forces on each other, properties include physical characteristics of given substances, and reactions are bound by laws of physics, like conservation of energy , mass , and charge . Fundamental physics seeks to better explain and understand phenomena in all spheres, without
6731-500: Is possible only in discrete steps proportional to their frequency. This, along with the photoelectric effect and a complete theory predicting discrete energy levels of electron orbitals , led to the theory of quantum mechanics improving on classical physics at very small scales. Quantum mechanics would come to be pioneered by Werner Heisenberg , Erwin Schrödinger and Paul Dirac . From this early work, and work in related fields,
6858-399: Is referred to as the principal value . The argument can be computed from the rectangular form x + yi by means of the arctan (inverse tangent) function. For any complex number z , with absolute value r = | z | {\displaystyle r=|z|} and argument φ {\displaystyle \varphi } , the equation holds. This identity
6985-590: Is referred to as the polar form of z . It is sometimes abbreviated as z = r c i s φ {\textstyle z=r\operatorname {\mathrm {cis} } \varphi } . In electronics , one represents a phasor with amplitude r and phase φ in angle notation : z = r ∠ φ . {\displaystyle z=r\angle \varphi .} If two complex numbers are given in polar form, i.e., z 1 = r 1 (cos φ 1 + i sin φ 1 ) and z 2 = r 2 (cos φ 2 + i sin φ 2 ) ,
7112-440: Is sometimes called " rationalization " of the denominator (although the denominator in the final expression might be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator. The argument of z (sometimes called the "phase" φ ) is the angle of the radius Oz with the positive real axis, and is written as arg z , expressed in radians in this article. The angle
7239-524: Is the distance from the origin to the point representing the complex number z in the complex plane. In particular, the circle of radius one around the origin consists precisely of the numbers z such that | z | = 1 {\displaystyle |z|=1} . If z = x = x + 0 i {\displaystyle z=x=x+0i} is a real number, then | z | = | x | {\displaystyle |z|=|x|} : its absolute value as
7366-831: Is the partial derivative of G {\displaystyle G} with respect to the first argument. The divergence of a vector-valued plane wave depends only on the projection of the vector G ( d , t ) {\displaystyle G(d,t)} in the direction n → {\displaystyle {\vec {n}}} . Specifically, ∇ ⋅ F → ( x → , t ) = n → ⋅ ∂ 1 G ( x → ⋅ n → , t ) {\displaystyle \nabla \cdot {\vec {F}}({\vec {x}},t)\;=\;{\vec {n}}\cdot \partial _{1}G({\vec {x}}\cdot {\vec {n}},t)} In particular,
7493-444: Is the usual (positive) n th root of the positive real number r .) Because sine and cosine are periodic, other integer values of k do not give other values. For any z ≠ 0 {\displaystyle z\neq 0} , there are, in particular n distinct complex n -th roots. For example, there are 4 fourth roots of 1, namely In general there is no natural way of distinguishing one particular complex n th root of
7620-599: Is to say: a + b = ( x + y i ) + ( u + v i ) = ( x + u ) + ( y + v ) i . {\displaystyle a+b=(x+yi)+(u+vi)=(x+u)+(y+v)i.} Similarly, subtraction can be performed as a − b = ( x + y i ) − ( u + v i ) = ( x − u ) + ( y − v ) i . {\displaystyle a-b=(x+yi)-(u+vi)=(x-u)+(y-v)i.} The addition can be geometrically visualized as follows:
7747-431: Is using physics or conducting physics research with the aim of developing new technologies or solving a problem. The approach is similar to that of applied mathematics . Applied physicists use physics in scientific research. For instance, people working on accelerator physics might seek to build better particle detectors for research in theoretical physics. Physics is used heavily in engineering. For example, statics,
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#17328764841437874-527: The Industrial Revolution as energy needs increased. The laws comprising classical physics remain widely used for objects on everyday scales travelling at non-relativistic speeds, since they provide a close approximation in such situations, and theories such as quantum mechanics and the theory of relativity simplify to their classical equivalents at such scales. Inaccuracies in classical mechanics for very small objects and very high velocities led to
8001-636: The Latin physica ('study of nature'), which itself is a borrowing of the Greek φυσική ( phusikḗ 'natural science'), a term derived from φύσις ( phúsis 'origin, nature, property'). Astronomy is one of the oldest natural sciences . Early civilizations dating before 3000 BCE, such as the Sumerians , ancient Egyptians , and the Indus Valley Civilisation , had a predictive knowledge and
8128-590: The Northern Hemisphere . Natural philosophy has its origins in Greece during the Archaic period (650 BCE – 480 BCE), when pre-Socratic philosophers like Thales rejected non-naturalistic explanations for natural phenomena and proclaimed that every event had a natural cause. They proposed ideas verified by reason and observation, and many of their hypotheses proved successful in experiment; for example, atomism
8255-628: The Scientific Revolution in the 17th century, these natural sciences branched into separate research endeavors. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry , and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in these and other academic disciplines such as mathematics and philosophy. Advances in physics often enable new technologies . For example, advances in
8382-601: The Standard Model of particle physics was derived. Following the discovery of a particle with properties consistent with the Higgs boson at CERN in 2012, all fundamental particles predicted by the standard model, and no others, appear to exist; however, physics beyond the Standard Model , with theories such as supersymmetry , is an active area of research. Areas of mathematics in general are important to this field, such as
8509-899: The arctan function can be approximated highly efficiently, formulas like this – known as Machin-like formulas – are used for high-precision approximations of π . The n -th power of a complex number can be computed using de Moivre's formula , which is obtained by repeatedly applying the above formula for the product: z n = z ⋅ ⋯ ⋅ z ⏟ n factors = ( r ( cos φ + i sin φ ) ) n = r n ( cos n φ + i sin n φ ) . {\displaystyle z^{n}=\underbrace {z\cdot \dots \cdot z} _{n{\text{ factors}}}=(r(\cos \varphi +i\sin \varphi ))^{n}=r^{n}\,(\cos n\varphi +i\sin n\varphi ).} For example,
8636-439: The camera obscura (his thousand-year-old version of the pinhole camera ) and delved further into the way the eye itself works. Using the knowledge of previous scholars, he began to explain how light enters the eye. He asserted that the light ray is focused, but the actual explanation of how light projected to the back of the eye had to wait until 1604. His Treatise on Light explained the camera obscura , hundreds of years before
8763-409: The commutative properties (of addition and multiplication) hold. Therefore, the complex numbers form an algebraic structure known as a field , the same way as the rational or real numbers do. The complex conjugate of the complex number z = x + yi is defined as z ¯ = x − y i . {\displaystyle {\overline {z}}=x-yi.} It
8890-579: The empirical world. This is usually combined with the claim that the laws of logic express universal regularities found in the structural features of the world, which may explain the peculiar relation between these fields. Physics uses mathematics to organise and formulate experimental results. From those results, precise or estimated solutions are obtained, or quantitative results, from which new predictions can be made and experimentally confirmed or negated. The results from physics experiments are numerical data, with their units of measure and estimates of
9017-436: The fundamental theorem of algebra , which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed field , where any polynomial equation has a root . Many mathematicians contributed to the development of complex numbers. The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by
9144-422: The ordered pair of real numbers ( ℜ ( z ) , ℑ ( z ) ) {\displaystyle (\Re (z),\Im (z))} , which may be interpreted as coordinates of a point in a Euclidean plane with standard coordinates, which is then called the complex plane or Argand diagram , . The horizontal axis is generally used to display the real part, with increasing values to
9271-539: The standard consensus that the laws of physics are universal and do not change with time, physics can be used to study things that would ordinarily be mired in uncertainty . For example, in the study of the origin of the Earth, a physicist can reasonably model Earth's mass, temperature, and rate of rotation, as a function of time allowing the extrapolation forward or backward in time and so predict future or prior events. It also allows for simulations in engineering that speed up
9398-435: The 16th and 17th centuries, and Isaac Newton 's discovery and unification of the laws of motion and universal gravitation (that would come to bear his name). Newton also developed calculus , the mathematical study of continuous change, which provided new mathematical methods for solving physical problems. The discovery of laws in thermodynamics , chemistry , and electromagnetics resulted from research efforts during
9525-771: The 1st century AD , where in his Stereometrica he considered, apparently in error, the volume of an impossible frustum of a pyramid to arrive at the term 81 − 144 {\displaystyle {\sqrt {81-144}}} in his calculations, which today would simplify to − 63 = 3 i 7 {\displaystyle {\sqrt {-63}}=3i{\sqrt {7}}} . Negative quantities were not conceived of in Hellenistic mathematics and Hero merely replaced it by its positive 144 − 81 = 3 7 . {\displaystyle {\sqrt {144-81}}=3{\sqrt {7}}.} The impetus to study complex numbers as
9652-499: The Italian mathematician Rafael Bombelli . A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton , who extended this abstraction to the theory of quaternions . The earliest fleeting reference to square roots of negative numbers can perhaps be said to occur in the work of the Greek mathematician Hero of Alexandria in
9779-450: The above equation, i was called an imaginary number by René Descartes . For the complex number a + b i {\displaystyle a+bi} , a is called the real part , and b is called the imaginary part . The set of complex numbers is denoted by either of the symbols C {\displaystyle \mathbb {C} } or C . Despite the historical nomenclature, "imaginary" complex numbers have
9906-502: The attacks from invaders and continued to advance various fields of learning, including physics. In the sixth century, Isidore of Miletus created an important compilation of Archimedes ' works that are copied in the Archimedes Palimpsest . In sixth-century Europe John Philoponus , a Byzantine scholar, questioned Aristotle 's teaching of physics and noted its flaws. He introduced the theory of impetus . Aristotle's physics
10033-587: The complex conjugate of a complex number cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division. For any complex number z = x + yi , the product is a non-negative real number. This allows to define the absolute value (or modulus or magnitude ) of z to be the square root | z | = x 2 + y 2 . {\displaystyle |z|={\sqrt {x^{2}+y^{2}}}.} By Pythagoras' theorem , | z | {\displaystyle |z|}
10160-434: The concepts of space, time, and matter from that presented by classical physics. Classical mechanics approximates nature as continuous, while quantum theory is concerned with the discrete nature of many phenomena at the atomic and subatomic level and with the complementary aspects of particles and waves in the description of such phenomena. The theory of relativity is concerned with the description of phenomena that take place in
10287-409: The constant speed predicted by Maxwell's equations of electromagnetism. This discrepancy was corrected by Einstein's theory of special relativity , which replaced classical mechanics for fast-moving bodies and allowed for a constant speed of light. Black-body radiation provided another problem for classical physics, which was corrected when Planck proposed that the excitation of material oscillators
10414-466: The development of a new technology. There is also considerable interdisciplinarity , so many other important fields are influenced by physics (e.g., the fields of econophysics and sociophysics ). Physicists use the scientific method to test the validity of a physical theory . By using a methodical approach to compare the implications of a theory with the conclusions drawn from its related experiments and observations, physicists are better able to test
10541-429: The development of modern physics in the 20th century. Modern physics began in the early 20th century with the work of Max Planck in quantum theory and Albert Einstein 's theory of relativity. Both of these theories came about due to inaccuracies in classical mechanics in certain situations. Classical mechanics predicted that the speed of light depends on the motion of the observer, which could not be resolved with
10668-407: The development of new experiments (and often related equipment). Physicists who work at the interplay of theory and experiment are called phenomenologists , who study complex phenomena observed in experiment and work to relate them to a fundamental theory . Theoretical physics has historically taken inspiration from philosophy; electromagnetism was unified this way. Beyond the known universe,
10795-856: The division of an arbitrary complex number w = u + v i {\displaystyle w=u+vi} by a non-zero complex number z = x + y i {\displaystyle z=x+yi} equals w z = w z ¯ | z | 2 = ( u + v i ) ( x − i y ) x 2 + y 2 = u x + v y x 2 + y 2 + v x − u y x 2 + y 2 i . {\displaystyle {\frac {w}{z}}={\frac {w{\bar {z}}}{|z|^{2}}}={\frac {(u+vi)(x-iy)}{x^{2}+y^{2}}}={\frac {ux+vy}{x^{2}+y^{2}}}+{\frac {vx-uy}{x^{2}+y^{2}}}i.} This process
10922-682: The errors in the measurements. Technologies based on mathematics, like computation have made computational physics an active area of research. Ontology is a prerequisite for physics, but not for mathematics. It means physics is ultimately concerned with descriptions of the real world, while mathematics is concerned with abstract patterns, even beyond the real world. Thus physics statements are synthetic, while mathematical statements are analytic. Mathematics contains hypotheses, while physics contains theories. Mathematics statements have to be only logically true, while predictions of physics statements must match observed and experimental data. The distinction
11049-865: The field of theoretical physics also deals with hypothetical issues, such as parallel universes , a multiverse , and higher dimensions . Theorists invoke these ideas in hopes of solving particular problems with existing theories; they then explore the consequences of these ideas and work toward making testable predictions. Experimental physics expands, and is expanded by, engineering and technology. Experimental physicists who are involved in basic research design and perform experiments with equipment such as particle accelerators and lasers , whereas those involved in applied research often work in industry, developing technologies such as magnetic resonance imaging (MRI) and transistors . Feynman has noted that experimentalists may seek areas that have not been explored well by theorists. Complex numbers In mathematics ,
11176-415: The field. His approach is entirely superseded today. He explained ideas such as motion (and gravity ) with the theory of four elements . Aristotle believed that each of the four classical elements (air, fire, water, earth) had its own natural place. Because of their differing densities, each element will revert to its own specific place in the atmosphere. So, because of their weights, fire would be at
11303-915: The first few powers of the imaginary unit i are i , i 2 = − 1 , i 3 = − i , i 4 = 1 , i 5 = i , … {\displaystyle i,i^{2}=-1,i^{3}=-i,i^{4}=1,i^{5}=i,\dots } . The n n th roots of a complex number z are given by z 1 / n = r n ( cos ( φ + 2 k π n ) + i sin ( φ + 2 k π n ) ) {\displaystyle z^{1/n}={\sqrt[{n}]{r}}\left(\cos \left({\frac {\varphi +2k\pi }{n}}\right)+i\sin \left({\frac {\varphi +2k\pi }{n}}\right)\right)} for 0 ≤ k ≤ n − 1 . (Here r n {\displaystyle {\sqrt[{n}]{r}}}
11430-799: The following de Moivre's formula : ( cos θ + i sin θ ) n = cos n θ + i sin n θ . {\displaystyle (\cos \theta +i\sin \theta )^{n}=\cos n\theta +i\sin n\theta .} In 1748, Euler went further and obtained Euler's formula of complex analysis : e i θ = cos θ + i sin θ {\displaystyle e^{i\theta }=\cos \theta +i\sin \theta } by formally manipulating complex power series and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities. The idea of
11557-400: The latter include such branches as hydrostatics , hydrodynamics and pneumatics . Acoustics is the study of how sound is produced, controlled, transmitted and received. Important modern branches of acoustics include ultrasonics , the study of sound waves of very high frequency beyond the range of human hearing; bioacoustics , the physics of animal calls and hearing, and electroacoustics ,
11684-490: The laws of classical physics accurately describe systems whose important length scales are greater than the atomic scale and whose motions are much slower than the speed of light. Outside of this domain, observations do not match predictions provided by classical mechanics. Einstein contributed the framework of special relativity, which replaced notions of absolute time and space with spacetime and allowed an accurate description of systems whose components have speeds approaching
11811-412: The manipulation of audible sound waves using electronics. Optics, the study of light, is concerned not only with visible light but also with infrared and ultraviolet radiation , which exhibit all of the phenomena of visible light except visibility, e.g., reflection, refraction, interference, diffraction, dispersion, and polarization of light. Heat is a form of energy, the internal energy possessed by
11938-500: The maximum field magnitude seen at the point x → {\displaystyle {\vec {x}}} . A plane wave can be studied by ignoring the directions perpendicular to the direction vector n → {\displaystyle {\vec {n}}} ; that is, by considering the function G ( z , t ) = F ( z n → , t ) {\displaystyle G(z,t)=F(z{\vec {n}},t)} as
12065-696: The modern development of photography. The seven-volume Book of Optics ( Kitab al-Manathir ) influenced thinking across disciplines from the theory of visual perception to the nature of perspective in medieval art, in both the East and the West, for more than 600 years. This included later European scholars and fellow polymaths, from Robert Grosseteste and Leonardo da Vinci to Johannes Kepler . The translation of The Book of Optics had an impact on Europe. From it, later European scholars were able to build devices that replicated those Ibn al-Haytham had built and understand
12192-462: The other Philoponus' criticism of Aristotelian principles of physics served as an inspiration for Galileo Galilei ten centuries later, during the Scientific Revolution . Galileo cited Philoponus substantially in his works when arguing that Aristotelian physics was flawed. In the 1300s Jean Buridan , a teacher in the faculty of arts at the University of Paris , developed the concept of impetus. It
12319-399: The other only on time. A plane standing wave , in particular, can be expressed as F ( x → , t ) = G ( x → ⋅ n → ) S ( t ) {\displaystyle F({\vec {x}},t)=G({\vec {x}}\cdot {\vec {n}})\,S(t)} where G {\displaystyle G} is
12446-459: The other, you will see that the ratio of the times required for the motion does not depend on the ratio of the weights, but that the difference in time is a very small one. And so, if the difference in the weights is not considerable, that is, of one is, let us say, double the other, there will be no difference, or else an imperceptible difference, in time, though the difference in weight is by no means negligible, with one body weighing twice as much as
12573-572: The particles of which a substance is composed; thermodynamics deals with the relationships between heat and other forms of energy. Electricity and magnetism have been studied as a single branch of physics since the intimate connection between them was discovered in the early 19th century; an electric current gives rise to a magnetic field , and a changing magnetic field induces an electric current. Electrostatics deals with electric charges at rest, electrodynamics with moving charges, and magnetostatics with magnetic poles at rest. Classical physics
12700-516: The point x → {\displaystyle {\vec {x}}} along the direction n → {\displaystyle {\vec {n}}} . The displacement is constant over each plane perpendicular to n → {\displaystyle {\vec {n}}} . The values of the field F {\displaystyle F} may be scalars, vectors, or any other physical or mathematical quantity. They can be complex numbers , as in
12827-596: The positions of the planets . According to Asger Aaboe , the origins of Western astronomy can be found in Mesopotamia , and all Western efforts in the exact sciences are descended from late Babylonian astronomy . Egyptian astronomers left monuments showing knowledge of the constellations and the motions of the celestial bodies, while Greek poet Homer wrote of various celestial objects in his Iliad and Odyssey ; later Greek astronomers provided names, which are still used today, for most constellations visible from
12954-1013: The product and division can be computed as z 1 z 2 = r 1 r 2 ( cos ( φ 1 + φ 2 ) + i sin ( φ 1 + φ 2 ) ) . {\displaystyle z_{1}z_{2}=r_{1}r_{2}(\cos(\varphi _{1}+\varphi _{2})+i\sin(\varphi _{1}+\varphi _{2})).} z 1 z 2 = r 1 r 2 ( cos ( φ 1 − φ 2 ) + i sin ( φ 1 − φ 2 ) ) , if z 2 ≠ 0. {\displaystyle {\frac {z_{1}}{z_{2}}}={\frac {r_{1}}{r_{2}}}\left(\cos(\varphi _{1}-\varphi _{2})+i\sin(\varphi _{1}-\varphi _{2})\right),{\text{if }}z_{2}\neq 0.} (These are
13081-568: The real and imaginary part of 5 + 5 i are equal, the argument of that number is 45 degrees, or π /4 (in radian ). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are arctan (1/3) and arctan(1/2), respectively. Thus, the formula π 4 = arctan ( 1 2 ) + arctan ( 1 3 ) {\displaystyle {\frac {\pi }{4}}=\arctan \left({\frac {1}{2}}\right)+\arctan \left({\frac {1}{3}}\right)} holds. As
13208-399: The related entities of energy and force . Physics is one of the most fundamental scientific disciplines. A scientist who specializes in the field of physics is called a physicist . Physics is one of the oldest academic disciplines . Over much of the past two millennia, physics, chemistry , biology , and certain branches of mathematics were a part of natural philosophy , but during
13335-472: The right, and the imaginary part marks the vertical axis, with increasing values upwards. A real number a can be regarded as a complex number a + 0 i , whose imaginary part is 0. A purely imaginary number bi is a complex number 0 + bi , whose real part is zero. As with polynomials, it is common to write a + 0 i = a , 0 + bi = bi , and a + (− b ) i = a − bi ; for example, 3 + (−4) i = 3 − 4 i . The set of all complex numbers
13462-461: The rule i 2 = − 1 {\displaystyle i^{2}=-1} along with the associative , commutative , and distributive laws . Every nonzero complex number has a multiplicative inverse . This makes the complex numbers a field with the real numbers as a subfield. The complex numbers also form a real vector space of dimension two , with { 1 , i } {\displaystyle \{1,i\}} as
13589-520: The rules for complex arithmetic, trying to resolve these issues. The term "imaginary" for these quantities was coined by René Descartes in 1637, who was at pains to stress their unreal nature: ... sometimes only imaginary, that is one can imagine as many as I said in each equation, but sometimes there exists no quantity that matches that which we imagine. [ ... quelquefois seulement imaginaires c'est-à-dire que l'on peut toujours en imaginer autant que j'ai dit en chaque équation, mais qu'il n'y
13716-440: The speed being proportional to the weight and the speed of the object that is falling depends inversely on the density object it is falling through (e.g. density of air). He also stated that, when it comes to violent motion (motion of an object when a force is applied to it by a second object) that the speed that object moves, will only be as fast or strong as the measure of force applied to it. The problem of motion and its causes
13843-412: The speed of light. Planck, Schrödinger, and others introduced quantum mechanics, a probabilistic notion of particles and interactions that allowed an accurate description of atomic and subatomic scales. Later, quantum field theory unified quantum mechanics and special relativity. General relativity allowed for a dynamical, curved spacetime, with which highly massive systems and the large-scale structure of
13970-569: The square roots of negative numbers , a situation that cannot be rectified by factoring aided by the rational root test , if the cubic is irreducible ; this is the so-called casus irreducibilis ("irreducible case"). This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around 1545 in his Ars Magna , though his understanding was rudimentary; moreover, he later described complex numbers as being "as subtle as they are useless". Cardano did use imaginary numbers, but described using them as "mental torture." This
14097-412: The study of probabilities and groups . Physics deals with a wide variety of systems, although certain theories are used by all physicists. Each of these theories was experimentally tested numerous times and found to be an adequate approximation of nature. For instance, the theory of classical mechanics accurately describes the motion of objects, provided they are much larger than atoms and moving at
14224-447: The sum of two complex numbers a and b , interpreted as points in the complex plane, is the point obtained by building a parallelogram from the three vertices O , and the points of the arrows labeled a and b (provided that they are not on a line). Equivalently, calling these points A , B , respectively and the fourth point of the parallelogram X the triangles OAB and XBA are congruent . The product of two complex numbers
14351-444: The top, air underneath fire, then water, then lastly earth. He also stated that when a small amount of one element enters the natural place of another, the less abundant element will automatically go towards its own natural place. For example, if there is a fire on the ground, the flames go up into the air in an attempt to go back into its natural place where it belongs. His laws of motion included: that heavier objects will fall faster,
14478-423: The understanding of electromagnetism , solid-state physics , and nuclear physics led directly to the development of technologies that have transformed modern society, such as television, computers, domestic appliances , and nuclear weapons ; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus . The word physics comes from
14605-423: The universe can be well-described. General relativity has not yet been unified with the other fundamental descriptions; several candidate theories of quantum gravity are being developed. Physics, as with the rest of science, relies on the philosophy of science and its " scientific method " to advance knowledge of the physical world. The scientific method employs a priori and a posteriori reasoning as well as
14732-573: The use of Bayesian inference to measure the validity of a given theory. Study of the philosophical issues surrounding physics, the philosophy of physics , involves issues such as the nature of space and time , determinism , and metaphysical outlooks such as empiricism , naturalism , and realism . Many physicists have written about the philosophical implications of their work, for instance Laplace , who championed causal determinism , and Erwin Schrödinger , who wrote on quantum mechanics. The mathematical physicist Roger Penrose has been called
14859-988: The validity of a theory in a logical, unbiased, and repeatable way. To that end, experiments are performed and observations are made in order to determine the validity or invalidity of a theory. A scientific law is a concise verbal or mathematical statement of a relation that expresses a fundamental principle of some theory, such as Newton's law of universal gravitation. Theorists seek to develop mathematical models that both agree with existing experiments and successfully predict future experimental results, while experimentalists devise and perform experiments to test theoretical predictions and explore new phenomena. Although theory and experiment are developed separately, they strongly affect and depend upon each other. Progress in physics frequently comes about when experimental results defy explanation by existing theories, prompting intense focus on applicable modelling, and when new theories generate experimentally testable predictions , which inspire
14986-777: The value of the field is then the same, and constant in time, at every one of its points. The term is also used, even more specifically, to mean a "monochromatic" or sinusoidal plane wave : a travelling plane wave whose profile G ( u ) {\displaystyle G(u)} is a sinusoidal function. That is, F ( x → , t ) = A sin ( 2 π f ( x → ⋅ n → − c t ) + φ ) {\displaystyle F({\vec {x}},t)=A\sin \left(2\pi f({\vec {x}}\cdot {\vec {n}}-ct)+\varphi \right)} The parameter A {\displaystyle A} , which may be
15113-432: The vertical axis. A complex number can also be defined by its geometric polar coordinates : the radius is called the absolute value of the complex number, while the angle from the positive real axis is called the argument of the complex number. The complex numbers of absolute value one form the unit circle . Adding a fixed complex number to all complex numbers defines a translation in the complex plane, and multiplying by
15240-573: The way vision works. Physics became a separate science when early modern Europeans used experimental and quantitative methods to discover what are now considered to be the laws of physics . Major developments in this period include the replacement of the geocentric model of the Solar System with the heliocentric Copernican model , the laws governing the motion of planetary bodies (determined by Kepler between 1609 and 1619), Galileo's pioneering work on telescopes and observational astronomy in
15367-399: The works of many scientists like Ibn Sahl , Al-Kindi , Ibn al-Haytham , Al-Farisi and Avicenna . The most notable work was The Book of Optics (also known as Kitāb al-Manāẓir), written by Ibn al-Haytham, in which he presented the alternative to the ancient Greek idea about vision. In his Treatise on Light as well as in his Kitāb al-Manāẓir , he presented a study of the phenomenon of
15494-538: Was a step toward the modern ideas of inertia and momentum. Islamic scholarship inherited Aristotelian physics from the Greeks and during the Islamic Golden Age developed it further, especially placing emphasis on observation and a priori reasoning, developing early forms of the scientific method . The most notable innovations under Islamic scholarship were in the field of optics and vision, which came from
15621-450: Was found to be correct approximately 2000 years after it was proposed by Leucippus and his pupil Democritus . During the classical period in Greece (6th, 5th and 4th centuries BCE) and in Hellenistic times , natural philosophy developed along many lines of inquiry. Aristotle ( Greek : Ἀριστοτέλης , Aristotélēs ) (384–322 BCE), a student of Plato , wrote on many subjects, including
15748-417: Was not scrutinized until Philoponus appeared; unlike Aristotle, who based his physics on verbal argument, Philoponus relied on observation. On Aristotle's physics Philoponus wrote: But this is completely erroneous, and our view may be corroborated by actual observation more effectively than by any sort of verbal argument. For if you let fall from the same height two weights of which one is many times as heavy as
15875-427: Was prior to the use of the graphical complex plane. Cardano and other Italian mathematicians, notably Scipione del Ferro , in the 1500s created an algorithm for solving cubic equations which generally had one real solution and two solutions containing an imaginary number. Because they ignored the answers with the imaginary numbers, Cardano found them useless. Work on the problem of general polynomials ultimately led to
16002-415: Was proved later that the use of complex numbers is unavoidable when all three roots are real and distinct. However, the general formula can still be used in this case, with some care to deal with the ambiguity resulting from the existence of three cubic roots for nonzero complex numbers. Rafael Bombelli was the first to address explicitly these seemingly paradoxical solutions of cubic equations and developed
16129-531: Was studied carefully, leading to the philosophical notion of a " prime mover " as the ultimate source of all motion in the world (Book 8 of his treatise Physics ). The Western Roman Empire fell to invaders and internal decay in the fifth century, resulting in a decline in intellectual pursuits in western Europe. By contrast, the Eastern Roman Empire (usually known as the Byzantine Empire ) resisted
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