What is mapping in pre-algebra?
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(8x+3)(4x+3)\n32x2+24x+24x+9\n32x2+48x+9
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HI: I tried to answer your email but it said that your email address has not been confirmed and so the email didn't go through.\n\nalpha and beta are scalers for the vector space V.  This problem is actually so easy that people don't believe what they've done.  You really just need to show that Span S behaves like V.  \n\nSpan S must be a vector space itself.  Two elements from the span is \nu=a1*v1 + a2*v2...ak*vk\nand \nw=b1*v1 + b2*v2...bk*vk\n\nshow that alpha*u+beta*w is in Span S where alpha, beta are scalers for your vector space.\n\nalpha*u+beta*w = (alpha*a1+beta*b1)*v1 + ....\nis a linear combination of the vectors in Span S, thus it's in Span S.  \n\nSo, Span S is not empty and is closed, thus its a subspace.
676624
Algebra (from Arabic: الجبر, al-ğabr) is a branch of mathematics concerning the study of structure, relation and quantity. Elementary algebra is often taught in high school and gives an introduction to the basic ideas of algebra: studying what happens when numbers are added or multiplied, and how to make polynomials and find their roots.\n\nAlgebra is much broader than elementary algebra and can be generalized. Rather than working directly with numbers, one can work with symbols or elements of some set. Addition and multiplication are viewed as general operations, and their precise definitions lead to structures such as groups, rings and fields.\n\nMathematics can be defined as the logically rigorous study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions.\n\nMathematics is used throughout the world in fields such as science, engineering, medicine, and economics. These fields both inspire and make use of new discoveries in mathematics. New mathematics is also created for its own sake, without any particular application in view.\n\nThe word "mathematics" comes from the Greek μάθημα (máthēma) meaning science, knowledge, or learning, and μαθηματικός (mathēmatikós), meaning fond of learning. It is often abbreviated math in the U.S. and Canada and maths in Britain and many Commonwealth countries.\n\nThe evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g., addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to knowledge of geometry.\n\nFurther steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse.\n\nFrom the beginnings of recorded history, the major disciplines within mathematics arose out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change.\n\nMathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both.\n\nMathematical discoveries have been made throughout history and continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proof."\n\nArithmetic or arithmetics (from the Greek word αριθμός = number) in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numbers, though professional mathematicians often treat arithmetic as a synonym for number theory. It is the oldest and simplest branch of mathematics, used widely by almost everyone from simple daily counting to more advanced science and business.\n\nGeometry (Greek γεωμετρία; geo = earth, metria = measure)
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can you say what you mean by imageset of the function?\ndo youi mean pre image and image,namely the ordered pairs?\nif so\nx=4,y=0;x=1,y=-9;x=0,y=-12\nx=-5,y=-9,x=-2,y=-12\nso the ordered pairs are\n(4,0);(1,-9);(0,-12);(-5,-9);(-2,-12)
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In conformal mapping, the complex plane is extended to include "the point at infinity". The mapping f(z) = 1/z maps the unit circle onto the unit circle and points that were inside the circle map onto points outside the circle and vise versa.\n\nSo you might say, yes, you can turn a circle inside out.
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By considering a change of base point, it is clear that π\n1\n(X, {x\n0\n} depends only on the\npath component of x\n0\n. Since π\n1\n(X × Y, {(x\n0\n, y\n0\n)} = π\n1\n(X, {x\n0\n}) × π\n1\n(Y, {y\n0\n}), we can\nassume the space is path connected and omit the base point.\nSimply connected: an arcwise-connected space with trivial fundamental group. Ho-\nmotopy equivalent spaces have the same fundamental group. Thus, contractible\nspaces, which are homotopy equivalent to a point, are simply connected. Actually,\nany loop S\n1\n→ X which is homotopically trivial extends to a map S\n\n\nSorry, copy pasted it from my e-book.
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http://www.algebra.com
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http://purplemath.com/modules/index.htm\n\nis an awesome resource for all kinds of algebra topics, even ones you may have forgotten and need to brush up on.\n\nTake a look at that one.\nUMRmathmajor
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Do a google search on "Descartes Rule of Signs" and you will learn all you need to know.\n\nAlso, the complex roots come in matched pairs known as "complex conjugates."  To make up an example, suppose your roots are x = 3 + 2i and x = 3 - 2i.\n\nThen x - (3 + 2i) = 0 and x - (3 - 2i) = 0.\n\n[x - (3 + 2i)] [x - (3 - 2i)] = 0\n\n[(x-3) + 2i] [(x-3) - 2i] = 0\n\n(x-3)^2 - (2i)^2 = 0\n\n(x^2 - 6x + 9) -  4i^2 = 0\n\n(x^2 - 6x + 9) - (-4) = 0\n\nx^2 - 6x + 13 = 0\n\nThis quadratic has no real roots.  Its pair of roots are called "complex conjugates."  If you graph this function y = f(x), you'll see that it never crosses the x-axis.  To get the minimum, take the derivative:\n\ndy/dx = 2x - 6 = 0  ==>  x = 3\n\nf(3) = 3^2 - 6 x 3 + 13 = 9 - 18 + 13 = 4 > 0\n\nYou can solve by the quadratic formula or by completing the square:\n\nx^2 - 6x + 9 = -13 + 9 = -4\n(x - 3)^2 = +/- sqrt(-4) = +/- 2i\nx = 3 +/- 2i\n\nThis solution may be represented as two points in the complex plane, (3, 2i) and (3, -2i), or as vectors 3i +/- i2j.  The magnitude of these vectors is sqrt(3^2 + 2^2) = sqrt(13).  Therefore, they can be expressed as complex numbers\n\nsqrt(13)(sin theta +/- i cos theta)\n\nwhere the angle theta is determined by tan theta = 2/3.\n\nFinally, to directly answer your question, "are complex roots like fake roots, but what does that mean?", I'd say that the complex roots are roots that satisfy the original equation.  If you plug in the complex roots, the original equation is satisfied.\n\nA complex number, such as 3 + 2i, typically has a real portion and an imaginary portion.  The imaginary portion is not a real number, and it involves the square root of negative one.  Nevertheless, complex numbers can be graphed in the complex plane, and complex numbers have properties that make them useful in mathematics and, for example, in the theory of electrical circuits.  If you study transient effects in electromagnetics, you learn the value of complex numbers in a big hurry.\n\nAnd one final thought about graphing.  You can graph a complex function in three dimensions, with x, y, and z coordinates.  Let x be the real portion, y the imaginary portion, and z the magnitude, i.e., sqrt(x^2 + y^2).  When you do that, you'll see that these complex solutions <do> cut some of the axes.
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Algebra is a branch of mathematics concerning the study of structure, relation and quantity. The name is derived from the treatise written by the Persian mathematician Muḥammad ibn Mūsā al-Ḵwārizmī titled Al-Kitab al-Jabr wa-l-Muqabala (meaning "The Compendious Book on Calculation by Completion and Balancing"), which provided symbolic operations for the systematic solution of linear and quadratic equations.\n\nTogether with geometry, analysis, and number theory, algebra is one of the several main branches of mathematics. Elementary algebra is often part of the curriculum in secondary education and provides an introduction to the basic ideas of algebra, including effects of adding and multiplying numbers, the concept of variable, definition of polynomials, along with factorization and determining their roots.\n\nAlgebra is much broader than elementary algebra and can be generalized. In addition to working directly with numbers, algebra covers working with symbols, variables, and set elements. Addition and multiplication are viewed as general operations, and their precise definitions lead to structures such as groups, rings and fields.
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algebra is an early form of calculus.\n\nbut what do you think about this\nhttp://answers.yahoo.com/question/index;...
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The Order of Operations rules as we know them could not have existed \nbefore algebraic notation existed; but I strongly suspect that they \nexisted in some form from the beginning - in the grammar of how people \ntalked about arithmetic when they had only words, and not symbols, to \ndescribe operations. It would be interesting to study that grammar in \nGreek and Latin writings and see how clearly it can be detected.\n\nAt the other end, I think that computers have influenced the subject, \nso that it is taught more rigidly now than it used to be, since \nprogramming languages have had to define how every expression is to be \ninterpreted. Before then, it was more acceptable to simply recognize \nsome forms, like x/yz, as ambiguous and ignore them - something I \nthink we should do more often today, considering some of the questions \nwe get on such issues.\n\nI spent some time researching this question, because it is asked \nfrequently, but I have not found a definitive answer yet. We can't say \nany one person invented the rules, and in some respects they have \ngrown gradually over several centuries and are still evolving.\n\nHere are my conclusions, perhaps in more detail than you want:\n\n1. The basic rule (that multiplication has precedence over addition) \nappears to have arisen naturally and without much disagreement as \nalgebraic notation was being developed in the 1600s and the need for \nsuch conventions arose. Even though there were numerous competing \nsystems of symbols, forcing each author to state his conventions at \nthe start of a book, they seem not to have had to say much in this \narea. This is probably because the distributive property implies a \nnatural hierarchy in which multiplication is more powerful than \naddition, and makes it desirable to be able to write polynomials with \nas few parentheses as possible; without our order of operations, we \nwould have to write\n\n     ax^2 + bx + c\nas\n     (a(x^2)) + (bx) + c\n\nIt may also be that the concept existed before the symbolism, perhaps \njust reflecting the natural structure of problems such as the \nquadratic.\n\nYou can see an example of early notation in "Earliest Uses of Grouping \nSymbols" at:\n\n   http://members.aol.com/jeff570/grouping.html   \n\nwhere the use of a vinculum (an early version of parentheses) shows, \nboth in its presence (around an additive expression) and its absence \n(around the multiplicative term "B in D") that the rules were \nimplicitly followed:\n                                                 ________________\n   In Van Schooten's 1646 edition of Vieta, B in D quad. + B in D\n   is used to represent B(D^2 + BD). \n\n2. There were some exceptions early in this development; in \nparticular, math historian Florian Cajori quotes many writers for \nwhom, in the special case of a factorial-like expression such as\n\n     n(n-1)(n-2)\n\nthe multiplication sign seems to have had some of the effect of an \naggregation symbol; they would write\n\n     n * n - 1 * n - 2\n\n(using a dot or cross where I have the asterisks) to express this. Yet \nCajori points out that this was an exception to a rule already \nestablished, by which "nn-1n-2" would be taken as the quadratic \n"n^2 - n - 2." \n\nThere was also an early notation in which a multiplication would be \nreplaced by a comma to indicate aggregation: \n\n     n, n - 1 \n\nwould mean\n     \n     n (n - 1) \n\nwhereas \n     \n     nn-1 \n\nmeant\n\n     n^2 - 1.\n\n3. Some of the specific rules were not yet established in Cajori's own \ntime (the 1920s). He points out that there was disagreement as to \nwhether multiplication should have precedence over division, or \nwhether they should be treated equally. The general rule was that \nparentheses should be used to clarify one's meaning - which is still \na very good rule. I have not yet found any twentieth-century \ndeclarations that resolved these issues, so I d
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If f : A-->B is surjective map, then its onto but not one-one .\nso you can write domain of f as set of all values that would give values in B.\n\nYou can write as\ndom(f) = {x : f(x)=y, x is in A, y is in B}\n\nSince the function is not one-one, there can be more than one x giving the same values of y, I dont think you need more conditions in set representation.
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This question! Isn't it under mathmatics? LOL!
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the way you communicate to one another using algebraic symbols?
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Is the symbol "&" just a part of the problem?\n(&2)m\nCommutative law of multiplication says you can switch the order in which things are multiplied.\n&2 = 2&\nThus,\n(&2)m = (2&)m\nThe associative law of multiplication says you can move the parentheses.\n(2&)m = 2(&m)
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http://mathworld.wolfram.com/PoincareConjecture.html