Significant figures · how many figures do you write

The display

0:00You measure two things. Your calculator gives: zero point three three seven five. You copy it all down. Four figures, and you measured two. Where do those other two come from?

Every measurement has play

0:17If you measure a quantity, you usually do not find exactly the right value. That is not your fault: every instrument has play, and that play is called the measurement uncertainty. Note the difference with a measurement error, because those two get mixed up constantly.

0:33Errors are things that go wrong: the pointer was not at zero. Those you can avoid. An uncertainty you cannot. It belongs to the instrument, and all you can do is write it down.

Between two marks

0:46Take an ordinary ruler with millimetre marks. Lay it beside and the end does not fall on a mark, because it almost never does. It sits in between. So you estimate. You look at where it lies between two marks and you note that down. And that estimate is sometimes too high and sometimes too low. That belongs to it.

1:08Someone beside you measuring the same estimates slightly differently, and you have both measured well. Look at where this end lies. What would you write down?

How big is that play

1:23Right, you estimate. But how big is that play? There is a guideline for that: you take a tenth of the smallest scale division. If there are millimetre marks, your uncertainty is about a tenth of a millimetre. And note what that guideline is and is not for.

1:42It holds for a scale with marks between which you must estimate. If you measure with a vernier calliper, you simply read that vernier off and you do not estimate at all. With a display the same: it rounds by itself, and you have no influence on it.

2:00Those instruments take the estimate out of your hands, and then the instrument sets the uncertainty and not your eye.

What a figure tells you

2:09And here the whole story sits. The number of figures you write something down with is a measure of the precision of your instrument. Those figures are called significant figures, and they are not styling: they are a statement. If you write two metres, you say it lies somewhere between one and a half and two and a half.

2:29If you write two point zero zero metres, you say: between one point nine nine five and two point zero zero five. The same length, two very different claims.

Which zeros count

2:42One rule you often need. When counting significant figures, zeros at the start of a number do not count, but zeros at the end do. Zero point zero zero eight has one significant figure: those zeros in front only say where the point is. Eight point zero zero has three, because those zeros at the end say that you measured to the hundredth.

3:09If you put a zero at the end that you did not measure, you lie about your instrument. Three numbers. How many significant figures has each?

The first rule of thumb

3:24Back to the beginning. Where did those four figures come from? I had measured the mat where I put my wet things. Seventy-five centimetres long, forty-five centimetres wide. Both two figures. Zero point seventy-five metres times zero point forty-five metres.

3:43The calculator gives zero point three three seven five. And then the rule: with multiplying and dividing the answer gets as many significant figures as the measured value with the fewest significant figures. Two and two, so two. Zero point thirty-four square metres.

4:05Write zero point three three seven five, you give four figures while you measured two.

The second rule of thumb

4:12With adding and subtracting it goes differently, and that is the trap. There it is not the significant figures that count but the number of figures after the point. The rule: with adding and subtracting the answer gets as many figures after the point as the measured value with the fewest figures after the point.

4:32And there everything must first be in the same unit, or you compare points that do not belong together.

Summary

4:40Three things. One: every measurement has play. Estimating between marks, that play is about a tenth of the smallest scale division; with a vernier or a display not, because there you do not estimate. Two: the number of figures you write something with is a statement about your instrument.

4:58Two metres and two point zero zero metres are not the same. Three: with multiplying you look at significant figures, with adding at figures after the point. Two rules, and they get mixed up constantly.

The two questions

5:13Two questions. Try them yourself first. First question: you measure a plate of zero point twenty-four metres by zero point six metres. How many significant figures has the area? Second question: someone writes his answer with eight figures because that looks more precise.

5:31What is wrong with that?

The answers

5:34Zero point twenty-four has two, zero point six has one. The fewest is one, so your answer gets one. Zero point one four four becomes zero point one square metre. That feels like throwing information away, and that is why it is hard. But your second measurement had only one figure, and then your answer cannot be more precise.

5:56And the second question: eight figures is not more precise. You are saying your instrument could manage eight figures, and it could not.

Closing

6:06You now know what a figure means in physics, and why you may not simply add them. Once measurements go in a table and make a graph of them, you use this at every point you plot.