#$&*
course mth 272
you actually put parenthesis around the 2, that part seems to get tricky on the computer.INSTRUCTOR RESPONSE: The expression was intentionally written to be misleading and make the point that, to avoid ambiguity, order of operations apply strictly, no matter what the expression looks like.
Normally, for clarity, the parentheses would be included. They aren't necessary, but when helpful it's a good idea to include them. You can, of course, have too many parentheses in an expression, making it harder than necessary to sort out. In practice we try to strike a balance.
The original expression was
(x - 5) ^ 2x-1 + 3 / x-2
White spaces make no difference in how an expression is evaluated, but they can help show the structure; e.g.,
(x - 5)^2 * x - 1 + 3 / x -2
is a visual improvement over the original. The * between the 2 and the x is not strictly necessary, but is also helpful.
((((x - 5) ^ 2)) * x) - 1 + (3 / x) - 2
verges on having too many parentheses at the beginning; it does help clarify the 3 / x.
STUDENT COMMENT
Although I read through your explanation and do see the point you are making, that 2x is actually 2 * x, I still think that
(-1) should be raised to 2x rather than 2. Kaking the answer -11/4, not 7/4.
INSTRUCTOR RESPONSE
When the expression (x - 5) ^ 2x-1 + 3 / x-2 is copied and pasted into a computer algebra system it is translated as
This notation is universal and unambiguous. Any deviation from strict interpretation (which does occur among some authors and among manufacturers of some calculators) tends to result in ambiguity and confusion.
STUDENT COMMENT
While I do understand what you are trying to relate, I will continue to make these mistakes on more than one occasion and will not penalize myself for not rewriting years of mathematics because of a syntax issue in an online class.
INSTRUCTOR RESPONSE
I don't penalize errors in typed notation when the intent is clear (though I will sometimes point out these errors), and when you take your tests you'll be writing them out by hand and this won't be an issue.
However this is not a syntax issue in an online class. This is the order of operations, as it has been since algebra was developed hundreds of years ago, and it's completely consistent with the mathematics you appear to know (quite well).
As stated here, if you use the wrong syntax in any computer algebra system, your expression will not be interpreted correctly. For this reason alone you need to understand the notation.
For this and other valid reasons you need to understand how the order of operations are represented in 'linear' fashion (i.e., 'typewriter notation') and to correctly interpret expressions written in this notatation.
Any mathematics that has been learned correctly is completely consistent with the order of operations and with the notation used in this course. If the mathematics you've learned was inconsistent with the order of operations (and I don't believe this is so in your case, but it is with many students), then you would need to adjust your thinking. Fortunately this is very easy to do. Interpret expressions literally, assume nothing, and everything works out.
You will also find that the notation quickly becomes easy to read and use, and that it expands your comprehension of all mathematical notation.
STUDENT COMMENT
I used -1^(2*4). I didn't realize that was doing multiplication before exponents. All of this typewriter notations seems ambiguous to me but I think that had I seen the expression in standard notation I would probably have made the same mistake in this instance. If I were writing this expression I would probably use a parenthesis or * to show the necessary separation.
INSTRUCTOR RESPONSE
Parentheses, even when they aren't strictly necessary, are often useful to clarify the expression. An parentheses, even when not necessary, are part of the order of operations.
Spacing is not part of the order of operations. An expression has the same meaning even if all spaces are removed.
However as long as an expression is correctly formed, spacing as well as parentheses can certainly be used to make it more readable.
I don't go to any trouble in this exercise to make the expressions readable, since my goal here is to make the point about order of operations, which give an expression its unambiguous meaning.
However in most of the documents you will be working with, I do make an effort to clarify the meanings of expressions through their formatting, often using unnecessary parentheses and spacing to help clarify meaning.
Certainly I encourage you to do the same.
STUDENT QUESTION
I didn’t separate the ¾ as a stand alone fraction, I am confused about why you don’t treat it as an equation that the
denominator isn’t treated as a denominator.
INSTRUCTOR RESPONSE
Your work was good throughout most of this problem. You did forget to copy down a -1 in one of the early steps, but otherwise followed the order of operations correctly until nearly the last step.
However near the end you said that 4+3/4-2=7/2.
You appear to have performed the addition 4 + 3 and the subtraction 4 - 2 before dividing. However the division has to be done first.
The division sign is between the 3 and the 4, so the division is 3/4, and that gives you the fraction 3/4.
Therefore the expression 4+3/4-2 tells you to 'add 3/4 to 4 then subtract 2'.
When actually writing this out we would probably include parentheses. That wasn't done here, as it would have defeated the point being made about order of operations, but for clarity we might have written
4 + (3/4) - 2.
The parentheses are not necessary around the 3/4, since the order of operations is sufficient to unambiguously define the result, but they do make the expression easier to read and reduce the likelihood of error.
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Self-critique (if necessary):
ok
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Self-critique Rating:
3
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Question: `q005. At the link
http://www.vhcc.edu/dsmith/genInfo/introductoryproblems/typewriter_notation_examples_with_links.htm
(copy this path into the Address box of your Internet browser; alternatively use the path
http://vhmthphy.vhcc.edu/ > General Information > Startup and Orientation (either scroll to bottom of page or click on Links to Supplemental Sites) > typewriter notation examples
and you will find a page containing a number of additional exercises and/or examples of typewriter notation.Locate this site, click on a few of the links, and describe what you see there.
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Your solution:
A couple of problems put in typewriter notation then in standard notation
confidence rating #$&*:
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3
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Given Solution:
`aYou should see a brief set of instructions and over 30 numbered examples. If you click on the word Picture you will see the standard-notation format of the expression. The link entitled Examples and Pictures, located in the initial instructions, shows all the examples and pictures without requiring you to click on the links. There is also a file which includes explanations.
The instructions include a note indicating that Liberal Arts Mathematics students don't need a deep understanding of the notation, Mth 173-4 and University Physics students need a very good understanding,
while students in other courses should understand the notation and should understand the more basic simplifications.
There is also a link to a page with pictures only, to provide the opportunity to translated standard notation into typewriter notation.
end program
STUDENT COMMENT (not quite correct)
I see a collection of typewriter problems, after looking at some of them I see that the slash mark is to create a fraction rather than to denote division.
INSTRUCTOR CORRECTION
A fraction is a division of the numerator by the denominator. The slash mark indicates division, which can often be denoted by a fraction.
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Self-critique (if necessary):
ok
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Self-critique Rating:
3
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Question: `q006 Standard mathematics notation is easier to look at; it's easier to see the meaning of the expressions.
However it's very important to understand order of operations, and students do get used to this way of doing it.
You should of course write everything out in standard notation when you work it on paper.
It is likely that you will at some point use a computer algebra system, and when you do you will probably have to enter expressions using a keyboard, so it is well worth the trouble to get used to this notation.
Indicate your understanding of why it is important to understand this notation.
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Your solution:
Not all programs will have a way to type in standard notation and there for to understand it you should learn typewriting notation
confidence rating #$&*:
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3
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Self-critique (if necessary):
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Self-critique rating: