Kia ora my junior scientists! If you think back to high school maths or science, you probably remember practicing significant figures. And then never using them again, ensuring that knowledge was safely disposed of along with other traumatic childhood events. In this video, I am going to explain why significant figures are actually important to know. I’ll refresh your memory on the rules, and show you why in science, 8 x 8 is not 64, but 60. As you move through your studies and your scientific career, you will be expected to know and use significant figures, so let’s talk about why. The very simple reason why we use significant figures is because science is not maths. Whereas maths deals with exact numbers, scientists can’t measure things exactly, not even using highly precise scientific instruments. Because of this, whenever we measure something to study it, there is a degree of uncertainty associated with it. When used properly, significant figures capture and communicate that uncertainty. When I look at a measurement of, for example, 1g I know that it is a less precise measurement than 1.0g or 1.00g, even though arithmetically they are equivalent.
So what is a significant figure? There are 5 super simple rules that you can follow to count the number of significant figures in your measurements.
1. All non-zero numbers ARE significant.
2. Zeros between two non-zero digits ARE significant.
3. Leading zeros are NOT significant.
4. Trailing zeros are only significant if there is a decimal point in the number.
5. Exact numbers (non-measurements) have infinite significant figures.
So now you know what a significant figure is, and using those rules, you should be able to work out how many significant figures there are in any given measurement. But why do you even need to do that? Because when we analyze data, we normally have to perform calculations on those measurements. Maybe we are reporting an average, or scaling the data so it is comparable with some other data. These calculations, if we are not very careful, can falsely increase the apparent precision of our measurements. We don’t want that, because we are lying.
So the really important question is - how do you know how many significant figures to use? Now that you know what significant figures are, and why we use them, you can probably work out the answer to that for yourself. But to make sure, we’ll go through it here. If we only have a certain level of precision in our measurements, we can’t get any more precision than that. Therefore, we can’t end up with any more significant figures than what we started with. We can’t coax more significant figures out any measurements with less significant figures, so we always use the smaller number of significant figures. It’s not more accurate to use more numbers, it’s less accurate because you’re falsifying the certainty.
That leaves us with two simple rules. If we are multiplying or dividing, then the number of significant figures in our answer can’t be more than the smallest number of significant figures that we started with.
If we are adding or subtracting, then the number of fractional digits can’t be more than the smallest number of fractional digits that we started with.
Let’s look at a few examples.
A simple addition problem: 7.9 + 3. The calculator says 10.9 is the answer, but we can’t have more fractional digits than the smallest number of fractional digits in the input, so that means that we have to round our number, and 11 is the correct answer.
Let’s say we wanted to calculate a ratio of two measurements: 3.7836 and 4.50, so our calculation is 3.7836 divided by 4.50. If we use a calculator, we get the answer of 0.8408. But the calculator assumes we are doing maths (which uses exact numbers), but we are doing science, which uses measurements, so we need to round the answer to the number of significant figures we have. 3.7836 has 5 significant digits, because there are no zeroes. 4.50 has three significant digits, because the trailing zeroes are significant when there is a decimal in the number. Therefore the smallest number of significant digits is three, and therefore our answer should have three significant digits. Leading zeroes are not significant, so we keep these three digits here, and then round. In this case we round up, and therefore the correct answer is 0.841.
Using this logic, I can now prove to you with science that 8 x 8 is 60, providing we are dealing with measurements and not exact numbers. The calculator tells us the answer is 64, but as long as we are dealing with measurements, then we need to take significant figures into account. In this case, there is only one significant figure in each number, so we keep the first digit of the answer, and round the remainder. Leaving our answer as 60!
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