Regarding operations that are opposite each other, it is fun for a student to discover but boring to listen to someone tell it. You could tell a first grader to add five to a number and then ask what to do to get the original number. But if you tried to tell a first grader this stuff, there would be no benefit. When you try to make first grade stuff sound like Ph.D. level, you kill brain cells.
I'm too lazy to look up "mathematical expression." I guess if you write a string of different operations, then you have "mixed operations."
Multiplication of whole numbers is repeated addition. After that, it gets more complicated.
This logic of why multiplication and addition take precedence over addition and subtraction doesn't make sense to me. We could just as easily agree that addition and subtraction take precedence over multiplication and addition and then use brackets as needed. But the way things work, such as a farmer having various plots of land and trying to find the total acreage or a customer buying numerous quantities of various items and trying to find the total price makes it so that less brackets are used if we agree that multiplication and division take precedence over addition and subtraction. We only make this agreement to cut down on brackets. There doesn't need to be any agreement at all but it would take a lot more brackets to write an equation.
You sure are determined to make division take precedence over multiplication and subtraction take precedence over addition. I think you have a very complex way of thinking that I would avoid teaching anyone else. We can get along just fine saying that division and multiplication have exactly equal precedence We can say that addition and subtraction have exactly equal precedence. We agree that they are performed left to right. We could have just as easily agreed that operations are performed right to left and probably on 50% of other planets they do this.
You are an intelligent guy and you make us think though.
regards,
Brian
__._,_.___----- Original Message -----From: VinaireSent: Thursday, December 06, 2007 9:09 AMSubject: Re: [Math4u] Mathematical notation
I have just written this summary for my "Milestone" on MIXED OPERATIONS. I shall be interested in receiving your comments.The operations of addition and subtraction are opposite of each other, but they are of the same order. When one adds and then subtracts the same quantity, the two effects cancel each other out.Similarly, the operations of multiplication and division are opposite of each other, but they are of the same order. When one multiplies and then divides by the same quantity, the two effects cancel each other out.When all these operations are present together in a mathematical expression we have mixed operations.A multiplication is "repeated addition." Similarly a division is "repeated subtraction." Therefore, the operations of multiplication and division are of a higher order and, in mixed operations, they take precedence over the operations of addition and subtraction.Those portions of a mathematical expression that contain a series of multiplication and division are called terms. The terms are separated from each other by plus (addition) or minus (subtraction). NOTE: Any operations within parentheses are considered a term. A term could also be a single number. In reducing mixed operations, one reduces the terms first, and then the mathematical expression.Normally, one reduces an individual term from left to right. Otherwise, within the term, division takes precedence over multiplication. Then, one reduces the mathematical expression from left to right. Otherwise, subtraction takes precedence over addition.Understanding the logic involved in reducing mixed operations is very important. Only when this logic is understood, do the "formulas" about precedence of operations make sense.Vinaire
----- Original Message ----
From: Buck Dick <kcubar34@yahoo.com>
To: Math For You <math4u@yahoogroups.com>
Sent: Sunday, December 2, 2007 9:07:14 AM
Subject: [Math4u] Mathematical notation
Brian wrote:"...but I would like everyone to speak the same language."Thank you (again) Brian (and Vinaire).One of the universal appeals of mathematics is that the notation used by those who know the subject is standard. It makes no difference what (meta)language is being used for the text, the symbolic language is the same all over this world. It makes no difference if you read left to right or right to left or top to bottom, the mathematical symbolism is standard.There are thousands of mathematical dictionaries and encyclopedias published in most of the written languages used in this world. One of them should be part of every person's library, especially if they are students of the subject.The internet has some good sites for mathematics, the ultimate being Wolfram's sites, but there is a lot of incorrect information out there, since anyone can post anything they wish to post, right or wrong.So look it up, use it correctly and then we will all be able to communicate with the same understandings, don't just try to pass a test or pass that course because it is needed for your degree. Learn something about the subject.By the way, my four mathematical dictionaries all define the colon symbol as used in mathematics to be the symbol for a "ratio," and a ratio as the comparison or division of two quantities that are expressed in the same units. If the units are different, the comparison is called a "rate."Also, the horizontal line which separates the "numerator" (Latin for "number") and "denominator" (Latin for "name") of a fraction has a name. It is a "vinculum." (Look it up; the name has also been used for other things since the original Latin, but in mathematics it is still the horizontal line separating the parts of a fraction.) Now, of course, the backslash is generally used since it is easier to put things all on one line, but then parentheses need to be used to indicate the grouping, etc., etc., etc.My point is simple: Learn the symbolic language used in mathematics if you want to try to communicate your ideas.Buck Dick
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