In physics problems, you use significant digits to express your answers. Significant digits, also often called significant figures, represent the accuracy with which you know your values.
For example, if you know only the values you’re working with to two significant digits, your answer should be 1.5, which has two significant digits, not 1.532984529045, which has 13!
Here’s how it works: Suppose you’re told that a skater traveled 10.0 meters in 7.0 seconds. Note the number of digits: The first value has three significant figures, the other only two.
The rule is that when you multiply or divide numbers, the result has the number of significant digits that equals the smallest number of significant digits in any of the original numbers. So if you want to figure out how fast the skater was going, you divide 10.0 by 7.0, and the result should have only two significant digits — 1.4 meters per second.
On the other hand, when you’re adding or subtracting numbers, the rule is that the last significant digit in the result corresponds to the last significant digit in the least accurate measurement. How does that work? Take a look at this addition example:
So is the result 24.83? No, it’s not. The 12 has no significant digits to the right of the decimal point, so the answer shouldn’t have any either. That means you should round the value of the result up to 25.
Zeros used just to fill out values down to (or up to) the decimal point aren’t considered significant. For example, the number 3,600 has only two significant digits by default. That’s not true if the value was actually measured to be 3,600, of course, in which case it’s usually expressed as 3,600.; the final decimal indicates that all the digits are significant.
Rounding numbers in physics does not quite work the same way as it does in math. For example, in math you would round 45 up to 50, but in physics you round it down to 40. However, you would round the number 35 up to 40 in both physics and math. In math, the rule is to always round the digit 5 up, but the rule in physics is to always the round the digit 5 towards the even. Thus, you would round the numbers 15, 35, 55, and 75 up, but you would round 25, 45, 65, and 85 down in physics. Physicists do this because always rounding the 5 up would create small biases in their data.
Sample question
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You’re multiplying 12.01 by 9.7. What should your answer be, keeping in mind that you should express it in significant digits?
The correct answer is 120.
The calculator says the product is 116.497.
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.
Practice questions
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What is 19.3 multiplied by 26.12, taking into account significant digits?
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What is the sum of 7.9, 19, and 5.654, taking into account significant digits?
Following are the answers to the practice questions:
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504
The calculator says the product is 504.116.
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.
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33
Here’s how you do the sum:
The value 19 has no significant digits after the decimal place, so the answer shouldn’t either, making it 33 (32.554 rounded up).
