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Algebra for dummies free
Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts") is one of the broad parts of
mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of
mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics.
It includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields.
The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra.
Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such
applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily
by professional mathematicians.
Algebra is the language that the field of mathematics uses to talk about abstract world numbers.
one or more v values of the variable are usually solutions to algebraic equation. Algebra was first combined with geometry by Rene Descates to study coordinate geometry. This was a revolution in mathematics and science. Coordinates are levels for points in two or more dimensional space.
RamanujanRamanujan was an Indian born mathematician and algebraist. He developed many theorems of algebra. One of his discoveries is the formula of number three(3).
He was invited to Oxford from great mathematician Tomas Hardy. Ramanujan had no idea how he invented the equations. He claimed that he saw those eqations in his dreams. Hardy was very much reluctant to accept all his formulas because those had no proof. Ramanujan was a man of great mystery.
GeneratorsGenerator can have different meaning in different field of study. In mathematics it has a unique definition. Generators are used in group theory. It arises in Lie algebra or Lie group. First let us see what Lie group is.
A Lie group is a group which is also a differentiable manifold. The group operations are smooth.
The rotation matrices form a subgroup of GL(2, R). It is a Lie group under it own right. It is a one-dimensional compact connected Lie group which is diffeomorphic to a circle.
A set of generators is the set of group elements such that possibly repeated application of the generators on themselves and each other is capable of producing all the elements of the same group.
Riemman sphere is simply the complex one dimensional manifold with infinity ∞ added to it.
Simply adjoining an extra point called '1' to the complex plane does not
make it completely clear that the required seamless structure holds in the
neighbourhood of 1, the same as everywhere else. The way that we can
address this issue is to regard the sphere to be constructed from two
'coordinate patches', one of which is the z-plane and the other the
w-plane. All but two points of the sphere are assigned both a z-coordinate
and a w-coordinate (related by the Mo¨bius transformation above). But
one point has only a z-coordinate (where w would be 'infinity') and
another has only a w-coordinate (where z would be '∞'). We use
either z or w or both in order to deWne the needed conformal structure
and, where we use both, we get the same conformal structure using
either, because the relation between the two coordinates is holomorphic.
In fact, for this, we do not need such a complicated transformation between z and w as the general Mo¨bius transformation. It suffices to consider the particularly simple Mo¨bius transformation given by
w = 1/z, z = 1/w ,
where z = 0 and w = 0, would each give 1 in the opposite patch. I have indicated in Figure below how this transformation maps the real and imaginary coordinate lines of z.
Patching the Riemann sphere from the complex z- and w-planes, via
w = 1/z, z = 1/w. (Here, the z grid lines are shown also in the w-plane.) The
overlap regions exclude only the origins, z = 0 and w = 0 each giving '1' in the
All this defines the Riemann sphere in a rather abstract way. We can see more clearly the reason that the Riemann sphere is called a ‘sphere’ by employing the geometry illustrated in Figure below. I have taken the z-plane to represent the equatorial plane of this geometrical sphere. The points of the sphere are mapped to the points of the plane by what is called stereographic projection from the south pole. This just means that I draw a straight line in the Euclidean 3-space (within which we imagine everything to be taking place) from the south pole through the point z in the plane. Where this line meets the sphere again is the point on the sphere that the complex number z represents. There is one additional point on the sphere, namely the south pole itself, and this represents z =1. To see how w fits into this picture, we imagine its complex plane to be inserted upside down (with w = 1, i, -1, -i matching z = 1, -i, -1, i, respectively), and we now project stereographically from the north pole (Figure b). An important and beautiful property of stereographic projection is that it maps circles on the sphere to circles (or straight lines) on the plane.
(a) Riemann sphere as unit sphere whose equator coincides with the
unit circle in z's (horizontal) complex plane. The sphere is projected (stereographically)
to the z-plane along straight lines through its south pole, which itself
gives z=1. (b) Re-interpreting the equatorial plane as the w-plane, depicted
upside down but with the same real axis, the stereographic projection is now
from the north pole (w=1), where w = 1/z. (c) The real axis is a great circle
on this Riemann sphere, like the unit circle but drawn vertically rather than
Plato's mathematical world
This was an extraordinary idea for its time, and it has turned out to be a very powerful one. But does the Platonic mathematical world actually exist, in any meaningful sense? Many people, including philosophers, might regard such a 'world' as a complete fiction—a product merely of our unrestrained imaginations. Yet the Platonic viewpoint is indeed an immensely valuable one. It tells us to be careful to distinguish the precise mathematical entities from the approximations that we see around us in the world of physical things. Moreover, it provides us with the blueprint according to which modern science has proceeded ever since. Scientists will put forward models of the world—or, rather, of certain aspects of the world—and these models may be tested against previous observation and against the results of carefully designed experiment. The models are deemed to be appropriate if they survive such rigorous examination and if, in addition, they are internally consistent structures. The important point about these models, for our present discussion, is that they are basically purely abstract mathematical models. The very question of the internal consistency of a scientific model, in particular, is one that requires that the model be precisely specified. The required precision demands that the model be a mathematical one, for otherwise one cannot be sure that these questions have well-defined answers.If the model itself is to be assigned any kind of 'existence', then this existence is located within the Platonic world of mathematical forms. Of course, one might take a contrary viewpoint: namely that the model is itself to have existence only within our various minds, rather than to take Plato's world to be in any sense absolute and 'real'. Yet, there is something important to be gained in regarding mathematical structures as having a reality of their own. For our individual minds are notoriously imprecise, unreliable, and inconsistent in their judgements. The precision, reliability, and consistency that are required by our scientific theories demand something beyond any one of our individual (untrustworthy) minds.
It may be helpful if I put the case for the actual existence of the Platonic world in a different form. What I mean by this ‘existence’ is really just the objectivity of mathematical truth. Platonic existence, as I see it, refers to the existence of an objective external standard that is not dependent upon our individual opinions nor upon our particular culture. Such ‘existence’ could also refer to things other than mathematics, such as to morality or aesthetics , but I am here concerned just with mathematical objectivity, which seems to be a much clearer issue. Let me illustrate this issue by considering one famous example of a mathematical truth, and relate it to the question of 'objectivity'. In 1637, Pierre de Fermat made his famous assertion now known as ‘Fermat’s Last Theorem’ (that no positive nth power3 of an integer, i.e. of a whole number, can be the sum of two other positive nth powers if n is an integer greater than 2), which he wrote down in the margin of his copy of the Arithmetica, a book written by the 3rd-century Greek mathematician Diophantos. In this margin, Fermat also noted: ‘I have discovered a truly marvellous proof of this, which this margin is too narrow to contain.’ Fermat’s mathematical assertion remained unconWrmed for over 350 years, despite concerted efforts by numerous superb mathematicians. A proof was Wnally published in 1995 by Andrew files (depending on the earlier work of various other mathematicians), and this proof has now been accepted as a valid argument by the mathematical community.
Now, do we take the view that Fermat's assertion was always true, long before Fermat actually made it, or is its validity a purely cultural matter, dependent upon whatever might be the subjective standards of the community of human mathematicians? Let us try to suppose that the validity of the Fermat assertion is in fact a subjective matter. Then it would not be an absurdity for some other mathematician X to have come up with an actual and specific counter-example to the Fermat assertion, so long as X had done this before the date of 1995.4 In such a circumstance, the mathematical community would have to accept the correctness of X’s counter-example. From then on, any effort on the part of Wiles to prove the Fermat assertion would have to be futile, for the reason that X had got his argument in Wrst and, as a result, the Fermat assertion would now be false! Moreover, we could ask the further question as to whether, consequent upon the correctness of X's forthcoming counter-example, Fermat himself would necessarily have been mistaken in believing in the soundness of his 'truly marvellous proof', at the time that he wrote his marginal note. On the subjective view of mathematical truth, it could possibly have been the case that Fermat had a valid proof (which would have been accepted as such by his peers at the time, had he revealed it) and that it was Fermat's secretiveness that allowed the possibility of X later obtaining a counter-example! I think that virtually all mathematicians, irrespective of their professed attitudes to 'Platonism', would regard such possibilities as patently absurd
Of course, it might still be the case that Wiles's argument in fact contains an error and that the Fermat assertion is indeed false. Or there could be a fundamental error in Wiles's argument but the Fermat assertion is true nevertheless. Or it might be that Wiles's argument is correct in its essentials while containing 'non-rigorous steps' that would not be up to the standard of some future rules of mathematical acceptability. But these issues do not address the point that I am getting at here. The issue is the objectivity of the Fermat assertion itself, not whether anyone's particular demonstration of it (or of its negation) might happen to be convincing to the mathematical community of any particular time.
Three 'worlds'— the Platonic mathematical, the physical, and the mental—and the three profound mysteries in the connections between them.
Gauss-Bonnet theoremGauss-Bonnet theorem relates curvature of a topological space with the Euler characterstic. In any given curved space the angles of a figure do not sum up to that of Euclidean space. In that case there is an excess of those angles. This is usually expressed by an indentity as follows:
If M is a compact surface with a Riemannian metric, then Where K =Gauss curvature , X = the Euler characteristic of M and dA=the area measure on determined by the Riemannian metric.
For an triangle the value of X(M) = V -E + F = 2 . This is different for different Euelerian shapes and figures.
Linear programming a method to solve linear inequalities and euations to find the best optimal condition. It is a mathematical method to finf minimum or
maximum of a linear function of several variables , such as output or cost.
Suppose a company produces two commodities of x and y. The total cost must come as x commodity and twice the total cost of y. So the total cost comes as C = x + 2y in a given time. This may serve as our objective function which extremum value we seek. There are other constraints as to how many commodities of x and y can be produced in the mean time simultaneously. So our mathematical argument is as follows:
The shadded area contains all the solutions of the objective function for which the cost is minimum or maximum.
Binomial theoremBionomial theorem was developed by Issac Newton. It is a theorem used to describe the expansion of powers of bionomial. As an example some powers of binomial are :
This can be generalized as :
The coefficient of each term can be computed using pascal triangle :
First row corresponds to n=0; Second row corresponds to n=1 expansion and so on to for all finite value of n. In pascal triangle all the values of a row other than 1's at the end are computed by two numbers just above it in the upper row. Pascal first discovered such a triangle and that is why it is called Pascal's triangle.
Where we assumed first term a = 1 for simplicity. Now when n is any positive integer the series always terminates. But if n is not any positive integer the series does not terminate and expansion becomes infinite. This series can be expressed by another explicit formula of the form quite similar to above:
Some examples are
Process of long divisionThere is a process of dividing a polynomial by another polynomial. This is called long division. This process is quite similar to the division rule of large numbers :
Matrix algebraMatrix algebra is an algebra which concerns with addition and multiplication of matrices. A matrix is a collection of numbers. Determinant of a 3X3 matrix can be calculated in terms of 2X2 matrix determinants.
Another application of matrix is found in solving a system of linear equations.
When we have a system of linear equations of several variables we need to find the determinant of the matrix of the coefficient of the variables.
In the example given above a system of two equations is solved by 2X2 matrix. Similar reasoning can be applied in case of more equations with more unknowns.
Matrix algebra is perhaps the most useful tool in theoretical physics. Tensors are special kind of matrices.
Trace and logarithmDeterminant of a 2X2 matrix is
The lograitm function of a matrix B is
Where I is the identity matrix .
Jacobi formulaIn matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A.
If A is a differentiable map from the real numbers to n × n matrices
By Jacobi formula for any matrix the trace identity hold.
Where adj A denotes the adjugate of matrix A. Adj of a matrix is defined as :
Magnitudes and quantities are different concepts. Magnitude is defined as whatever greater or less than something. Quantity is a particular contained in magnitude. Quantity is an instance of magnitude. What can be greater or less is surely a magnitude. Magnitude is a many-one relation whereas quantity is one-to-one relation. Many terms always belong to the same magnitude. As for example the 5 magnitude earthquake and 3 magnitude earthquake belong to the same kind of magnitude. Where quantity is one to one relation. 5 and 3 in this case are quanities or particular. This is the basic difference between qauntity and magnitude.
Mathematics is like poetry. You can only learn mathematics by doing mathematics. I have done enough mathematics in my life. There is no greater pleasure than doing or understanding mathematics. Besides mathematics is more beautiful than girls and does not cheat with us. Mathematics is probably the only language of saying the same thing in different words. Whatever let us get back to algebra for dummies :
Quadratic irrationalTo understand quadratic irrational we first need to learn what a continued fraction is.
A continued fraction is a finite or infinite expression a + (b + (c + (d + ... )^-11)^-1)^-1, where a, b, c, d, . . . are positive integers
Any rational number can be expressed as a terminating such expression.
To express a real number we allow the continued fraction to run on forever. Some examples are
Each number mentioned above is represented as a continued fraction. In the first two of these infinite examples, the sequences of natural numbers that appear—namely 1, 2, 2, 2, 2, . . . in the first case and 5, 3, 1, 2, 1, 2, 1, 2, . . . in the second—have the property that they are ultimately periodic (the 2 repeating indefinitely in the first case and the sequence 1, 2 repeating indefinitely in the second). Recall that, as already noted above, in the familiar decimal notation, it is the rational numbers that have (Wnite or) ultimately periodic expressions. We may regard it as a strength of the Greek 'continued-fraction' representation, on the other hand, that the rational numbers now always have a Wnite description. A natural question to ask, in this context, is: which numbers have an ultimately periodic continued-fraction representation? It is a remarkable theorem, first proved, to our knowledge, by the great 18thcentury mathematician Joseph C. Lagrange that the numbers whose representation in terms of continued fractions are ultimately periodic are what are called quadratic irrationals.
What is a quadratic irrational and what is its importance for Greek geometry? It is a number that can be written in the form a + √b , where a and b are fractions where b is not a perfect square. In case when a=0 some quadratic irrationals are √2, √3 , √5
Although incorporating the quadratic irrationals gets us some way towards numbers adequate for Euclidean geometry, it does not do all that is needed. In the tenth (and most diYcult) book of Euclid, numbers like √(a + √b);
are considered (with a and b positive rationals). These are not generally quadratic irrationals, but they occur, nevertheless, in ruler-and-compass constructions. Numbers sufficient for such geometric constructions would be those that can be built up from natural numbers by repeated use of the operations of addition, subtraction, multiplication, division, and the taking of square roots. But operating exclusively with such numbers gets extremely complicated, and these numbers are still too limited for considerations of Euclidean geometry that go beyond ruler-and-compass constructions. It is much more satisfactory to take the bold step. This provided the Greeks with a way of describing numbers that does turn out to be adequate for Euclidean geometry.
Ruler - compass construction is a construction involving only drawing straight lines and circles. Another problem involving this ruler-compass construction is that of squaring a cirle. The problem formally states as follows:
Using only a ruler and a compass it is not possible to square a circle or to turn the circle into a square whose perimeter is the same as that of the circumference of the circle.
Discrete numbers in natural worldDiscrete numbers are the natural numbers (0, 1, 2, 3,... ). These numbers are used for counting purpose. There is no other number in between the two consecutive numbers. These numbers are very useful in physical science , especially in physics. In physics ther are certain quantities like , spin which is expressed by these natural numbers. In quantum mechanics various quantum numbers are integer multiple of certain quantities. Positive integers are also natural numbers. Positive integers are 0 , 1, 2, 3, 4,... . These are the same as the natural numbers.
Algebra skillsThere are some basic rules for algebraic manipulations.
Rational functionRational functions can be expressed as a ratio of two algebraic polynomials: one being numerator and other being denominator. Such fractions are called rational. The coefficients of the polynomials need not be rational numbers , they can be taken over any field K.
There is another method partial- fraction decomposition of rational function. It is the process of decomposing a complex rational function into a sum of a polynomial and one or several fractions with simpler denominator.
An example of this decomposition can be given.
Partial fraction decomposition is very usefull in doing integration of complex functions whose usual formula is not known. As an example :
In summary the general rule of decomposition will be :
Improper rational function can be factored like this:
In some cases there are repeated roots. Rules are somewhat different in these cases:
Giving is a great virtue..
Middle term factorMiddle term factoring is a method to decompose a complex polynomial into simple factor expressions. It is like factoring a big number into two or three small numbers:
Interval and inequalityInterval is the set of real numbers lying between two real numbers ( may be inclusive). The concept of interval has many use in measure theory. It can be used to define distance. This use can be seen in theory of relativity. There are many kinds of intervals.
A closed interval is the set of all the real numbers between two numbers including its endpoints which are those two numbers. Whereas the open interval excludes those two endpoints.
A degenerate real number is an interval with only one real number. Various kinds of interval are
A more elaborate description can be given
Measure theoryMeasure theory is related to the interval discussed above. Measure is the function that assigns real numbers to subsets of real numbers.
Measure must satify these properties:
Where l(E) is the interval of set E. There is a theorem of measure as follows:
Vector projection is defined by the usual dot product of two vectors. We can project one vector onto other vector in this way:
With this definition of projection comes Gram-Schmidt law.
We define projection operator by
The Gram–Schmidt process then works as follows:
Vector parallelogram law is the vector addition law. Using this law we can find the resultant of two vectors.
Unit vectorConcept of unit vector is necessary to understand vector operation in curved and Euclidean space. A unit vector is any vector divided by its length.
After defining the unit vector we can represent any vector in the three dimensional space.
Linear algebraEigenvalue can be very useful concept in quantum mechanics problem , especially solving Schrodinger equation
Summary and linear algebraThe most common operations of vector algebra are :
Where v1, v2, v3 and u1, u2, u3 are the components of vector v and u.
Directional cosineDirectional cosines are the angles between the vector and the coordinate axes :
LatticeLattice is basically a concept to be studied in abstract algebra. As abstract algebra is also a kind of algebra , it can be mentioned here. Lattice is any partially ordered set in which every two elements have a unique supremum (greatest lower bound) and a unique minimum (least upper bound). An example can be a natural numbers , partially ordered by divisibility. In this particular lattice the least upper bound is the greatest common divisor and supremum is the least common multiple.
In group theory lattice has a different meaning. In group theory a lattice in R(n) is a subgroup of additive group R(n) which is isomorphic to additive group Z(n). This lattice spans the entire real vector space R(n). In our house , especially on the floor there are sometimes seen special shapes repeated periodically over it. These are called tiling with primitive cells. A lattice can be viewed exactly like the tilings.
Formally a lattice λ in R(n) has the form :
Various lattices in Euclidean space can be visualized intuitively as follows:
When R(n) is n dimensional Euclidean space.
Axiom of ArchimedesAxiom of Archimedes is an important property of real number. Earlier Archimedes found that two magnitudes always have ratio between them so one can find a number particularly an integer which multiplied with the one will exceed the other. The exact formal statement would be :
If x and y are two elements of linearly ordered group then for every natural number n we have
x + x + ..... + x (n times) < y
or nx < y This is so called Archimedean axiom. Real numbers follow this property but the rationals do not. There is an alternative formulation of the same principle.
Let η is a small arbitrary number. Then there is a natural number n such that 1/n < η