Constructible number

In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length |r| can be constructed with compass and straightedge in a finite number of steps. Equivalently, r is constructible if and only if there is a closed-form expression for r using only the integers 0 and 1 and the operations for addition, multiplication and square roots; the geometric definition of constructible numbers motivates a corresponding definition of constructible points, which can again be described either geometrically or algebraically. A point is constructible if it can be produced as one of the points of a compass and straight edge construction, starting from a given unit length segment. Alternatively and equivalently, taking the two endpoints of the segments to be the points and of a Cartesian coordinate system, a point is constructible if and only if its Cartesian coordinates are both constructible numbers; the set of constructible numbers forms a field: applying any of the four basic arithmetic operations to members of this set produces another constructible number.

This field is a field extension of the rational numbers and in turn is contained in the field of algebraic numbers. It is the real quadratic closure of the rational numbers, the smallest field extension of the rationals that includes the square roots of all of its positive numbers; the proof of the equivalence between the algebraic and geometric definitions of constructible numbers has the effect of transforming geometric questions about compass and straightedge constructions into algebra. This transformation leads to the solutions of many famous mathematical problems, which defied centuries of attack. Let O and A be two given distinct points in the Euclidean plane, define S to be the set of points that can be constructed with compass and straightedge starting with O and A; the points of S are called constructible points. O and A are, by definition, elements of S. To more describe the remaining elements of S, make the following two definitions: a line segment whose endpoints are in S is called a constructed segment, a circle whose center is in S and which passes through a point of S is called a constructed circle.

The points of S, besides O and A are: the intersection of two non-parallel constructed segments, or lines through constructed segments, the intersection points of a constructed circle and a constructed segment, or line through a constructed segment, or the intersection points of two distinct constructed circles. As an example, the midpoint of constructed segment OA is a constructible point. One construction for it is to construct two circles with OA as radius, the line through the two crossing points of these two circles; the midpoint of segment OA is the point where this segment is crossed by the constructed line. This geometric formulation can be used to define a Cartesian coordinate system in which the point O is associated to the origin having coordinates and in which the point A is associated with the coordinates; the points of S may now be used to link the geometry and algebra by defining a constructible number to be a coordinate of a constructible point. An equivalent definition is that a constructible number is the length of a constructible line segment.

If a constructible number is represented as the x-coordinate of a constructible point P the segment from O to the perpendicular projection of P onto line OA is a constructible line segment with length x. And conversely, if x is the length of a constructible line segment the intersection of line OA and a circle centered at O with radius equal to the length of this segment gives a point whose first Cartesian coordinate is x. Given any two constructible numbers x and y, one can construct the points P = and Q = as above, as the points at distances x and y from O along line OA and its perpendicular axis through O. Then, the point can be constructed as the intersection of two lines perpendicular to the axes through P and Q. Therefore, the constructible points are the points whose Cartesian coordinates are constructible numbers. If a and b are the non-zero lengths of constructed segments elementary compass and straightedge constructions can be used to obtain constructed segments of lengths a + b, a − b, ab and a/b.

The latter two can be done with a construction based on the intercept theorem. A less elementary construction using these tools is based on the geometric mean theorem and will construct a segment of length √a from a constructed segment of length a, it follows from these considerations that the numbers defined as constructible in this geometric way include all of the numbers in the quadratic closure of the rationals. Compass and straightedge constructions for constructible numbers If a and b are constructible numbers with b ≠ 0 a ± b, a×b, a/b, √a, for non-negative a, are constructible. Thus, the set of constructible real numbers form a field. Furthermore, since 1 is a constructible number, all rational numbers are constructible and ℚ is a subfield of the field of constructible numbers. Any constructible number is an algebraic number. More if γ is a constructible real number and γ ∉ ℚ there is a finite sequence of real numbers α1... αn = γ such that ℚ is an extension of ℚ of degree 2. In particular, = 2r for some integer r ≥ 0.

Using different terminology, a real number is constructible if and only if it lies in a field at the top of a finite tower of quadratic extensions, starting with the rational field ℚ

Induction generator

An induction generator or asynchronous generator is a type of alternating current electrical generator that uses the principles of induction motors to produce electric power. Induction generators operate by mechanically turning their rotors faster than synchronous speed. A regular AC induction motor can be used as a generator, without any internal modifications. Induction generators are useful in applications such as mini hydro power plants, wind turbines, or in reducing high-pressure gas streams to lower pressure, because they can recover energy with simple controls. An induction generator draws its excitation power from an electrical grid; because of this, induction generators cannot black start a de-energized distribution system. Sometimes, they are self-excited by using phase-correcting capacitors. An induction generator produces electrical power when its rotor is turned faster than the synchronous speed. For a typical four-pole motor operating on a 60 Hz electrical grid, the synchronous speed is 1800 rotations per minute.

The same four-pole motor operating on a 50 Hz grid will have a synchronous speed of 1500 RPM. The motor turns slower than the synchronous speed. For example, a motor operating at 1450 RPM that has a synchronous speed of 1500 RPM is running at a slip of +3.3%. In normal motor operation, the stator flux rotation is faster than the rotor rotation; this causes the stator flux to induce rotor currents, which create a rotor flux with magnetic polarity opposite to stator. In this way, the rotor is dragged along behind stator flux, with the currents in the rotor induced at the slip frequency. In generator operation, a prime mover drives the rotor above the synchronous speed; the stator flux still induces currents in the rotor, but since the opposing rotor flux is now cutting the stator coils, an active current is produced in stator coils and the motor now operates as a generator, sending power back to the electrical grid. An induction machine requires an externally-supplied armature current; because the rotor field always lags behind the stator field, the induction machine always consumes reactive power, regardless of whether it is operating as a generator or a motor.

A source of excitation current for magnetizing flux for the stator is still required, to induce rotor current. This can be supplied from the electrical grid or, once it starts producing power, from the generator itself; the generating mode for induction motors is complicated by the need to excite the rotor, which begins with only residual magnetization. In some cases, that residual magnetization is enough to self-excite the motor under load. Therefore, it is necessary to either snap the motor and connect it momentarily to a live grid or to add capacitors charged by residual magnetism and providing the required reactive power during operation. Similar is the operation of the induction motor in parallel with a synchronous motor serving as a power factor compensator. A feature in the generator mode in parallel to the grid is that the rotor speed is higher than in the driving mode. Active energy is being given to the grid. Another disadvantage of induction motor generator is that it consumes a significant magnetizing current I0 = %.

An induction machine can be started by charging the capacitors, with a DC source, while the generator is turning at or above generating speeds. Once the DC source is removed the capacitors will provide the magnetization current required to begin producing voltage. An induction machine, operating may spontaneously produce voltage and current due to residual magnetism left in the core. Active power delivered to the line is proportional to slip above the synchronous speed. Full rated power of the generator is reached at small slip values. At synchronous speed of 1800 rpm, generator will produce no power; when the driving speed is increased to 1860 rpm, full output power is produced. If the prime mover is unable to produce enough power to drive the generator, speed will remain somewhere between 1800 and 1860 rpm range. A capacitor bank must supply reactive power to the motor; the reactive power supplied should be equal or greater than the reactive power that the machine draws when operating as a motor.

The basic fundamental of induction generators is the conversion from mechanical energy to electrical energy. This requires an external torque applied to the rotor to turn it faster than the synchronous speed. However, indefinitely increasing torque doesn't lead to an indefinite increase in power generation; the rotating magnetic field torque excited from the armature works to counter the motion of the rotor and prevent over speed because of induced motion in the opposite direction. As the speed of the motor increases the counter torque reaches a max value of torque that it can operate until before the operating conditions become unstable. Ideally, induction generators work best in the stable region between the no-load condition and maximum torque region; the maximum power that can be produced by an induction motor operated as a generator is limited by the rated current of the machine's windings. In induction generators, the reactive power required to establish the air gap magnetic flux is provided by a capacitor bank connected to the machine in case of stand-alone system and in case of grid connection it draws reactive power from the grid to maintain its air gap flux.

For a grid-connected system and voltage at the


Normalsi is a Polish rock band from Łódź formed in 1999 by a group of friends. Their music can be described as a blend of classical rock and grunge with poetic lyrics touching upon existential problems; the name comes from the title of David Gilbert's novel The Normals that tells a story of a young university graduate, Billy Schine, who decides to take part in an experimental testing of a new anti-psychotic drug in order to earn some money needed to pay his student loan. Gilbert shows the change in the behaviour of the normal, that is, the physically healthy, mentally stable, sober people, under the influences of the drug and constant testing, become everything but normal; the founders of the band, Piotr Pachulski and Adam Marszałkowski, made their first musical steps in Colorado Band. The music of that band oscillated between rock and blues. Soon, however and Marszałkowski decided to leave the group and create a new project closer to their musical inclinations; as a result, in 1999 Normalsi came into being.

Shortly afterwards, the band became an essential part of the rock underground in Łódź. After a series of difficulties with the publisher, six years after the establishment of the band, Normalsi released their first album entitled Soliloquium; the whole album is an internal monologue of an artist – a musician at the edge of sanity, unable to find himself in the world where people are numb and indifferent. It is a story of a lonely man who tries to preserve his sanity while being entangled in the web of contradictory feelings; when everything fails, the music seems to be the only antidote to his madness. A year after the release of the band’s debut, Normalsi prepared the second album entitled Dekalog, czyli piekło muzykantów; this time it is a concept album based on the Decalogue, in which each song is an interpretation of one of the Ten Commandments. The album can be seen as a continuation of the themes taken up in Soliloquium at both lyrical and musical level. Not only does it present direct and intimate conversations with God but some songs feature stories of such biblical characters as Pontius Pilate, Matthew the Evangelist, or Cain and Abel.

In Dekalog… Normalsi do not instruct how to live and deal with everyday problems – they only ask questions which the listeners have to answer themselves. The third album of the band, entitled Pokój z widokiem na wojnę, by similarity to their two previous albums, proves that Normalsi consciously decided not to change their musical style. Despite the lack of musical surprises, the album is not monotonous. Once again the listeners are challenged by the band that does not allow them to stay indifferent and forces them to take their stand on the messages conveyed in the songs. Current members Piotr "Chypis" Pachulski – lead vocalist, lyrics Mirek "Koniu" Mazurczyk – guitar Marcin "Rittus" Ritter – bass Former members Adam "Marszałek" Marszałkowski - drums Marcin "Rittus" Ritter – bass Krzysztof Szewczyk – drums Studio albums Soliloquium Dekalog, czyli piekło muzykantów Pokój z widokiem na wojnę Myspace Facebook