1 angstrom to centimeters:
Use a second conversion factor to convert that into inches:
5. 8,760 hr. Convert 1 year into days:
Use a second conversion factor to convert that into hours:
6. 0.18 km/hr. One foot is 12 inches, so convert that into centimeters:
Use a second conversion factor to convert centimeters into meters and then into kilometers:
Convert 1 hour into minutes:
Use your results to convert 10 feet per minute into kilometers per hour:
Checking the Accuracy and Precision of Measurements
Accuracy and precision are important when making (and analyzing) measurements in physics. You can’t imply that your measurement is more precise than you know it to be by adding too many significant digits, and you have to account for the possibility of error in your measurement system by adding a
This section is all about how to properly account for the known precision of the measurements and carry that through the calculations, how to represent numbers in a way that is consistent with their known precision, and what to do with calculations that involve measurements with different levels of precision.
Finding the number of significant digits
In a measurement, significant digits (or significant figures) are those that were actually measured. Say you measure a distance with your ruler, which has millimeter markings. You can get a measurement of 10.42 centimeters, which has four significant digits (you estimate the distance between markings to get the last digit). But if you have a very precise micrometer gauge, then you can measure the distance to within one-hundredth of that, so you may measure the same thing to be 10.4213 centimeters, which has six significant digits.
By convention, zeroes that simply fill out values down to (or up to) the decimal point aren’t considered significant. When you see a number given as 3,600, you know the 3 and 6 are included because they’re significant. However, knowing which, if any, of the zeros are significant can be tricky.
Tip: The best way to write a number so you leave no doubt about how many significant digits there are is to use scientific notation. For example, if you read of a measurement of 1,000 meters, you don’t know if there are one, two, three, or four significant figures. But if it were written as
Rounding answers to the correct number of digits
When you do calculations, you often need to round your answer to the correct number of significant digits. If you include any more digits, you claim a precision that you don’t really have and haven’t measured.
For example, if someone tells you that a rocket traveled 10.0 meters in 7.0 seconds, the person is telling you that the distance is known to three significant digits and the seconds are known to two significant digits (the number of digits in each of the measurements). If you want to find the rocket’s speed, you can whip out a calculator and divide 10.0 meters by 7.0 seconds to come up with 1.428571429 meters per second, which looks like a very precise measurement indeed. But the result is too precise – if you know your measurements to only two or three significant digits, you can’t say you know the answer to ten significant digits. Claiming as such would be like taking a meter stick, reading down to the nearest millimeter, and then writing down an answer to the nearest ten-millionth of a millimeter. You need to round your answer.
Remember: The rules for determining the correct number of significant digits after doing calculations are as follows:
✔ When you multiply or divide numbers: The result has the same number of significant digits as the original number that has the fewest significant digits. In the case of the rocket, where you need to divide, the result should have only two significant digits (the number of significant digits in 7.0). The best you can say is that the rocket is traveling at 1.4 meters per second, which is 1.428571429 rounded to one decimal place.
✔
When you add or subtract numbers: Line up the decimal points; the last significant digit in the result corresponds to the right-most column where all numbers still have significant digits. If you have to add 3.6, 14, and 6.33, you’d write the answer to the nearest whole number – the 14 has no significant digits after the decimal place, so the answer shouldn’t, either. You can see what we mean by taking a look for yourself:
When you round the answer to the correct number of significant digits, your answer is 24.
Remember: When you round a number, look at the digit to the right of the place you’re rounding to. If that right-hand digit is 5 or greater, round up. If it’s 4 or less, round down. For example, you round 1.428 up to 1.43 and 1.42 down to 1.4.
Examples
Q. You’re multiplying 12.01 centimeters by 9.7 centimeters. What should your answer be, keeping in mind that you should express it in significant digits?
A. The correct answer is 120 centimeters squared.
1. The calculator says the product of 12.01 and 9.7 is 116.497.
2. The number of significant digits in your result is the same as the smallest number of significant digits in any of the values being multiplied. That’s two here (because of 9.7), so your answer rounds up to 120 centimeters squared.
Q. You’re squaring 17.3 and then subtracting 79.9134. What is the result, with the correct number of significant digits?
A. The correct answer is 219.
1. The calculator says the square of 17.3 is 299.29.
2. The number of significant digits in a product is the same as the smallest number of significant digits in any of the values being multiplied (when you square 17.3, you’re multiplying 17.3 by itself). There are three significant digits in 17.3, so you round your result to three digits, or 299.
3. The calculator says 299 minus 79.9134 is 219.0866.
4. The 299 has no significant digits after the decimal place, so the answer shouldn’t either. You round your result to 219.
Practice Questions
1. What is 19.3 multiplied by 26.12, taking into account significant digits?
2. What is the sum of 7.9 grams, 19 grams, and 5.654 grams, taking into account significant digits?
3. What do you get if you divide 1.93 meters by 0.069 seconds, keeping the correct number of significant digits?
4. What do you get if you add 5.2 square meters to the result of 1.36 meters times 0.7130 meters, keeping the correct number of significant digits?