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0.10 Significant figures by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.

An engineer is designing a car engine, and has been told that the diameter of the pistons (which are being designed by someone else) is 5.0 cm. He knows that 0.02 cm of clearance is required for a piston of this size, so he designs the cylinder to have an inside diameter of 5.02 cm. Luckily, his supervisor catches his mistake before the car goes into production. She explains his error to him, and mentally puts him in the “do not promote” category.

What was his mistake? The person who told him the diameter of the piston intentionally said “5.0 cm,” not “5.00 cm,” specifically to avoid creating the impression that the number was extremely accurate. In reality, the piston's diameter was 5.04 cm. They would never have fit in the 5.02 cm cylinders.

In most cases, the result of a calculation involving several pieces of data can be no more accurate than the least accurate piece of data. “Garbage in, garbage out.” Since the 5.0 cm figure for the diameter of the pistons was not very accurate, the result of the engineer's calculation, 5.04 cm, was really not as accurate as he thought.

For calculations involving multiplication and division, a given fractional or “percent” error in one of the inputs causes the same fractional error in the output. The number of digits in a number provides a rough measure of its possible fractional error. These are called significant figures or “sig figs.” Examples:

3.14 | 3 sig figs |

3.1 | 2 sig figs |

0.03 | 1 sig fig, because the zeroes are just placeholders |

3.0×10^{1} |
2 sig figs |

30 | could be 1 or 2 sig figs, since we can’t tell if the 0 is a placeholder or a real sig fig |

In such calculations, your result should not have more than the number of sig figs in the least accurate piece of data you started with.

`=>` A triangle has an area of 6.45 `m^2` and a base with a width of 4.0138 m. Find its height.

`=>` The area is related to the base and height by `A=bh"/"2`.

`h=(2A)/b`

3.21391200358762 m (calculator output)

3.21 m

The given data were 3 sig figs and 5 sig figs. We're limited by the less accurate piece of data, so the final result is 3 sig figs. The additional digits on the calculator don't mean anything, and if we communicated them to another person, we would create the false impression of having determined `h` with more precision than we really obtained.

*self-check:* The following quote is taken from an editorial by Norimitsu Onishi in the New York Times, August 18, 2002.

Consider Nigeria. Everyone agrees it is Africa's most populous nation. But what is its population? The United Nations says 114 million; the State Department, 120 million. The World Bank says 126.9 million, while the Central Intelligence Agency puts it at 126,635,626.

What should bother you about this?

(answer in the back of the PDF version of the book)

Dealing correctly with significant figures can save you time! Often, students copy down numbers from their calculators with eight significant figures of precision, then type them back in for a later calculation. That's a waste of time, unless your original data had that kind of incredible precision.

*self-check:*How many significant figures are there in each of the following measurements?

(1) 9.937 m

(2) 4.0 s

(3) 0.0000000000000037 kg

(answer in the back of the PDF version of the book)

The rules about significant figures are only rules of thumb, and are not a substitute for careful thinking. For instance, $20.00 + $0.05 is $20.05. It need not and should not be rounded off to $20. In general, the sig fig rules work best for multiplication and division, and we sometimes also apply them when doing a complicated calculation that involves many types of operations. For simple addition and subtraction, it makes more sense to maintain a fixed number of digits after the decimal point.

When in doubt, don't use the sig fig rules at all. Instead, intentionally change one piece of your initial data by the maximum amount by which you think it could have been off, and recalculate the final result. The digits on the end that are completely reshuffled are the ones that are meaningless, and should be omitted.

`=>` How many sig figs are there in sin 88.7°?

`=>` We're using a sine function, which isn't addition, subtraction, multiplication, or division. It would be reasonable to guess that since the input angle had 3 sig figs, so would the output. But if this was an important calculation and we really needed to know, we would do the following:

sin 88.7° =0.999742609322698

sin 88.8° =0.999780683474846

Surprisingly, the result appears to have as many as 5 sig figs, not just 3:

sin 88.7° = 0.99974,

where the final 4 is uncertain but may have some significance. The unexpectedly high precision of the result is because the sine function is nearing its maximum at 90 degrees, where the graph flattens out and becomes insensitive to the input angle.

0.10 Significant figures by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.