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Algebra can essentially be considered as doing computations similar to that of arithmetic with non-numerical mathematical objects. First sentence Algebra Statements based on these variables are manipulated using the rules of operations that apply to numbers, such as addition.
This can be done for a variety of reasons, including equation solving. Initially, these objects were variables representing either numbers that were not yet known unknowns or unspecified numbers indeterminates or parameters , allowing one to state and prove properties that are true no matter which numbers are involved.
Elementary algebra is the part of algebra that is usually taught in elementary courses of mathematics. Addition and multiplication can be generalized and their precise definitions lead to structures such as groups , rings and fields , studied in the area of mathematics called abstract algebra.
Abstract algebra is a name usually given to the study of the algebraic structures such as groups , rings , fields and algebras themselves. Here are my thoughts on the first comparison. Line 1 is obviously one of the hardest things: Saying what algebra actually is. I think 1A is a better stab than 1B. This could be similar to 1A or different, if someone has a better suggestion. Anyone got any thoughts on this, or any of the other comparisons?
Lazard, can you say which bits of column A you thought were controversial and why? Yaris talk , 11 December UTC Here are the details of what is controversial: A1: concerning the study of the rules of operations and relations : This is not specific to algebra, it applies also to mathematical analysis which studies the functional relations and the operations of derivation and integration.
A1 is too technical for a first sentence in the lead: To be understood the reader must know what is a term at this level of generality, this word has no commonly accepted meaning in mathematics, except in the very technical field of mathematical logic , a polynomial, an equation the reader should guess that differential equations are not considered here and an algebraic structure circular definition.
A1: This "definition" does not take into account the multiplicity of subareas of algebra, mentioned in lead B, but not quoted in the table: Presently, algebra is divided in several subareas including linear algebra, group theory, ring theory and combinatorics see below for more subareas. The definition of "pure mathematics" is controversial. The opinion of many mathematicians is that the concept does not exists. Even if it would exist, one may not include in it computer algebra nor "applied algebra" for a reliable source for the existence of this concept, see Journal of Algebra and Its Applications , for example.
A3, elementary algebra: In Yaris this receives an undue weight by the place which is devoted to him. As such, it deserves to be cited in the lead, but it does not needs more place than that given in version B. A4, abstract algebra. This is also a badly defined concept whose existence itself is as controversial as the concept of "pure mathematics".
If it exists, does linear algebra and its applications for example in weather forecast belong to it? However, as many people use this term, it deserves a brief mention, like that in B4. I agree that version B is not sufficiently sourced. But version A is also unsourced. It most controversial aspect it that it seems reduce "algebra" to "elementary algebra" and "abstract" algebra. Mathematics Subject Classification is a reliable source showing that it is non-controversially wrong. Yaris, please explain what is wrong or controversial in lead B.
In your post, the only given argument is "I do not like it", which is an argument to avoid in such a discussion. Thanks for explaining your reasoning. Yes I agree that it is perhaps starting too technical but I definitely think it would be useful to say what algebra is.
Can you think of a better way of putting it? Similarly, differential equations are types of equations but they need the concept of limit in there too assuming you define calculus in terms of limits. A3 and A4. I think we need to remember the readers. The idea that algebra can deal with things other than numbers will seem weird. If we start with "elementary algebra" and then progress to "abstract algebra" it should lead the reader in.
You say this is an "educational concept" as if this is a bad thing but the reason educators use this concept is because it makes things easier to explain - which is exactly what we want to do. Mathematics Subject Classification is designed for classifying mathematical research, rather than explaining mathematics to the lay reader.
Yaris talk , 11 December UTC IMO, defining "algebra" as as "doing computations similar to that of arithmetic with non-numerical mathematical objects" is a perfect definition that can be understood by kids that know what are the operations of arithmetics.
It has also the advantage, in the next sentence, to explain in a few simple words why "variables" is an essential notion of algebra. Moreover, this definition cover all the aspects of algebra, abstract as well as applied, at research level as well as at educational level. This definition is maybe "weird", but it is true: working with numbers is not algebra, but Number theory or in case of real numbers mathematical analysis.
Your sentence " They suggest that educators have to change mathematics to make it easier, and that taught mathematics is different of research mathematics. IMHO, the task of teachers of mathematics is to teach mathematics, not to teach other things, because they seem easier to teach. Educators have certainly to choose what is taught, and how it has to be presented for being understood.
What is taught in elementary courses may certainly be called "elementary algebra", but it remains a part of algebra, which is exactly what version B says. A very important task of educators is also to open the way for learning more mathematics not only for future mathematicians but also for future engineers, computer scientists and searchers in other sciences.
This wider opening on the mathematical world is difficult to give in elementary classes and in textbooks, and is thus an important role of WP. It is thus a bad advice to give to the kids that opposing "educational mathematics" and "research mathematics".
There is a continuity and any temptation of breaking it is bad for students as well for mathematics itself for me mathematics is not plural, but singular. Here is a practical example illustrating these considerations. Before August , polynomial greatest common divisor was a stub containing only what you call "elementary mathematics". This can be explained without using more sophisticated mathematics than that is taught in elementary courses in which polynomial GCD is taught.
This explanation needs research results that are not older than 50 years. The present version is, maybe, not detailed enough for being understood by kids except the best ones , and deserve certainly to be expanded, at least by adding examples. However, as it only uses "elementary algebra" methods, it can certainly be understood by their teachers, and this may be very useful to improve their course.
The present version of the algorithm is thus "elementary mathematics" by the involved methods and "research results" by the date of the results that it contains. How you class it, if you break the continuity of mathematics. You might have variables which represent numbers. Arithmetic is all about algorithms that work with positional notation. Algebra is not. What does the statement mean? Do you agree that this is the issue?
From your response now, you seem to have the idea that there is only one real way of subdividing algebra, which is the way that research papers on algebra are divided. I have no desire impose a separation of research and educational mathematics.
I like the example you give of the improvements to polynomial greatest common divisor. I guess my main point is that saying version A gives undue weight to elementary algebra misses the point that elementary algebra is exactly what most people want to find out about. I happily agree that algebra can not be split into the elementary and the abstract. Linear algebra is a good example of something that can not easily be classified as one or the other.
Similarly, complex numbers come about as soon as you start looking for roots to polynomials, but are they elementary? These concepts are just useful concepts in explaining what algebra is. The role of the article is to explain algebra to the lay reader. If this can include summarising how current research in algebra is classified, that is great, but the purpose of the article is more than that. Maybe I have remarked it, but I did not answered because my opinion it that the problem cannot be solved by a discussion with only two editors.
As Yaris has reinserted his controversial definitions and assertion in the article, I answer by inserting punctual answers inside his post. On the other hand, none of the assertions of my version has been shown controversial. I have moved it below. Yaris talk , 16 December UTC B1: This is exactly what is said in the next sentence: "Initially these objects were variables representing either numbers that were not yet known unknowns or unspecified numbers indeterminates or parameters , allowing one to state and prove properties that are true no matter which numbers are involved.
Your definition of arithmetic is very different from that can be fount at arithmetic. The issue with your "definition" is that it involves notions that are not known by the layman "rules of operations", "relations", "constructions" "terms", "polynomial", "algebraic structure" and does not explain how algebra differs from the other parts of mathematics, while everybody knows and has done arithmetics computations, even if he does not know what is computing with variables which is exactly what algebra is.
Lazard talk , 16 December UTC A3 and A4: If algebra is not what the majority of our audience "thought algebra was", the first task is to give a correct view of what algebra is really.
The second one is to give the access to what they "thought algebra was". This is fulfilled by links and brief definitions of elementary algebra and abstract algebra. As there are sections in this article and specific articles devoted to them, more details are misplaced in the lead.
Lazard talk , 16 December UTC Third opinion This discussion is difficult to add to, since there are so many things being discussed at once. Anyway, I am broadly in favour of D. For example, the article mathematics , or even geometry , have much nicer lead sections. I think it is difficult to nail down what algebra is for the lay person unlike geometry which is "the study of shape", whatever that means. Describing it as a "broad part of mathematics" first of all, seems like a good idea.
Mark M talk , 16 December UTC Hi Mark, I think there are three points of contention: A1 I agree this that this could be made more accessible but think it is a decent stab for now. I think D. Lazard thinks that it is far too complicated a place to start.
Lose all the other more complicated stuff. B1 I think this is potentially misleading and D.
Algebra can essentially be considered as doing computations similar to that of arithmetic with non-numerical mathematical objects. First sentence Algebra Statements based on these variables are manipulated using the rules of operations that apply to numbers, such as addition. This can be done for a variety of reasons, including equation solving.
Modern algebra and trigonometry.
He wrote the Katyayana Sulba Sutra, which presented much geometry , including the general Pythagorean theorem and a computation of the square root of 2 correct to five decimal places. Jain mathematicians are important historically as crucial links between the mathematics of the Vedic period and that of the "classical period. In particular, their fascination with the enumeration of very large numbers and infinities led them to classify numbers into three classes: enumerable, innumerable and infinite. Not content with a simple notion of infinity, their texts define five different types of infinity: the infinite in one direction, the infinite in two directions, the infinite in area, the infinite everywhere, and the infinite perpetually. In addition, Jain mathematicians devised notations for simple powers and exponents of numbers like squares and cubes, which enabled them to define simple algebraic equations beejganita samikaran. Jain mathematicians were apparently also the first to use the word shunya literally void in Sanskrit to refer to zero.
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Mary P. Today, both an education center and an endowment bear her name in recognition her legacy. This is a review of the and editions of the book. The only substantive differences I noticed were in the last chapter on p. Answers to exercises are in the back of the book. Student Version "With Answers" is printed on the spine - Answers to odd and even-numbered exercises are in the back of the book.
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