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.
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