Significant Figures Help (page 2)
Introduction to Significant Figures
When multiplication or division is done using power-of-10 notation, the number of significant figures (also called significant digits ) in the result cannot legitimately be greater than the number of significant figures in the least-exact expression.
Consider the two numbers x = 2.453 × 10 4 and y = 7.2 × 10 7 . The following is a valid statement in pure arithmetic:
But if x and y represent measured quantities, as they would in experimental science or engineering, the above statement needs qualification. We must pay close attention to how much accuracy we claim.
How Accurate Are We?
When you see a product or quotient containing a bunch of quantities in scientific notation, count the number of individual numerals (digits) in the decimal portions of each quantity. Then identify the quantity with the smallest number of digits, and count the number of individual numerals in it. That’s the number of significant figures you can claim in the final answer or solution.
In the above example, there are four single digits in the decimal part of x, and two single digits in the decimal part of y. So we must round off the answer, which appears to contain six significant figures, to two. (It is important to use rounding, and not truncation!) We should conclude that:
No More Squigglies
In science and engineering, approximation is the rule, not the exception. If you want to be rigorous, therefore, you must use squiggly equals signs whenever you round off any quantity, or whenever you make any observation. But writing these squigglies can get tiresome. Most scientists and engineers are content to use ordinary equals signs when it is understood there is an approximation or error involved in the expression of a quantity. From now on, let us do the same. No more squigglies!
Suppose we want to find the quotient x/y in the above situation, instead of the product xy. Proceed as follows:
What About Zeros?
Sometimes, when you make a calculation, you’ll get an answer that lands on a neat, seemingly whole-number value. Consider x = 1.41421 and y = 1.41422. Both of these have six significant figures. The product, taking significant figures into account, is:
This looks like it’s exactly equal to 2. In pure mathematics, 2.00000 = 2. But not in physics! (This is the sort of thing that drove the purist G.H. Hardy to write that mathematicians are in better contact with reality than are physical scientists.) Those five zeros are important. They indicate how near the exact number 2 we believe the resultant to be. We know the answer is very close to a mathematician’s idea of the number 2, but there is an uncertainty of up to ±0.000005. If we chop off the zeros and say simply that xy = 2, we allow for an uncertainty of up to ±0.5, and in this case we are entitled to better than that. When we claim a certain number of significant figures, zero gets as much consideration as any other digit.
In Addition And Subtraction
When measured quantities are added or subtracted, determining the number of significant figures can involve subjective judgment. The best procedure is to expand all the values out to their plain decimal form (if possible), make the calculation as if you were a pure mathematician, and then, at the end of the process, decide how many significant figures you can reasonably claim.
In some cases, the outcome of determining significant figures in a sum or difference is similar to what happens with multiplication or division. Take, for example, the sum x + y, where x = 3.778800 × 10 –6 and y = 9.22 × 10 –7 . This calculation proceeds as follows:
But in other instances, one of the values in a sum or difference is insignificant with respect to the other. Let’s say that x = 3.778800 × 10 4 , while y = 9.22 × 10 –7 . The process of finding the sum goes like this:
In this case, y is so much smaller than x that it doesn’t significantly affect the value of the sum. Here, it is best to regard y, in relation to x or to the sum x + y, as the equivalent of a gnat compared with a watermelon. If a gnat lands on a watermelon, the total weight does not appreciably change in practical terms, nor does the presence or absence of the gnat have any effect on the accuracy of the scales. We can conclude that the “sum” here is the same as the larger number. The value y is akin to a nuisance or a negligible error:
x + y = 3.778800 × 10 4
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
Using a calculator, find the value of sin (0° 0′ 5.33″), rounded off to as many significant figures as are justified.
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
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?
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:
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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