Monday, February 16, 2009

Significant Figures and Scientific Notation

Question:
What are significant figures? What is scientific notation? Why do I have to report my answers in general chemistry using significant figures and scientific notation?

Answer:

The short answer for all three questions is that significant figures and scientific notation make numbers easier to read, and make your answers make more sense. You probably need more information than that, though. Let's start with the general concept of significant figures, see how they relate to scientific notation, and then see how we combine the two concepts to answer questions.

Significant Figures

It is almost always impossible to make an exact measurement. For example, imagine that we measured an object with a ruler that was only marked every centimeter. We could be sure that the length was, say, between 11 cm and 12 cm, and we could guess that it was about 11.2 cm (rather than 11.3 cm). However, we'd have no idea beyond the tenths place; the actual length could be anywhere from 11.15 cm to 11.25 cm. Significant figures are a way of logging how much we had to guess. To find the significant figures in our measurement, we simply count the digits that mean something. Our measurement only has three significant figures, since we can't log anything past the 2.

This gives us our first rule about significant figures:
  1. Any non-zero digit is always significant.
Significant figures are a little more complicated when zeros are involved. They aren't a lot more complicated, especially if you remember that significant figures are the digits that mean something, but zeros can take some getting used to.

Our next rule about significant figures should be easy enough to understand:
  1. If a measurement includes a decimal place, any zeros that follow non-zeros are significant. For example, in 11.20000 cm, all of the zeros are significant; we wouldn't report them if we hadn't measured the number out that far.
That rule becomes important when you combine it with our next rule:
  1. Leading zeros are not significant. Leading zeros are the placeholders at the start of a decimal number, such as the zeros in 0.000001. Only the 1 is significant, since all of the zeros are just placeholders.
That only leaves us with two types of zeros:
  1. Zeros between significant digits are significant. For example, the zeros in 10001 are significant.
  2. Any zeros we didn't mention above are not significant. This is the rule people forget. For example, the zeros in 100 don't follow a decimal place, and they aren't between significant digits, so they aren't significant. "100 m" only has 1 significant figure.
We've seen that significant figures are important when we make a measurement. Any significant figure we report in a measurement is exactly that; it's significant. Our measurement is already a guess in the last digit, so we don't want to tack on extra digits that don't mean  anything. When we perform calculations using that guess, we shouldn't pretend we were able to measure more than we actually did measure. We'll look at that more in a moment, but first let's see why scientific notation is useful, and how it makes things easier.

Scientific Notation

Scientific notation, such as "2.0214 × 104," might look complicated, but it actually makes things easier. When you report a number in scientific notation, you only include the significant figures. You also end up with a number that's a lot easier to work with for calculations, once you learn how to use it.

To convert a numer to scientific notation:
  1. Write the significant digits as a number between 1 and 10.
  2. Count the digits between the new decimal point and the old decimal point.
  3. Write that number as a positive exponent of 10 if the original number was bigger than 1 (or smaller than -1), and as a negative exponent of 10 if the original number was between 0 and 1 (or between 0 and -1).
For example, 10,120,000 is 1.012 × 107 (since the decimal place moved 7 places to the left) and 0.00000267 is 2.67 × 10-6 (since the decimal place moved 6 places to the right).

When you multiply numbers in scientific notation, you can add the exponents of the 10 part to estimate your final answer. For example, 1.012 × 107 times 2.67 × 10-6 is roughly 10-1, or 0.1. Keeping that in mind will help you avoid a lot of "I typed this number into my calculator incorrectly" errors.

Figuring Out Significant Figures in a Calculated Number

When you add, subtract, multiply, or divide measurements, you have to make sure your result doesn't add to or lose the precision in the original measurement. To help with this, there are some simple rules:
  • When you multiply or divide, the answer has the least number of significant figures of the numbers being multiplied or divided. For example 123 × 1234 × 2 × 21315879 × 867879451 only has 1 significant figure, since "2" only has 1 significant figure.
  • When you add or subtract, the answer is significant to the least significant decimal place in the numbers you added or subtracted. For example, 100 + 1 + 5.123155 + 101 is only significant in the hundreds place, since "100" is only significant in the hundreds place.
  • You should apply these rules at the end of a calculation. If you apply the rules too early, you can lose some of the information you started with. 
Work through the question below to see if you understand how to apply these rules.


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1 comment:

  1. ahm?????what is scientific notation in calculations

    ReplyDelete