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Significant Figures Help (page 2)

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By McGraw-Hill Professional
Updated on Oct 3, 2011

A Bit Of Confusion

When a value is not in power-of-10 notation, it is best to convert it to that form before deciding on the number of significant figures it contains. If the value begins with 0 followed by a decimal point, for example 0.0004556, it’s not too difficult to figure out the number of significant digits (in this case four). But when a number is large, it might not be clear unless the authors tell you what they have in mind.

Have you heard that the speed of light is 300,000,000 meters per second? To how many significant digits do they claim this? One? Two? Three? More? It turns out that this expression is accurate to three significant figures; they mean to say 3.00 × 10 8 meters per second. A more accurate value is 299,792,000. This expression happens to be accurate to six significant figures: in scientific notation it is 2.99792 × 10 8 .

If this confuses you, you are not alone. It can befuddle the best scientists and engineers. You can do a couple of things to avoid getting into this quagmire of uncertainty. First, always tell your audience how many significant figures you claim when you write an expression as a large number. Second, if you are in doubt about the accuracy in terms of significant figures when someone states or quotes a figure, ask for clarification. It is better to look a little ignorant and get things right, than to act smart and get things wrong.

Significant Figures Practice Problems

Practice 1

Using a calculator, find the value of sin (0° 0′ 5.33″), rounded off to as many significant figures as are justified.

Solution 1

Our angle is specified to three significant figures, so that is the number of significant figures to which we can justify an answer. We are looking for the sine of 5.33 seconds of arc. First, let’s convert this angle to degrees. Remember that 1″ = (1/3600)°. So:

• sing a calculator, we get:

sin 0.00148° = 2:58 × 10 –5

Practice 2

Suppose a building is 205.55 meters high. The sun is shining down from an angle of 33.5° above the horizon. If the ground near the building is perfectly flat and level, how long is the shadow of the building?

Solution 2

The height of the building is specified to five significant figures, but the angle of the sun above the horizon is specified to only three significant figures. Therefore, our answer will have to be rounded to three significant figures.

The situation is illustrated in Fig. 7-2. We assume the building is perfectly flat on top, and that there are no protrusions such as railings or antennas. From the right-triangle model, it is apparent that the height of the building (205.55 meters) divided by the length of the shadow (the unknown, s ) is equal to tan 33.5°. Thus:

Fig. 7–2. Illustration for Solution 2.

This is 311 meters, rounded to three significant figures. When performing this calculation, the tangent of 33.5° was expanded to more than the necessary number of significant figures, and the answer rounded off only at the end. This is in general a good practice, because if rounding is done at early stages in a calculation, the errors sometimes add together and produce a disproportionate error at the end. You might want to try rounding the tangent of 33.5° in the above problem to only three significant figures, and see how, or if, that affects the final, rounded-off answer.

Practice Problems for these concepts can be found at:  Scientific Notation Practice Test

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