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Scientific notation

DCJLEE
 
The following multiple-choice item appeared in a campus-wide quiz in an Intermediate Algebra course.
 
Write in scientific notation: 255,000.
(1)  255 x 10^3
(2) 25.5 x 10^4
(3) 2.55 x 10^5
(4) 255,000
 
When I expressed my opinion that the presumed correct answer (3) is incorrect as it doesn't preserve the precision implied by the original 255,000 , a fellow instructor asserted "Since 2.55 = 2.55000, I don't see anything wrong with our answers." Please help us clarify this matter. 
Thank you in advance.
 
John Lee
 
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Re: Scientific notation

Robert Hansen
I would say that in a mathematical sense, your friend is right. In a physical (measurement) sense, you are right. It is hard to talk about precision without some context.

For example, 255,000 might only have 3 digits of precision. You really don't know how many digits of precision that number has because it wasn't indicated (which is an advantage of scientific notation). Generally, they would indicate the precision separately if they are just using standard notation.

But you are correct to say that a number written as 2.55000 * 10^5 has 6 digits of precision.

Anyways, that is how I would describe it.
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Re: Scientific notation

Bishop, Wayne
In reply to this post by DCJLEE
(3) is correct based on standard usage. Unless additional information
is given, one should assume that the original number was only
accurate to three significant digits. Including one or more zeros in
the solution would indicate a level of accuracy that is not evident
in the original. If the number of arose in integer-only arithmetic,
or some such, scientific notation would be inappropriate because,
even with the zeros, it would indicate an approximate real number,
not an integer.

Wayne

At 02:38 AM 8/16/2009, [hidden email] wrote:

>
>The following multiple-choice item appeared in a campus-wide quiz in
>an Intermediate Algebra course.
>
>Write in scientific notation: 255,000.
>(1)  255 x 10^3
>(2) 25.5 x 10^4
>(3) 2.55 x 10^5
>(4) 255,000
>
>When I expressed my opinion that the presumed correct answer (3) is
>incorrect as it doesn't preserve the precision implied by the
>original 255,000 , a fellow instructor asserted "Since 2.55 =
>2.55000, I don't see anything wrong with our answers." Please help
>us clarify this matter.
>Thank you in advance.
>
>John Lee
>
>
>----------
>
>
>__________ Information from ESET NOD32 Antivirus, version of virus
>signature database 4340 (20090816) __________
>
>The message was checked by ESET NOD32 Antivirus.
>
><http://www.eset.com>http://www.eset.com

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Re: Scientific notation

Dave L. Renfro
In reply to this post by DCJLEE
John Lee wrote:

http://mathforum.org/kb/message.jspa?messageID=6816836

> The following multiple-choice item appeared in a campus-wide
> quiz in an Intermediate Algebra course.
>
> Write in scientific notation: 255,000.
> (1) 255 x 10^3
> (2) 25.5 x 10^4
> (3) 2.55 x 10^5
> (4) 255,000
>
> When I expressed my opinion that the presumed correct
> answer (3) is incorrect as it doesn't preserve the
> precision implied by the original 255,000 , a fellow
> instructor asserted "Since 2.55 = 2.55000, I don't see
> anything wrong with our answers." Please help us clarify
> this matter.
> Thank you in advance.

In my opinion, this is the kind of question that should
be avoided in tests. If it is important for the student
to know what "scientific notation" means, then it shouldn't
be tested in a multiple choice format. It should be in a
fill-in-the-blank format (which should be very easy to
grade). It's also difficult for me to determine what is
being tested, aside from simply knowing the meaning of
a certain word, which to me isn't something that I'd
consider all that important when making placement decisions
or whatever the purpose of the test was. I'd be more
interested in knowing whether the student could select
the correct value of an expression such as

[(3.6 x 10^5)(4 x 10^7)]  / [(6 x 10^3)(0.04 x 10^5)]

(without a calculator, of course) or put into numerical
order several values written in scientific notation or
something else that actually tested mathematics.

Dave L. Renfro
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Re: Scientific notation

Dave L. Renfro
In reply to this post by DCJLEE
John Lee wrote (in part):

http://mathforum.org/kb/message.jspa?messageID=6816836

>> Write in scientific notation: 255,000.
>> (1) 255 x 10^3
>> (2) 25.5 x 10^4
>> (3) 2.55 x 10^5
>> (4) 255,000

Dave L. Renfro wrote (in part):

http://mathforum.org/kb/message.jspa?messageID=6817852

> In my opinion, this is the kind of question that should
> be avoided in tests. If it is important for the student
> to know what "scientific notation" means, then it shouldn't
> be tested in a multiple choice format. It should be in a
> fill-in-the-blank format (which should be very easy to
> grade). It's also difficult for me to determine what is
> being tested, aside from simply knowing the meaning of
> a certain word, which to me isn't something that I'd
> consider all that important when making [...]

Here's another thing that bothers me about this question.
Notice how all the options give the same number, only
written in different ways, the first two of which are
just minor variations of what is desired. In fact, if
the student knows nothing more than that you're supposed
to have a mantissa between 1 and 9.999...9, no math
insight or background is needed to select (3) as correct.
Perhaps the other options should have been things like
2.55 x 10^4, 2.55 x 10^6, (2.55)^5, (255)^3, etc.

Dave L. Renfro
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