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### Significant Digits in Measurement

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Date: 08/13/2001 at 13:38:01
From: Tricia
Subject: Significant digits in measurement

I understand that there are rules to determine the significant digits
in measurement; however, I do not comprehend the basic concept behind
the use of significant digits. It seems to me that all digits are
significant, especially zeros for place value. An example of my
confusion would be: How can 3000 have only one significant digit (3)?
or How can 0.0050 have only two significant digits (50)?

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Date: 08/13/2001 at 15:34:17
From: Doctor Peterson
Subject: Re: Significant digits in measurement

Hi, Tricia.

The basic idea is that all digits that are not there ONLY for place
value purposes are significant.

Suppose I built a building that rose 10 stories high, but the first
two stories were just columns holding the rest up above a highway. The
first two "stories" would be important, certainly; but they wouldn't
really count as part of the building, because they only hold it in
place, without actually containing anything useful. The building has
only eight "significant stories"! Zeros used only for place value at
the right of a number are mere scaffolding holding it "above" the
decimal point, and don't really contain any information.

It's a lot easier to talk about significant digits when you write
numbers in scientific notation, which is designed to neatly separate
significant digits from the size of the number, by ensuring that no
extra zeros are required in order to write it.

If we write 3000 as 3 * 10^3, we can see we have only one significant
digit. If I write it instead as 3.000 * 10^3, it has four, because I
am explicitly telling you all four digits. When I write it as 3000,
you can't tell whether the 0's are there because I know they are
correct, or just because I had to put them there to write the number
(placeholders). You really can't say just from looking at the number
what I meant.

Similarly, if we write 5.0 * 10^-3, we clearly have two significant
digits. In this case, that is just as clear from 0.0050, because you
know the zeros before the 5 are there only as placeholders. If those
digits were non-zero, I would have had to write their correct values,
whereas if the zeros in 3000 were really non-zero, I have the option
of writing them or rounding them off.

Scientific notation eliminates the need to write any digits as mere
placeholders, so all the digits you see are significant (unless you
write unnecessary zeros on the left, which would be silly).

Here are several answers in our archives that are relevant to this
question:

Significant Figures and Scientific Notation
http://mathforum.org/library/drmath/view/56291.html

Rules for Significant Figures and Decimal Places
http://mathforum.org/library/drmath/view/58335.html

Significant Digits
http://mathforum.org/library/drmath/view/57160.html

I searched the archives (http://mathforum.org/mathgrepform.html   ) for
"significant digits" to find them.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
Elementary Place Value
Elementary Square Roots
Middle School Exponents

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