# Physics Lesson 1.3.1 - What are significant figures?

Welcome to our Physics lesson on What are significant figures?, this is the first lesson of our suite of physics lessons covering the topic of Significant Figures and their Importance, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.

## What are significant figures?

Let's start this section by giving answer to the above questions. Thus, for the flying distance London-Birmingham, we cannot round further the result as it is already rounded. Therefore, it is meaningless to say "163.000 km" when 163 km is a rounded value. Also, we cannot round the elephant mass more than to the nearest kg as it is obvious the original value (6000 kg) is already rounded. Finally, we cannot round up the dimensions of the door to the nearest metre and order the carpenter to produce a 2m × 1m door because it will not fit to the wall.

From the above examples, it is obvious that the numerical information must be meaningful and it must help the reader understand the point.

In physics problems, we use significant digits (figures) to express the answers. Significant figures represent the accuracy with which we know the values.

The accuracy of an apparatus is equal to the smallest unit it can measure. For example, the accuracy of a ruler is 1mm as it represents the smallest division of the ruler. The accuracy of an electronic balance is 1g as it shows the smallest value above zero appearing in the screen and so on.

Thus, for example if there are 3 tiles in each 1 m of a room, and we are interested to know the length of a tile, it is obvious we divide 100 cm by 3. However, the calculator displays the value

100cm/3 tiles = 33.333333. cm/tile

The first decimal place represents millimetres, the second one tenths of millimetres and so on. But is it really meaningful to write the result with such an accuracy? Of course, not. First of all, we measure the length of the room using a meter stick, a ruler or something similar. The precision of these devices extends up to millimetres. Hence, it is meaningless if we express a result in a higher order of accuracy than the precision provided by the apparatus itself. We can either say 33 cm or 33.3 cm for the length of a single tile (33.3 cm is the best version but 33 cm is also acceptable considering the fact that the gaps between the tiles can contribute in the decrease of the actual value. But it is wrong to say "33.33 cm" as the apparatus offers a precision up to 0.1 cm (1mm), so the result must not contain more than one decimal place.

If we want to express the abovementioned result as "33 cm", it contains two significant figures: two 3s and if you express the result as 33.3 cm, it has 3 significant figures: three 3s.

### Example 1

Round up the results of the measurements shown below to meaningful values and then write the number of significant figure each value contains after making the corrections.

1. The distance between two cities A and B is 364.000 km.
2. The dimensions of a notebook measured using a simple ruler are 35.425 cm and 24.1 cm

### Solution 1

As stated before, it is meaningless to express the distance between two cities at a higher order of accuracy then kilometres as often it is not clear where a certain city begins or ends. Therefore, we must write simply 364 km for the distance between cities A and B. In this case, the result is expressed in three significant figures: 3, 6 and 4.

Since the accuracy of the second measurement is up to one tenth of a centimeter (or up to 1 mm), it is meaningless to consider the hundredths and thousandths of centimeter in the first measurement. Therefore, we can write 35.4 cm for the first measurement (dimension) and 24.1 cm for the second one. Both numbers have 3 significant figures each (3, 5 and 4 for the first and 2, 4 and 1 for the second measurement).

## More Significant Figures and their Importance Lessons and Learning Resources

Units and Measurements Learning Material
Tutorial IDPhysics Tutorial TitleTutorialVideo
Tutorial
Revision
Notes
Revision
Questions
1.3Significant Figures and their Importance
Lesson IDPhysics Lesson TitleLessonVideo
Lesson
1.3.1What are significant figures?
1.3.2How to find the number of significant figures in a given numerical value?
1.3.3How to round up a number to the required number of significant figures?
1.3.4Why are significant figures so important in measurements?

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