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.
----- 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
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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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