## Principle of Uncertainty & Significant Figures

Principle of Uncertainty

Scientists report their measurements as accurately as possible. However, no measurement is 100% precise. As a result, scientists have developed a method of reporting their level of precision (or potential for error) with each measurement they make. This level of precision is called *uncertainty*. The more precise a measurement is the less uncertainty there is related to it.

Astronomers, like other scientists, always have some *uncertainty* with their measurements. Why do you suppose that scientists report the average distance between the earth and the sun at 93,000,000 miles? Why don’t they report it at 93,285, 321.12343168 miles or perhaps 93,284,321.12 miles? They report it to this level of accuracy, because the distance may vary hundreds of thousands of miles depending on the time of the year. So when they say the earth is 93,000,000 million miles away from the sun, effectively scientists are saying that the earth is 93,000,000 million ± 500,000 miles from the sun or that the earth is between 92,500,001 and 93,500,000 miles from the sun.

Consider another example of uncertainty. Suppose that Andrew weighs himself on his bathroom scale at home. This scale, which measures to the nearest pound, indicates that he weighs 151 pounds. When he is weighed on his doctor’s scale, which measures to the nearest 1/10th of pound, he weighs 151.3 pounds. However, when he is weighed on a truck scale, which weighs to the nearest 100 pounds, he weighs 200 pounds. So what does Andrew really weigh? He weighs somewhere close to 151.3 pounds. Does this mean that the other measurements were inaccurate? No, not if you consider the *principle of uncertainty*. Properly reported Andrew’s weight on the truck scale would be 200 ± 50 pounds. The ± 50 lbs. is an indication of the uncertainty of the measurement on the truck scale. Therefore 151 pounds and 151.3 pounds are both within that level of uncertainty. In the same way when Andrew is measured on the bathroom scale he weighs 151 ± 0.5 pounds. In that case 151.3 is within that level of uncertainty.

Significant Figures

When scientists complete computations with measurements that involve uncertainty, they use a method of “rounding” that is called *significant figures*. Why are they referred to as significant figures? If Amy weighed 121 pounds on her bathroom scale would she report her weight at 121.214543 pounds? Of course she wouldn’t, those additional digits of “0.214543” could not be actually measured on the scale and therefore would not be considered significant. In the same way if your phone bill was $181.00 for three month would you report that your monthly bill was $ 60.3333333333/month? Since you can not make a payment less than 0.01 dollars you would simply report your monthly phone bill as $ 60.33 rather than $ 60.3333333333/month.

Why do scientists use *significant figures*? It provides them with a method of reporting the level of accuracy of their measurement. It also provides a systematic method of rounding calculations that involve scientific measurements.

### Rules for Determining the Number of Significant Figures

The rules for significant figures are relatively basic with the exception of a couple of the special rules for zeros.

#### Rules for Non-zero Digits

- Non-zero digits are always significant.
*Example: 232 has three significant digits.*

#### Rules for Zeros

- Any zero/s between two significant digits are significant.
*Example: in 303 the zero is significant.* - Any zero/s that comes
*after the last non-zero digit*and to the right of the decimal point are significant.*Example: in 3.4400 the zeros are significant.* - For numbers smaller than 1, any zero/s which appear to the right of the decimal point and before the first significant digit is
*not*significant.*Example: in .05 the zero is not significant.* - Any zero/s which appear to the right of a decimal point when no other digits are present are considered significant.
*Example: in 6.00 the zeros are significant.* - Any zero/s which appear to the left of the decimal point are significant.
*Example: in 7800. the zeros are significant.* - Any zero/s which are part of a whole number that appear after all significant digits are not significant, if there is no decimal point present.
*Example: in 98,000 the zeros are not significant.*

### Determining Significant Figures When Multiplying or Dividing

*After completing the multiplication or division* of two or more measured items, determine which multiplier or dividend/divisor has the *fewest significant figures* and then round the product or quotient so that it has the same number of significant figures. *Example: 3.0 grams x 12.1 grams = 36.3 grams. However “3.0 grams” has only two significant figures, therefore the answer would be reported as 36 grams (two significant figures).*

### Determining Significant Figures When Adding or Subtracting

*After completing the addition or subtraction* of two or more measured items, round the result to the least significant place value which is in common with all of the numbers being added or subtracted. *Example: 3.015 grams + 5.1 grams + 2.16 grams = 10.275 grams which would be rounded to 10.3 grams because “5.1 grams” is only significant to the 10 ^{th}s place and therefore 10.275 should be rounded to the 10^{th}s place.* If the numbers do not have any significant place value in common, then round to the next place value up. For example, if you were adding 10 and 4, the zero in the 10 is not significant and therefore there is no place value in common. As a result the tens place is the next available place value and should be used to determine what the answer is in significant figures. In this case, 10 + 4 = 10 (in significant figures).

### Special Rules for Determining Significant Figures

- The rules only apply to
*measured quantities*, not discrete numbers or a constant within a formula. - When a series of calculations are being done, the number of significant figures should be completed at the last step and
*not between*each step. - If a number cannot be given in significant figures in standard form, it should be reported in scientific notation.

Questions Related to Uncertainty & Significant Figures

- Scientists consider uncertainty when they make m_________________.

- Emily measured 3.152 grams of NaCl on her balance.

What was her level of uncertainty? ± _______________________.

- Mark found that NASA reported the distance between the earth and the moon at 406,700 kilometers.

What was the level of Mark’s uncertainty? ± ___________________.

- Why are scientists concerned about potential uncertainty related to their measurement?

- What problems could occur if scientists did not report their findings using significant figures?

- How are significant figures and uncertainty related?

- Why do scientists use significant figures?

- Is the zero in “3.0” significant? Explain.

- How many significant figures are there in 7.0605?

- How many significant figures are there in 0.0050120?

- What is the answer to 3.21 grams x 4.6 gram in significant figures?

- What is the answer to 9.15 inches + 10 inches in significant figures?