Significant Figures

Significant Figures

Sometimes, a question will specify a number of digits to round to. When it doesn't, you need to always make sure to simply round to the correct number of significant digits. If you don't round correctly, you may have an answer that’s wrong even if your math was right.

To calculate the number of significant digits in a number, ignore all 0s before the main chunk of number (the non-0 digits) and ignore all 0s after it. For example, there are the same number of significant digits in the number 1730 as there are in the number 000001730.000000 -- each number has three significant digits.

When you do any mathematical operation in science, consider the number of significant digits in all the numbers the question gives you. The most important number to bear in mind is the smallest number of significant digits of all the numbers given by the question.

Do all the math with all the digits available to you, and then, once you have the answer in the end, your very last step will be to round your answer to that smallest number of significant digits given to you.

For example, if a question tells you to multiply:

92.5718 x 27.0 x 0.007770 x 13.9728 x 0.009089

…you would first multiply all those complete numbers together, for a result of something like 2.466397307... Then, as your last step, round that to the smallest number of significant digits in the numbers given to you: 27.0 has two significant digits, so you would round your result to two significant digits, for a final answer of 2.5.

The theory behind significant figures has to do with precision and certainty. The number of digits in a number expresses the precision of that number; if I tell you a number with a large number of digits, I’m telling you I know the number precisely to that number of digits; simple mathematical operations, even with small integers, can return results with large, even infinite, numbers of digits after the decimal. Using the same number of significant figures in your answer as you measured, or as you were given, has to do with "garbage in, garbage out".

An example: If I measure a circle's radius as about 3ish centimeters, and then I multiply 32 by 𝝅, which computers have calculated to more than 100 trillion digits, I’m not  justified in saying that I know the circle's area any more precisely than 30ish centimeters, even if I use a supercomputer and my math spits out a 28.2733388 followed by a hundred trillion digits.

Another example, if I take a ruler and I measure 11 centimeters, I'm never justified in claiming any more precision than that, no matter what math I do to it. If I happen to know that each gzortaplex is 1.7 centimeters long, and I want to know how many gzortaplexes I have…

  11 centimeters    x     1.0 gzortaplex      =

                      1.7 centimeters

I can then cancel out the centimeters and divide 11 by 1.7, BUT I am not justified in saying I have exactly 6.4705882353… gzortaplexes. I only got into this situation with a precision of 2 significant figures, so I'm only justified in claiming a precision of 2 significant figures as an answer: I've got about 6.5 gzortaplexes.

Another example: If I have a spectacularly sophisticated electron microscope that can measure distance accurately and precisely down to the nanometer, but some of the other numbers in my equation are approximations, then I’m not justified in claiming my results aren't approximate.

If any of the numbers in my math is an approximation, then the final result is always an approximation. It doesn't matter how precise any of the other numbers are. “Garbage in, garbage out”.