Talk:Arithmetic
From Academic Kids
"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
- 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.)
- 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.
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 )
Can anyone explain what on earth "arithmetic" (in the sense of this article) has to do with "change"??
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.
