# Units, Scientific Notation, and Significant Figures

### Significant Figures

#### 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 decimeters, 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:

1. Non-zero digits are always significant; 1.121 has four significant digits.
2. Any zeros between two significant digits are significant; 1.08701 has six significant digits.
3. 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.
4. Zeros after the decimal point and after figures are significant; in the number 0.2540, the 2, 4, 5 and last 0 are significant.
5. Exponential digits in scientific notation are not significant; 1.12x106 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.

#### Significant Figures in Operations

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

#### 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.

For example, let's say you have to add the numbers 1.121, 48.00679392, and 6.3457:

1.121 + 48.00679392 + 6.3457 = 55.47349392
But, because 1.121 has only three decimal places, the answer must actually be: 55.473, since it is at the third decimal place that uncertainty begins to enter the picture.

#### Significant Figures in Multiplication and Division

The rule governing multiplication and division of significant figures is slightly different than that for addition and subtraction, but just as simple:

The final value can only have as many significant figures as the original value with the least significant figures.

For example, consider the following situation: a scientist needs to calculate a constant value, K, based on the following equation:

K = (D x E) / B
where B, D and E are measured values that the scientist observed (weight, volume, temperature, pressure).

B=6.00 g D= 22 C E= 22.457 mL

22.457 mL has 5 significant figures and 22 C has 2 significant figures. The number that the calculator gives is 82.34233... However this has 7 significant figures and none of the measurements were this accurate. In fact, we must reduce the answer to only two significant figures, since that's how many 22 has. The answer, K, must be truncated to 82 ml C/g to reflect the 2 significant figures in the D value.

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