then you would know that there are four significant figures.
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?
Practice Answers
1. 504. The calculator says the product is 504.116. However, 19.3 has three significant digits, and 26.12 has four, so you use three significant digits in your answer. That makes the answer 504.
2. 33 g. Here’s how you do the sum of 7.9, 19, and 5.654:
The value 19 has no significant digits after the decimal place, so the answer shouldn’t either, making it 33 grams (32.554 grams rounded up).
3. 28 m/s. The calculator says the quotient of 1.93 and 0.069 is 27.9710144928, and your answer should have units of meters divided by seconds, or meters per second. Because 1.93 has three significant digits and 0.069 has two (the leading zero doesn’t count), you use two significant digits in your answer. That makes the answer 28 meters per second.
4. 6.2 m2. The calculator says the product of 1.36 and 0.7130 is 0.96968, and if you add 5.2, you get 6.16968. Your answer has units of meters squared. Because 5.2 has only one significant digit after the decimal place, which is less than the three significant digits after the decimal place you keep after multiplying 1.36 and 0.7130, you use one significant digit after the decimal place in your answer. That makes the answer 6.2 meters squared.
Physicists don’t always rely on significant digits when recording measurements. Sometimes, you see measurements that use plus-or-minus signs to indicate possible error in measurement, as in the following:
The
Arming Yourself with Basic Algebra
Physics deals with plenty of equations, and to be able to handle them, you should know how to move the variables in them around. Note that algebra doesn’t just allow you to plug in numbers and find values of different variables; it also lets you rearrange equations so you can make substitutions in other equations, and these new equations show different physics concepts. If you can follow along
