Talk:Arithmetic

Arithmetic and number theory

According to the wikipedia entry on number theory, (the sense of the term) arithmetic is not to be confused with (the sense of the term) elementary arithmetic. It seems that a lot of the material under arithmetic pertains solely to elementary arithmetic.

(corrections made)

mikeliuk 04:58, 22 May 2006 (UTC)

Actually what the number theory article cautions is not to confuse the various senses of the term arithmetic:

The term "arithmetic" is also used to refer to number theory. .... This sense of the term arithmetic should not be confused either with elementary arithmetic, or with the branch of logic which studies Peano arithmetic as a formal system.

I do think there is a lot of good stuff in elementary arithmetic and am concerned that this article might evolve to duplicate much of what is there.

So we should probably try to be clear about what should go in which article. Jake 17:59, 22 May 2006 (UTC)

Probably not a lot of progress to be made until someone turns up who actually does arithmetic in a sense other than the sense of elementary arithmetic.

mikeliuk 16:36, 28 May 2006 (UTC)

There is currently no material here that cannot be put under either elementary number theory or elementary arithmetic. As such, this topic is superfluous. Arithmetic - in the broadest sense - is the set of properties one might associate with the integers. Yet arithmetic is in no way restricted to the integers. For instance, "arithmetic of elliptic curves" describes the point of view of treating points on elliptic curves as integers, whereas "arithmetic in rings" describes the use of number theoretic properties (such as divisibility and irreducibles) in general rings (usually in Dedekind domains where the definitions make most sense). If we are to make the topic of arithmetic different from that of number theory and elementary (aka pre-school) arithmetic, I suggest this is the most appropriate route to take. 217.155.61.70 18:01, 7 June 2006 (UTC)

"Exponentiation" & "square roots"?

Binary operations exists in pairs of inverses. The first- addition and subtraction; the second- multiplication and division; the third- involution (also called "exponentiation") and evolution.

The term "exponentiation" is awkward because there is no linguistically-logical inverse term available. OmegaMan

“taking the logarithm of”? mfc

Although exponential and logarithmic functions are certainly inverses, there is no binary operation known as or similar to "taking the logarithm". OmegaMan

Assuming, from the discussion below, you are using binary in the sense dyadic, then that rather depends on how one defines the operation.

For example, for almost any practical implementation of a function there is a defined context: division without a context is only tractable if the answer is a rational pair—where one might argue that the division has not been effected. So division is really a trinary operation. And a log function needs to provide the base (and often other information, in practice).

x=y log 10 might be one way of requesting logarithm of y in base 10.

The term "square roots" is clearly inadequate to describe a binary operation whereby roots can be extracted by any arbitrary amount. OmegaMan

square root is a bit of a special case, as it is included in IEEE 754, which is often thought of as an arithemtic standard. I'll see if I can rework that paragraph to make it clearer. mfc

binary operations usually refer to those done on 0 and 1. Please do not confuse the issue. Dori 17:33, Nov 25, 2003 (UTC)

"Base 2" or "a binary base" or simply "binary" are what you are inaccurately referring to. Note that "binary operation" is a compound term with a distinctly different meaning.

"Binary operations" are defined at this moment as such in their Wikipedia entry. [Perhaps it carries some weight with you?]

"In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. It is sometimes called a dyadic operation as well.

More precisely, a binary operation on a set S is a binary function from S and S to S, in other words a function f from the Cartesian product S × S to S." _________________________________________________________________________

"Typical examples of binary operations are the addition and multiplication of numbers and matrices as well as composition of functions on a single set." __________________________________________________________________________

"Binary operations are often written using infix notation such as a * b, a + b, or a · b rather than by functional notation of the form f(a,b)." ________________________________________________________________

The following definition is currently on the Wikipedia entry for "arithmetic"-

"Arithmetic is a branch of mathematics which records elementary properties of certain arithmetical operations on numbers."

This is an empty definition because it relies upon the term "arithmetical operations" to define the term "arithmetic".

(Omitted an inappropriate remark.)

OmegaMan

I realize that there is more than one definition of binary, but the 1 and 0 one is more prominent and it is likely to confuse the readers. If you could explain it better, than maybe it could be used. You have to remember that this is an encyclopedia and the readers are not likely to be well versed in math. The first paragraph at least should probably be a general idea that describes the subject in the least confusing terms possible. Perhaps if you explained the term binary in this context later in the article, it would be more helpful and it could be used. I did not mean to imply that you do not understand the field (I am a math minor myself).

P.S. Consider getting an account if you would like to be credited with your attributions. It also makes communication easier.

regards, Dori 19:56, Nov 25, 2003 (UTC)

I approve of your revision, Dori. Thank you. I took your advice. From this day forward ... I am OmegaMan.

OmegaMan

How about putting up something like this :

Examples

• Addition: 2+2=4 and Subtraction: 4-2=2
• Multiplication: 4×4=16 and Division: 16÷4=4
• Logarithms: log_10 1000 = 3 and Exponentials 10^3=1000 (maybe add 3rd root of 1000 = 10)

(Maybe with a comment about the duality of these operations, and something about roots (perhaps just mentioning the n-th root of x is x^(1/n))

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Hm. I tend to think of arithmetic as the symbol-manipulating procedures on numerals ... but I admit that the fundamental theorem of arithmetic is about something deeper than that, so it may be too narrow. 142.177.23.79 23:40, 14 May 2004 (UTC) (My degree was in maths but all my education's Canadian, so ignore it. =p )

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Can anyone explain what on earth "arithmetic" (in the sense of this article) has to do with "change"??

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Arithmetic is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals.

Completely inaccurate. "Arithmetic" has 2 distinct senses:

• The study and practice of computational algorithms involving certain operations on integers and other numbers. These typically include +, -, *, ...etc., etc. The focus here is on the ALGORITHMIC and COMPUTATIONAL nature. This type of arithmetic is not an exploration of "properties of certain operations". It is simply the application of algorithms which implement the operations.
• Number theory (MODERN, as well as "elementary"). The study of the properties of the integers, esp. related to primality, divisibility, etc., etc., as well as any of the outgrowths of modern research that have developed as a result of this study.

These are 2 different things. Saying "which records elementary properties of certain operations", makes it sound like a fuzzy combination of both. The first sense only records results of applying algorithms, not "properties", the second records properties of operations, but much more (not just "elementary properties", not just "operations", etc.) The definition at the start manages not to get EITHER sense correct.

Pronunciation

It seems absurd to include a section on pronunciation here. Can't someone just look it up in the dictionary if they want to know? The pronunciation of the word "arithmetic" has nothing to do with arithmetic itself. I'm deleting it.

You really should. Asdfv 00:42, 2 February 2006 (UTC)

130.13.73.232 00:50, 18 January 2006 (UTC)

Agreed, it is unnecessary for this article, as pronunciation does not drastically affect meaning. This belongs in a dictionary, not an encyclopedia. I'll remove it. —siroχo 16:25, 2 February 2006 (UTC)

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What about original meaning of word arithmetic and name "modular arithmetic"? I cannot see that at this page.--Čikić Dragan 15:50, 3 February 2006 (UTC)

How about including "finger math"?

Numeralization

Don't you think "numeralization of the null concept" should be changed to "invention of zero"? If not, I think at least "numeralization of the null concept (invention of zero)" should be written instead. This would be much clearer. A.Z.

History

I would like to work on the history of arithmetic, but I'm worried that it will take me too far afield. I think the scope should be limited to arithmetic that can be performed with up to only pencil & paper (e.g. no Napier's bones, no multiplication using trig functions, and certainly nothing electronic.) This would include finger math, the Russian peasant algorithm algorithm, and (a few others after I think about it.) And also, limited to addition, subtraction, multiplication, division, and square roots. Thoughts? --M a s 01:34, 5 May 2006 (UTC)

Good edits, 35 / Pmagyar. --M a s 00:47, 9 May 2006 (UTC)

I agree. Meekohi 00:53, 9 May 2006 (UTC)

Have deleted the starting paragraph, "Recent experiments in cognitive science have shown that even infant humans have an innate ability to add and subtract small numbers (up to about five.) Linguistic evidence suggests that all cultures have had the concept of numbers greater than five, and the words for various numbers are in many languages simple additions (or in rare cases subtractions) of small numbers."

The phrase "recent experiments" does not give a good introduction to history. "Cognitive science" can tell us about history but would need rather more discussion. The sentence on "linguistic evidence" is rather short considering it heads the section on "history".

mikeliuk 18:20, 20 May 2006 (UTC)

Thanks for the comments Mike. I was trying to create a case (which many people believe, and have argued better than I have) that concepts of arithmetic are innate or at least predate written or recorded history - hence the connection to the experiments in subitizing, and also the connection to linguistic evidence. Is it your sense that such an argument should be integrated into the history section, or are you saying that a "prehistory" preamble doesn't belong? Thanks! --M a s 16:48, 22 May 2006 (UTC)

Sorry that the original comments were terse :) There should certainly be something about the observation that humans have a tendency to show understanding of number; both at very young ages (an innate understanding or an innate predisposition to formulating concepts of number), and this understanding is seen to develop independently in geographically separated civilizations. The proper place for ideas of innateness should probably be in the unfortunately titled elementary arithmetic (which is elementary in the sense that this branch of mathematics takes so much paper to outline that it could be in no way innate) where there is discussion of mental arithmetic and the arithmetical operations that one could argue are innate.

mikeliuk 16:29, 28 May 2006 (UTC)

Greek "Arithmetic"

At the beginning, the Greek spelling "αριθμός" appears (on my Firefox 1.0.4, Linux FC 2) to have two non-greek letters at the end. The next to the last letter appears to be letter o with an accute accent, and the last letter appears to be a c with a cedilla. Was it meant to be "omicron" "zeta"?--Todd 17:43, 9 May 2006 (UTC)

The last two letters are "omicron with an acute accent" and "sigma", which indeed appear as you described (the letter sigma has an alternate form when it comes at the end of a word). -- Jitse Niesen (talk) 01:19, 10 May 2006 (UTC)

Is Arithmetic Trivial?

Research mathematicians like my colleagues and I generally consider the arithmetic algorithms not as a tedious school subject, but as the living heart of algebra. Abstract algebra was not cut out of whole cloth by Galois or Kummer or Hilbert. The ring axioms of associativity, distributivity, etc, are what make the standard algorithms work: understanding the algorithms essentially means understanding these properties, which is half-way to abstract algebra. Furthermore, the recent movement toward computational algebra and algebraic geometry (Grobner bases), is a direct continuation of the genius of the arithmetic algorithms.

I agree that there's a lot of beauty and elegance in the algorithms themselves, and the relationship to abstract algebra is clear. There's plenty of documentation in the computational complexity studies to indicate that some computer algorithms for multiplication (Strassen etc.) were only developed after a more full understanding of abstract algebra. But is there any reliable documentation to show that the development of ring theory, field theory, and aa in general was influenced by a desire to understand the school algoirthms for arithmetic in general? In other words, which preceded the other? Thanks. --M a s 22:16, 12 May 2006 (UTC)

Thanks

I wanted to thank those people who have worked on this "core topic", it now looks so much better than when I read it in October! I think we'll probably be able to include it on the test CD release when you're finished with it. Thanks a lot! Walkerma 07:36, 27 May 2006 (UTC)