Explanation of Significant Figures

Date: 09/11/2007 at 17:55:10
From: Forest
Subject: Significant figures

My Physical Sciences teacher introduced our Honors class to the
concept of "Significant Figures".  At first it wasn't a problem.  She
made us take notes and include the book-given definition.  But then we
got a graded worksheet back, and almost everyone got a certain
question on significant figures wrong.  So, for the past two days,
there has been a heated debate going on over significant figures.

Ultimately, here's my question: How can 1,000,000 have only one
significant figure, but 1,000,000. have 7?  I understand that it's the
decimal point, but saying that 1,000,000 has only one significant
figure is coming across to me the same as saying that 1 = 1,000,000
because none of the zeros are important or "significant".  Also, my
teacher asked two other teachers for information, and one mentioned
the Atlantic-Pacific rule which "works most of the time", and the
other said it works "only some of the time". 

What's going on here? And what's up with the whole 1,000,000 thing?
And how many significant figures does .0549 have, and how about
0.0549?  I read someone else's question on this site, and it was
answered that 390 has only 2 SF, and 390. has 3.  However, 3900 has
three also, but 3900. has four.  What??????  Why???

To me, saying that 1,000,000 has only 1 SF, but 1,000,000. has 7 SF 
seems to me to mean the same as 1 = 1,000,000; but this doesn't make 
sense to me.  Having $1,000,000 is a lot different than having one 
dollar.  Is there a decent, constant rule about significant figures 
that I can remember?  And what makes "placeholders" unimportant, and 
what makes a number a "placeholder"?  0.0008 is a lot different than 
0.08.  Again, how are these zeros unimportant???  I'm very confused on 
a lot of this, and right now I'm so confused that I can't even 
explain everything that I'm confused about!

I understand this:
 1) That 0.435 has only three significant figures.
 2) That 435 has three significant figures as well.
 3) That 4.035 has four significant figures because the zero has a 
digit of a value of one or higher on each side.

I'm not sure about the rest of the significant figures thing...If 
you have time, in your answer, would it be out of your way to touch 
on all "combinations" at least a little?  (By this I mean: 0.03 has 
___ amount of SF because______, and 3.03 has... and 303 has... and 
try to get all of those?  Thanks!)

Date: 09/11/2007 at 22:50:13
From: Doctor Peterson
Subject: Re: Significant figures

Hi, Forest.

Have you looked at all we've said about significant digits?  Some
comments are found here:

  Selected Answers: Significant Digits
  http://mathforum.org/library/drmath/sets/select/dm_sig_digits.html 

I can't tell for sure which one(s) you are referring to.  I think most
of what you're asking is in there somewhere.  I'll also reply to your
questions.

>...what's up with the whole 1,000,000 thing?  And how many
>significant figures does .0549 have, and how about 0.0549?  I read
>someone else's question on this site, and it was answered that 390
>has only 2 SF, and 390. has 3.  However, 3900 has three also, but
>3900. has four.  What??????  Why???

A significant digit is one that we can properly assume to have been
actually measured.  We make such an assumption based on a set of
conventions about how to write measurements: we only write digits that
we have actually measured, unless the digits are zeros that we just
have to write in order to write the number at all.  For example, if I
use a ruler that shows millimeters (tenths of a centimeter), then if I
wrote 12.34 cm I would be lying, because I can't really see whether
that 4 (tenths of a millimeter) is correct.  If I wrote 12.30 cm, I
would still be lying, because I would be claiming that I know that
last digit is exactly zero, and I don't.  I have to write it as 12.3
cm.  All three of these digits are significant--they really represent
digits I measured.  They are still significant if I write it as 0.123
m (just changing the units to meters), or as 0.000123 km (using
kilometers this time).  Here, the zeros I added are there ONLY to make
it mean millionths; they are placeholders.

It isn't really quite true, as far as I am concerned, that 1,000,000
has one significant digit.  It MIGHT have only one, or it might have
seven.  The problem is that you can't be sure; the rule of thumb that
is being used doesn't allow you to indicate that a number like this
has 2, or 3, or 6 significant digits.  Maybe I know that it's between
95,000 and 1,005,000, so that the first three digits are all correct,
but the rules don't let me show that just from the way I write it. 
I'd have to explicitly tell you, the value is 1,000,000 to three
significant digits.

Significant digits are really best used only with scientific notation,
which avoids this problem.  We can write 1,000,000 with any number of
significant digits:

  1: 1 * 10^6
  2: 1.0 * 10^6
  3: 1.00 * 10^6
  4: 1.000 * 10^6
  5: 1.0000 * 10^6
  6: 1.00000 * 10^6
  7: 1.000000 * 10^6

The trick is that we don't need any "placeholders" in this notation;
every digit written is significant.

>Is there a decent, constant rule about significant figures 
>that I can remember?  And what makes "placeholders" unimportant, and 
>what makes a number a "placeholder"?  0.0008 is a lot different than 
>0.08.  Again, how are these zeros unimportant???  I'm very confused
>on a lot of this, and right now I'm so confused that I can't even 
>explain everything that I'm confused about! 

The main idea of significant digits is as a rough representation of
the RELATIVE precision of the number; that is, it is related to the
percentage error that is possible.  For example, 0.08 indicates that
the measurement might be anywhere between 0.075 and 0.085; a more
sophisticated way to indicate this is to write it as 0.08 +- 0.005,
meaning that there might be an error of 0.005 in either direction.
This error is 0.005/0.08 = 0.0625 = 6.25% of the nominal value.  If
you do the same thing with 0.0008, you get the same relative error;
although the value itself is very different, the error is proportional
to the value.  Both have one significant digit, which likewise
indicates the relative error, NOT the actual size of the error.

>I understand this:
> 1) That 0.435 has only three significant figures.
> 2) That 435 has three significant figures as well.
> 3) That 4.035 has four significant figures because the zero has a 
>digit of a value of one or higher one each side.
>
>I'm not sure about the rest of the significant figures thing...If 
>you have time, in your answer, would it be out of your way to touch 
>on all "combinations" at least a little?  (By this I mean: 0.03 has 
>___ amount of SF because______, and 3.03 has... and 303 has... and 
>try to get all of those?  Thanks!)

The significant digits of a number are all the digits starting at the
leftmost non-zero digit, through the rightmost digit, if there is a
decimal point.  If there is no decimal point, then you count only
through the rightmost nonzero digit, because zeros beyond that MIGHT
be there only to give other digits the correct place value.  (Zeros to
the left of the first nonzero digit serve a similar purpose, but
aren't counted anyway.)

Here are the cases I can think of:

Decimal point to the left of all nonzero digits: Count from the
leftmost nonzero digit all the way to the end.

  .103   0.103  0.00103  0.0010300
   \_/     \_/      \_/      \___/
    3       3        3         5

Decimal point in the middle: Count from the leftmost nonzero digit all
the way to the end.

  10.305  00103.0500
  \____/    \______/
    5          7

Decimal point on the right: Count from the leftmost nonzero digit all
the way to the end.

  10305.  1030500.
  \___/   \_____/
    5        7

No decimal point: Count from the leftmost nonzero digit to the
rightmost nonzero digit.

  10305  1030500 [here we don't know whether the last 2 zeros are
  \___/  \___/..  measured or just placeholders]
    5      5?   

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 09/12/2007 at 08:22:00
From: Forest
Subject: Thank you (Significant figures)

Dr. Peterson,

Thank you so much for the help on Significant Figures!  I think I
understand it now completely, and I liked the way you described
everything...especially the examples.  Oh, and sorry for making you
write all of those out...that probably took up a lot of your time. 
But thanks again for the help!  The Dr. Math service is amazing and I
always come out understanding whatever I had a problem with
completely.  Great job!

Forest