1.
Electromagnetism
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Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles. The electromagnetic force usually exhibits electromagnetic fields such as fields, magnetic fields. The other three fundamental interactions are the interaction, the weak interaction, and gravitation. The word electromagnetism is a form of two Greek terms, ἤλεκτρον, ēlektron, amber, and μαγνῆτις λίθος magnētis lithos, which means magnesian stone. The electromagnetic force plays a role in determining the internal properties of most objects encountered in daily life. Ordinary matter takes its form as a result of forces between individual atoms and molecules in matter, and is a manifestation of the electromagnetic force. Electrons are bound by the force to atomic nuclei, and their orbital shapes. The electromagnetic force governs the processes involved in chemistry, which arise from interactions between the electrons of neighboring atoms, there are numerous mathematical descriptions of the electromagnetic field. In classical electrodynamics, electric fields are described as electric potential, although electromagnetism is considered one of the four fundamental forces, at high energy the weak force and electromagnetic force are unified as a single electroweak force. In the history of the universe, during the epoch the unified force broke into the two separate forces as the universe cooled. Originally, electricity and magnetism were considered to be two separate forces, Magnetic poles attract or repel one another in a manner similar to positive and negative charges and always exist as pairs, every north pole is yoked to a south pole. An electric current inside a wire creates a corresponding magnetic field outside the wire. Its direction depends on the direction of the current in the wire. A current is induced in a loop of wire when it is moved toward or away from a field, or a magnet is moved towards or away from it. While preparing for a lecture on 21 April 1820, Hans Christian Ørsted made a surprising observation. As he was setting up his materials, he noticed a compass needle deflected away from north when the electric current from the battery he was using was switched on. At the time of discovery, Ørsted did not suggest any explanation of the phenomenon. However, three later he began more intensive investigations
2.
International System of Units
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The International System of Units is the modern form of the metric system, and is the most widely used system of measurement. It comprises a coherent system of units of measurement built on seven base units, the system also establishes a set of twenty prefixes to the unit names and unit symbols that may be used when specifying multiples and fractions of the units. The system was published in 1960 as the result of an initiative began in 1948. It is based on the system of units rather than any variant of the centimetre-gram-second system. The motivation for the development of the SI was the diversity of units that had sprung up within the CGS systems, the International System of Units has been adopted by most developed countries, however, the adoption has not been universal in all English-speaking countries. The metric system was first implemented during the French Revolution with just the metre and kilogram as standards of length, in the 1830s Carl Friedrich Gauss laid the foundations for a coherent system based on length, mass, and time. In the 1860s a group working under the auspices of the British Association for the Advancement of Science formulated the requirement for a coherent system of units with base units and derived units. Meanwhile, in 1875, the Treaty of the Metre passed responsibility for verification of the kilogram, in 1921, the Treaty was extended to include all physical quantities including electrical units originally defined in 1893. The units associated with these quantities were the metre, kilogram, second, ampere, kelvin, in 1971, a seventh base quantity, amount of substance represented by the mole, was added to the definition of SI. On 11 July 1792, the proposed the names metre, are, litre and grave for the units of length, area, capacity. The committee also proposed that multiples and submultiples of these units were to be denoted by decimal-based prefixes such as centi for a hundredth, on 10 December 1799, the law by which the metric system was to be definitively adopted in France was passed. Prior to this, the strength of the magnetic field had only been described in relative terms. The technique used by Gauss was to equate the torque induced on a magnet of known mass by the earth’s magnetic field with the torque induced on an equivalent system under gravity. The resultant calculations enabled him to assign dimensions based on mass, length, a French-inspired initiative for international cooperation in metrology led to the signing in 1875 of the Metre Convention. Initially the convention only covered standards for the metre and the kilogram, one of each was selected at random to become the International prototype metre and International prototype kilogram that replaced the mètre des Archives and kilogramme des Archives respectively. Each member state was entitled to one of each of the prototypes to serve as the national prototype for that country. Initially its prime purpose was a periodic recalibration of national prototype metres. The official language of the Metre Convention is French and the version of all official documents published by or on behalf of the CGPM is the French-language version
3.
Superconductivity
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It was discovered by Dutch physicist Heike Kamerlingh Onnes on April 8,1911, in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a mechanical phenomenon. It is characterized by the Meissner effect, the ejection of magnetic field lines from the interior of the superconductor as it transitions into the superconducting state. The occurrence of the Meissner effect indicates that superconductivity cannot be simply as the idealization of perfect conductivity in classical physics. The electrical resistance of a metallic conductor decreases gradually as temperature is lowered, in ordinary conductors, such as copper or silver, this decrease is limited by impurities and other defects. Even near absolute zero, a sample of a normal conductor shows some resistance. In a superconductor, the resistance drops abruptly to zero when the material is cooled below its critical temperature, an electric current flowing through a loop of superconducting wire can persist indefinitely with no power source. In 1986, it was discovered that some cuprate-perovskite ceramic materials have a temperature above 90 K. Such a high temperature is theoretically impossible for a conventional superconductor. There are many criteria by which superconductors are classified, by theory of operation, It is conventional if it can be explained by the BCS theory or its derivatives, or unconventional, otherwise. By material, Superconductor material classes include chemical elements, alloys, ceramics, on the other hand, there is a class of properties that are independent of the underlying material. For instance, all superconductors have exactly zero resistivity to low applied currents when there is no magnetic field present or if the field does not exceed a critical value. The resistance of the sample is given by Ohms law as R = V / I, if the voltage is zero, this means that the resistance is zero. Superconductors are also able to maintain a current with no applied voltage whatsoever, experiments have demonstrated that currents in superconducting coils can persist for years without any measurable degradation. Experimental evidence points to a current lifetime of at least 100,000 years, theoretical estimates for the lifetime of a persistent current can exceed the estimated lifetime of the universe, depending on the wire geometry and the temperature. In a normal conductor, an electric current may be visualized as a fluid of electrons moving across an ionic lattice. As a result, the energy carried by the current is constantly being dissipated and this is the phenomenon of electrical resistance and Joule heating. The situation is different in a superconductor, in a conventional superconductor, the electronic fluid cannot be resolved into individual electrons
4.
Crystal
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A crystal or crystalline solid is a solid material whose constituents are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macroscopic single crystals are usually identifiable by their geometrical shape, the scientific study of crystals and crystal formation is known as crystallography. The process of crystal formation via mechanisms of crystal growth is called crystallization or solidification, the word crystal derives from the Ancient Greek word κρύσταλλος, meaning both ice and rock crystal, from κρύος, icy cold, frost. Examples of large crystals include snowflakes, diamonds, and table salt, most inorganic solids are not crystals but polycrystals, i. e. many microscopic crystals fused together into a single solid. Examples of polycrystals include most metals, rocks, ceramics, a third category of solids is amorphous solids, where the atoms have no periodic structure whatsoever. Examples of amorphous solids include glass, wax, and many plastics, Crystals are often used in pseudoscientific practices such as crystal therapy, and, along with gemstones, are sometimes associated with spellwork in Wiccan beliefs and related religious movements. The scientific definition of a crystal is based on the arrangement of atoms inside it. A crystal is a solid where the form a periodic arrangement. For example, when liquid water starts freezing, the change begins with small ice crystals that grow until they fuse. Most macroscopic inorganic solids are polycrystalline, including almost all metals, ceramics, ice, rocks, solids that are neither crystalline nor polycrystalline, such as glass, are called amorphous solids, also called glassy, vitreous, or noncrystalline. These have no periodic order, even microscopically, there are distinct differences between crystalline solids and amorphous solids, most notably, the process of forming a glass does not release the latent heat of fusion, but forming a crystal does. A crystal structure is characterized by its cell, a small imaginary box containing one or more atoms in a specific spatial arrangement. The unit cells are stacked in three-dimensional space to form the crystal, the symmetry of a crystal is constrained by the requirement that the unit cells stack perfectly with no gaps. There are 219 possible crystal symmetries, called space groups. These are grouped into 7 crystal systems, such as cubic crystal system or hexagonal crystal system, Crystals are commonly recognized by their shape, consisting of flat faces with sharp angles. Euhedral crystals are those with obvious, well-formed flat faces, anhedral crystals do not, usually because the crystal is one grain in a polycrystalline solid. The flat faces of a crystal are oriented in a specific way relative to the underlying atomic arrangement of the crystal. This occurs because some surface orientations are more stable than others, as a crystal grows, new atoms attach easily to the rougher and less stable parts of the surface, but less easily to the flat, stable surfaces
5.
Alternating current
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Alternating current, is an electric current which periodically reverses direction, whereas direct current flows only in one direction. A common source of DC power is a cell in a flashlight. The abbreviations AC and DC are often used to mean simply alternating and direct, the usual waveform of alternating current in most electric power circuits is a sine wave. In certain applications, different waveforms are used, such as triangular or square waves, audio and radio signals carried on electrical wires are also examples of alternating current. These types of alternating current carry information encoded onto the AC signal and these currents typically alternate at higher frequencies than those used in power transmission. Electrical energy is distributed as alternating current because AC voltage may be increased or decreased with a transformer, use of a higher voltage leads to significantly more efficient transmission of power. The power losses in a conductor are a product of the square of the current and this means that when transmitting a fixed power on a given wire, if the current is halved, the power loss will be four times less. Power is often transmitted at hundreds of kilovolts, and transformed to 100–240 volts for domestic use, high voltages have disadvantages, such as the increased insulation required, and generally increased difficulty in their safe handling. In a power plant, energy is generated at a convenient voltage for the design of a generator, near the loads, the transmission voltage is stepped down to the voltages used by equipment. Consumer voltages vary somewhat depending on the country and size of load, the voltage delivered to equipment such as lighting and motor loads is standardized, with an allowable range of voltage over which equipment is expected to operate. Standard power utilization voltages and percentage tolerance vary in the different mains power systems found in the world, high-voltage direct-current electric power transmission systems have become more viable as technology has provided efficient means of changing the voltage of DC power. HVDC systems, however, tend to be expensive and less efficient over shorter distances than transformers. Three-phase electrical generation is very common, the simplest way is to use three separate coils in the generator stator, physically offset by an angle of 120° to each other. Three current waveforms are produced that are equal in magnitude and 120° out of phase to each other, if coils are added opposite to these, they generate the same phases with reverse polarity and so can be simply wired together. In practice, higher pole orders are commonly used, for example, a 12-pole machine would have 36 coils. The advantage is that lower rotational speeds can be used to generate the same frequency, for example, a 2-pole machine running at 3600 rpm and a 12-pole machine running at 600 rpm produce the same frequency, the lower speed is preferable for larger machines. If the load on a system is balanced equally among the phases. Even in the worst-case unbalanced load, the current will not exceed the highest of the phase currents
6.
Temperature
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A temperature is an objective comparative measurement of hot or cold. It is measured by a thermometer, several scales and units exist for measuring temperature, the most common being Celsius, Fahrenheit, and, especially in science, Kelvin. Absolute zero is denoted as 0 K on the Kelvin scale, −273.15 °C on the Celsius scale, the kinetic theory offers a valuable but limited account of the behavior of the materials of macroscopic bodies, especially of fluids. Temperature is important in all fields of science including physics, geology, chemistry, atmospheric sciences, medicine. The Celsius scale is used for temperature measurements in most of the world. Because of the 100 degree interval, it is called a centigrade scale.15, the United States commonly uses the Fahrenheit scale, on which water freezes at 32°F and boils at 212°F at sea-level atmospheric pressure. Many scientific measurements use the Kelvin temperature scale, named in honor of the Scottish physicist who first defined it and it is a thermodynamic or absolute temperature scale. Its zero point, 0K, is defined to coincide with the coldest physically-possible temperature and its degrees are defined through thermodynamics. The temperature of zero occurs at 0K = −273. 15°C. For historical reasons, the triple point temperature of water is fixed at 273.16 units of the measurement increment, Temperature is one of the principal quantities in the study of thermodynamics. There is a variety of kinds of temperature scale and it may be convenient to classify them as empirically and theoretically based. Empirical temperature scales are historically older, while theoretically based scales arose in the middle of the nineteenth century, empirically based temperature scales rely directly on measurements of simple physical properties of materials. For example, the length of a column of mercury, confined in a capillary tube, is dependent largely on temperature. Such scales are only within convenient ranges of temperature. For example, above the point of mercury, a mercury-in-glass thermometer is impracticable. A material is of no use as a thermometer near one of its phase-change temperatures, in spite of these restrictions, most generally used practical thermometers are of the empirically based kind. Especially, it was used for calorimetry, which contributed greatly to the discovery of thermodynamics, nevertheless, empirical thermometry has serious drawbacks when judged as a basis for theoretical physics. Theoretically based temperature scales are based directly on theoretical arguments, especially those of thermodynamics, kinetic theory and they rely on theoretical properties of idealized devices and materials
7.
Paramagnetism
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In contrast with this behavior, diamagnetic materials are repelled by magnetic fields and form induced magnetic fields in the direction opposite to that of the applied magnetic field. Paramagnetic materials include most chemical elements and some compounds, they have a magnetic permeability greater than or equal to 1. The magnetic moment induced by the field is linear in the field strength. It typically requires a sensitive balance to detect the effect. Paramagnetic materials have a small, positive susceptibility to magnetic fields and these materials are slightly attracted by a magnetic field and the material does not retain the magnetic properties when the external field is removed. Paramagnetic properties are due to the presence of unpaired electrons. Paramagnetic materials include magnesium, molybdenum, lithium, and tantalum, unlike ferromagnets, paramagnets do not retain any magnetization in the absence of an externally applied magnetic field because thermal motion randomizes the spin orientations. Thus the total magnetization drops to zero when the field is removed. Even in the presence of the field there is only a small induced magnetization because only a fraction of the spins will be oriented by the field. This fraction is proportional to the strength and this explains the linear dependency. Constituent atoms or molecules of paramagnetic materials have permanent magnetic moments, the permanent moment generally is due to the spin of unpaired electrons in atomic or molecular electron orbitals. In pure paramagnetism, the dipoles do not interact with one another and are oriented in the absence of an external field due to thermal agitation. When a magnetic field is applied, the dipoles will tend to align with the applied field, however, the true origins of the alignment can only be understood via the quantum-mechanical properties of spin and angular momentum. Paramagnetic behavior can also be observed in materials that are above their Curie temperature. At these temperatures, the thermal energy simply overcomes the interaction energy between the spins. In conductive materials the electrons are delocalized, that is, they travel through the more or less as free electrons. Conductivity can be understood in a band structure picture as arising from the filling of energy bands. In an ordinary nonmagnetic conductor the band is identical for both spin-up and spin-down electrons
8.
Pressure
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Pressure is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure is the relative to the ambient pressure. Various units are used to express pressure, Pressure may also be expressed in terms of standard atmospheric pressure, the atmosphere is equal to this pressure and the torr is defined as 1⁄760 of this. Manometric units such as the centimetre of water, millimetre of mercury, Pressure is the amount of force acting per unit area. The symbol for it is p or P, the IUPAC recommendation for pressure is a lower-case p. However, upper-case P is widely used. The usage of P vs p depends upon the field in one is working, on the nearby presence of other symbols for quantities such as power and momentum. Mathematically, p = F A where, p is the pressure, F is the normal force and it relates the vector surface element with the normal force acting on it. It is incorrect to say the pressure is directed in such or such direction, the pressure, as a scalar, has no direction. The force given by the relationship to the quantity has a direction. If we change the orientation of the element, the direction of the normal force changes accordingly. Pressure is distributed to solid boundaries or across arbitrary sections of normal to these boundaries or sections at every point. It is a parameter in thermodynamics, and it is conjugate to volume. The SI unit for pressure is the pascal, equal to one newton per square metre and this name for the unit was added in 1971, before that, pressure in SI was expressed simply in newtons per square metre. Other units of pressure, such as pounds per square inch, the CGS unit of pressure is the barye, equal to 1 dyn·cm−2 or 0.1 Pa. Pressure is sometimes expressed in grams-force or kilograms-force per square centimetre, but using the names kilogram, gram, kilogram-force, or gram-force as units of force is expressly forbidden in SI. The technical atmosphere is 1 kgf/cm2, since a system under pressure has potential to perform work on its surroundings, pressure is a measure of potential energy stored per unit volume. It is therefore related to density and may be expressed in units such as joules per cubic metre. Similar pressures are given in kilopascals in most other fields, where the prefix is rarely used
9.
Nuclear magnetic resonance
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Nuclear magnetic resonance is a physical phenomenon in which nuclei in a magnetic field absorb and re-emit electromagnetic radiation. NMR allows the observation of quantum mechanical magnetic properties of the atomic nucleus. Many scientific techniques exploit NMR phenomena to study physics, crystals. NMR is also used in advanced medical imaging techniques, such as in magnetic resonance imaging. The most commonly studied nuclei are 1H and 13C, although nuclei from isotopes of other elements have been studied by high-field NMR spectroscopy as well. A key feature of NMR is that the frequency of a particular substance is directly proportional to the strength of the applied magnetic field. Since the resolution of the technique depends on the magnitude of magnetic field gradient, many efforts are made to develop increased field strength. The effectiveness of NMR can also be improved using hyperpolarization, and/or using two-dimensional, three-dimensional and higher-dimensional multi-frequency techniques, the principle of NMR usually involves two sequential steps, The alignment of the magnetic nuclear spins in an applied, constant magnetic field B0. The perturbation of this alignment of the nuclear spins by employing an electro-magnetic, the required perturbing frequency is dependent upon the static magnetic field and the nuclei of observation. The two fields are chosen to be perpendicular to each other as this maximizes the NMR signal strength. The resulting response by the magnetization of the nuclear spins is the phenomenon that is exploited in NMR spectroscopy. NMR phenomena are also utilized in low-field NMR, NMR spectroscopy and MRI in the Earths magnetic field, in 1946, Felix Bloch and Edward Mills Purcell expanded the technique for use on liquids and solids, for which they shared the Nobel Prize in Physics in 1952. Yevgeny Zavoisky likely observed nuclear magnetic resonance in 1941, well before Felix Bloch and Edward Mills Purcell, russell H. Varian filed the Method and means for correlating nuclear properties of atoms and magnetic fields, U. S. Patent 2,561,490 on July 24,1951, Varian Associates developed the first NMR unit called NMR HR-30 in 1952. Purcell had worked on the development of radar during World War II at the Massachusetts Institute of Technologys Radiation Laboratory. His work during that project on the production and detection of radio frequency power, when this absorption occurs, the nucleus is described as being in resonance. Different atomic nuclei within a molecule resonate at different frequencies for the magnetic field strength. The observation of magnetic resonance frequencies of the nuclei present in a molecule allows any trained user to discover essential chemical and structural information about the molecule
10.
Diamagnetism
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Diamagnetic materials are repelled by a magnetic field, an applied magnetic field creates an induced magnetic field in them in the opposite direction, causing a repulsive force. In contrast, paramagnetic and ferromagnetic materials are attracted by a magnetic field, diamagnetism is a quantum mechanical effect that occurs in all materials, when it is the only contribution to the magnetism the material is called diamagnetic. In paramagnetic and ferromagnetic substances the weak force is overcome by the attractive force of magnetic dipoles in the material. The magnetic permeability of diamagnetic materials is less than μ0, the permeability of vacuum, diamagnetism was first discovered when Sebald Justinus Brugmans observed in 1778 that bismuth and antimony were repelled by magnetic fields. In 1845, Michael Faraday demonstrated that it was a property of matter and he adopted the term diamagnetism after it was suggested to him by William Whewell. Diamagnetism, to a greater or lesser degree, is a property of all materials, for materials that show some other form of magnetism, the diamagnetic contribution becomes negligible. Substances that mostly display diamagnetic behaviour are termed diamagnetic materials, or diamagnets, the magnetic susceptibility values of various molecular fragments are called Pascals constants. This means that diamagnetic materials are repelled by magnetic fields, however, since diamagnetism is such a weak property, its effects are not observable in everyday life. For example, the susceptibility of diamagnets such as water is χv = −9. 05×10−6. The most strongly diamagnetic material is bismuth, χv = −1. 66×10−4, nevertheless, these values are orders of magnitude smaller than the magnetism exhibited by paramagnets and ferromagnets. Note that because χv is derived from the ratio of the magnetic field to the applied field. All conductors exhibit an effective diamagnetism when they experience a magnetic field. The Lorentz force on electrons causes them to circulate around forming eddy currents, the eddy currents then produce an induced magnetic field opposite the applied field, resisting the conductors motion. Superconductors may be considered perfect diamagnets, because they expel all fields due to the Meissner effect, however this effect is not due to eddy currents, as in ordinary diamagnetic materials. If a powerful magnet is covered with a layer of water then the field of the magnet significantly repels the water and this causes a slight dimple in the waters surface that may be seen by its reflection. Diamagnets may be levitated in stable equilibrium in a magnetic field, Earnshaws theorem seems to preclude the possibility of static magnetic levitation. However, Earnshaws theorem applies only to objects with positive susceptibilities and these are attracted to field maxima, which do not exist in free space. Diamagnets are attracted to field minima, and there can be a minimum in free space
11.
Magnetic field
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A magnetic field is the magnetic effect of electric currents and magnetic materials. The magnetic field at any point is specified by both a direction and a magnitude, as such it is represented by a vector field. The term is used for two distinct but closely related fields denoted by the symbols B and H, where H is measured in units of amperes per meter in the SI, B is measured in teslas and newtons per meter per ampere in the SI. B is most commonly defined in terms of the Lorentz force it exerts on moving electric charges, Magnetic fields can be produced by moving electric charges and the intrinsic magnetic moments of elementary particles associated with a fundamental quantum property, their spin. In quantum physics, the field is quantized and electromagnetic interactions result from the exchange of photons. Magnetic fields are used throughout modern technology, particularly in electrical engineering. The Earth produces its own field, which is important in navigation. Rotating magnetic fields are used in electric motors and generators. Magnetic forces give information about the carriers in a material through the Hall effect. The interaction of magnetic fields in electric devices such as transformers is studied in the discipline of magnetic circuits, noting that the resulting field lines crossed at two points he named those points poles in analogy to Earths poles. He also clearly articulated the principle that magnets always have both a north and south pole, no matter how finely one slices them, almost three centuries later, William Gilbert of Colchester replicated Petrus Peregrinus work and was the first to state explicitly that Earth is a magnet. Published in 1600, Gilberts work, De Magnete, helped to establish magnetism as a science, in 1750, John Michell stated that magnetic poles attract and repel in accordance with an inverse square law. Charles-Augustin de Coulomb experimentally verified this in 1785 and stated explicitly that the north and south poles cannot be separated, building on this force between poles, Siméon Denis Poisson created the first successful model of the magnetic field, which he presented in 1824. In this model, a magnetic H-field is produced by magnetic poles, three discoveries challenged this foundation of magnetism, though. First, in 1819, Hans Christian Ørsted discovered that an electric current generates a magnetic field encircling it, then in 1820, André-Marie Ampère showed that parallel wires having currents in the same direction attract one another. Finally, Jean-Baptiste Biot and Félix Savart discovered the Biot–Savart law in 1820, extending these experiments, Ampère published his own successful model of magnetism in 1825. This has the benefit of explaining why magnetic charge can not be isolated. Also in this work, Ampère introduced the term electrodynamics to describe the relationship between electricity and magnetism, in 1831, Michael Faraday discovered electromagnetic induction when he found that a changing magnetic field generates an encircling electric field
12.
Complex number
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A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, 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 in the complex plane, a complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a number a + 0i whose imaginary part is 0