Significant figures

Making measurements in physics has restrictions on the accuracy that is possible. Also, sometimes it is more useful to restrict the amount of detail stated in a measurement. The NASA website actually gives the range of distances from the Sun to Pluto as:

4,436,820,000,000 m

to

7,315,930,000,000 m

The exact details of these numbers may not be necessary, or indeed accurate, so scientists round off the numbers. The rounding of a number to the level of importance in the details that is most useful is known as using significant figures.

The number of significant figures (s.f.) used equals the number of digits that are kept as important during the rounding of the number.

For example, the closest distance of Pluto may vary over time, so that in each orbit the value is not the same. To give an approximate value for the distance from the Sun to Pluto, an astronomer might simply give the average of the closest and furthest distances. Calculated exactly, this average is 5,876,375,000,000 m but, given the variations in the distance, it is only reasonable to quote this value approximately. For example, the astronomer might say it is 5,900,000,000,000 m or they might approximate even more and say it is 6 teremetres (Tm).

The average distance from the Sun to Pluto is:

5,876,400,000,000 m to five significant figures (s.f.)

5,900,000,000,000 m to two s.f.

6 Tm to one s.f.

Example

What is the furthest distance from the Sun to Pluto to three significant figures?

The NASA website says the distance is 7,315,930,000,000 m.

This is 7,320,000,000,000 m (3 s.f.)

curriculum-key-fact
It is always important to state units and number of significant figures for any calculated answer.
Question

What is the closest distance to the Sun to Pluto to 2 significant figures?

The NASA website says the distance is 4,436,820,000,000 m.

This is 4,400,000,000,000 m (2 s.f.)

Decimal significant figures

The number of significant figures (s.f.) used equals the number of digits that are important in the rounding of the number in order to keep its value. This means that for a number with a decimal point, zeros before the first non-zero digit can be ignored in counting the number of significant figures. Zeros after the last non-zero digit are counted though.

4.76000 has three significant figures.

0.0000476 also has three significant figures.

But trailing zeros are important in decimal numbers and are counted as significant figures.

0.00006100 has four significant figures - the six, the one and then the two trailing zeros are all significant.

Example

What is the value of 0.00206392 mol if quoted to four significant figures?

Ignore the zeros before the first value digit, and round off to four value digits, so this is:

0.002064 mol (4 s.f.)

Question

What is the value of 0.00246392 amps if quoted to five significant figures?

Ignore the zeros before the first value digit, and round off to five value digits, so this is:

0.0024639 A (5 s.f.)

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