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Defining Significant Figures

No experimental measurement can possibly be perfectly precise. Take, for
example, a wooden stick that is approximately two meters long. If a scientist
were to measure that stick with a ruler marked only with meters, then he could
only conclude with certainty that the stick measured 1 meter (though of course
he would recognize that his measurement was inexact). If his ruler was marked
with deciliters, then he could see with certainty that the stick measured 1.1
meters. If he could measure centimeters, he might see that the stick actually
measured 1.12 meters. Using a ruler with millimeters he could see the stick is
actually 1.121 meters long. Each smaller measurement allows the scientist to
determine the length of the stick with a bit more accuracy. But no scientist can
use a ruler to great effect for distances much smaller than a millimeter; such
small distances are simply beyond the ability of the scientist's ability to see.
At some point his measurements will necessarily become slightly inaccurate.

Scientists account for this unavoidable uncertainty in measurement through the
use of significant digits. Significant digits do not remove the
uncertainty; instead they alert others as to where the uncertainty lies. In the
case of our measurement of the stick, the value 1.121 meters alerts the next
scientist to come along that the last 1 digit on the right might be slightly
inaccurate.

Five rules govern significant figures:

- Non-zero digits are always significant; 1.121 has four significant digits.
- Any zeros between two significant digits are significant; 1.08701 has six
significant digits.
- Zeros before the decimal point are placeholders and
*not* significant;
in the number .00254, only the 2,5 and 4 are significant, meaning the number has
3 significant figures.
- Zeros after the decimal point and after figures are significant; in the
number 0.2540, the 2, 4, 5 and last 0 are significant.
- Exponential digits in scientific notation are not significant; 1.12
*x*10^{6}
has three significant digits, 1, 1, and 2.

These rules ensure accurate representation and interpretation of data. If, for
example, you were to read of an experimental reaction in which the resulting
chemical weighed 0.0254 g, you would know that the measurement is accurate to
0.0001 g and contains 3 significant figures.

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Significant Figures in Operations

When making calculations, significant figures become very important. You must
always be careful to remember how many significant figures you're separate
values have. The rules governing addition and subtraction, and those governing
multiplication and division are a little different.

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Addition and Subtraction of Significant Figures

Addition and subtraction of significant figures follows a simple rule:

The final value must have only as many decimals as the original value with the
least number of decimal places.