Significant Digits in Numbers Written in Scientific Notation

Date: 06/20/2005 at 19:17:35
From: Geoffrey
Subject: significant digits & scientific notation

Say you have the number 2.34 * 10^4.  Why does my textbook say there
are 3 significant digits, rather than 5 (from 23400--I'm assuming that 
the last two zeros are also significant)?

Date: 06/20/2005 at 22:53:17
From: Doctor Peterson
Subject: Re: significant digits & scientific notation

Hi, Geoffrey.

You're just making a wrong assumption: the whole point of significant
digits is that zeros you write only to get the other digits in the
right places are NOT significant.  When we write a number in 
scientific notation, we write ONLY the digits whose values we really 
know, so that all the digits written are in fact signficant, and no 
others.  In ordinary notation, you can't be sure of that.  That is one 
major advantage of scientific notation.

Now, IF you had measured something as exactly 23400 meters, say (to
the nearest meter), then you would write it in scientific notation as
2.3400 * 10^4.  Then the zeros would be significant.  But when it is
written as 2.34 * 10^4, we take that to mean that, even if you write
it as 23400, the zeros are not known to be accurate, and should not be
taken as being significant.

See this page for more:

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

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

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

Date: 06/21/2005 at 17:04:11
From: Geoffrey
Subject: significant digits & scientific notation

Thanks.  Here's another example from my book that I still don't
understand:

"Given the radius of a sphere, 8.6 cm, calculate the volume. 

So V = 4/3pi(0.086m)^3 = 2.6643 * 10^(-3) m^3 = 2.7 * 10^(-3) m^3 "

"Here, the result has been rounded to two digits because the radius
has two significant digits."

But I would first multiply 2.6643 * 10^(-3) = 0.006643, and then round
to the required two significant digits giving the number of "0.0".
What's wrong here?

Date: 06/21/2005 at 22:28:20
From: Doctor Peterson
Subject: Re: significant digits & scientific notation

Hi, Geoffrey.

You'll want to read some of the references in the link I gave you last
time, which explain what significant digits are.  You are using the
wrong digits.  Significant digits are digits from the first to the 
last NONZERO digit, except that in some cases zeros on the RIGHT may 
be significant if they are obtained from an actual measurement to be
exactly zero.

In the number 0.0026643, the zeros are NOT significant, because they
are there only to keep the nonzero digits in the right place.  If you
write it in scientific notation as 2.6643 * 10^-3, it is clearer that
the first two significant digits are the 2 and 6, and rounding it to
two significant digits gives you 2.7 * 10^-3 = 0.0027.

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


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

Date: 06/22/2005 at 17:02:56
From: Geoffrey
Subject: significant digits & scientific notation

I don't understand when you say, "the zeros are NOT significant,
because they are there only to keep the nonzero digits in the right
place."

If I'm correct, the significant digits in a number are the digits that
have been measured accurately, so I would say that the two zeroes in
0.024 are also significant digits.(?)

Date: 06/22/2005 at 22:40:21
From: Doctor Peterson
Subject: Re: significant digits & scientific notation

Hi, Geoffrey.

It can be hard to explain the precise definition of significant digits
clearly, and also to grasp the concept for the first time!  I hope if
you read more of our explanations you will find one that works for you.

Zeros on the left are not significant because they are not written for
the same reason as zeros on the right.  I decide to cut off a number
after a certain digit because any digits to the right are not known
accurately: 0.0120 differs from 0.012 in that, for the former, I know
that the zero on the right is correct, and for the latter, I don't.
Zeros on the left do, I suppose, tell you that those digits are not
something other than zero, but you wouldn't change them if you made a
more accurate measurement.  The change from 0.012 to 0.112 is not just
an improvement in accuracy, but a complete change in the value.

But ultimately, you have to recognize that we define significant
digits as we do not because the word "significant" forces us to define
it that way, but just because, as some of the explanations we have
given explain, that is the definition that works to give us a rule of
thumb for determining the precision of the result of a multiplication.
The leftmost nonzero digit indicates the size of the number, and the
rightmost one indicates the finest detail measured.  The distance
between those digits indicates the relative accuracy, and that is what
we need to use.

So the significant digits are all the digits starting at the first
non-zero digit, and ending at the last digit that would be written in
scientific notation--the last non-zero digit, or possibly the last 
zero that is written because it represents an accurate measurement.

Does that help?


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

Date: 06/23/2005 at 17:08:20
From: Geoffrey
Subject: significant digits & scientific notation

Thanks, significant digits are definitely not a simple subject (I
thought it was easy at first sight, but that's an assumption that
apparently can never be made in mathematics), but now I'm leaving the
infinite loop of confusion.