SIGNIFICANT FIGURES

The number of significant figures (SF) in a number is equal to the number of digits used to express the number.

Quantity
Value Number of
Sig. Figs

Speed of light 299,792,458 m/s 9

Tropical Year 365.2422 days 7

Inch .0254 m 3

Density of the Earth 5.515x10³ kg/m³ 4

Day 8.6400x10⁴ sec 5

Mass of Proton 1.672x10^-27 kg 4

The location of the decimal point does not effect the number of significant figures needed to represent a number.
The digits in the exponent of the power of ten representation of a number are not counted as significant since they are only used to locate the position of the decimal point.
Zeros may or may not be significant depending upon their location relative to the other digits.

Zero's Location Significance

Interior of a series of digits.
103, .1208 Counts as part of the number.
It is Significant

To the Right of a series of digits that is before or after the decimal point. 10., 1.10 Counts as part of the number.
It is Significant

To the Left of a series of digits
.029, 0.598, 038 Not Significant

Example: All of the following have only 3 Sig. Figs.

2.54x10⁴μm = 25.4 mm = 2.54 cm = .0254 m = 2.54x10^-5 km

SIGNIFICANT FIGURES AND UNCERTAINTY

If the value of a quantity was known with complete accuracy, then it would take an infinite number of significant figures to express the number.
Truncating and rounding the number of digits to, for example, 3 SF implies that the resulting number is only accurate to a certain degree.
The number of significant digits in the value of a quantity is a rough estimate of the accuracy to which that quantity is known.

# SF Approximate
Accuracy π Actual
% Error Year Actual
% Error

1 1/10 = 10% 3 4.5% 4x10² days 9.5%

2 1/100 = 1% 3.1 1.3% 3.7x10² days 1.3%

3 1/1000 = .1% 3.14 .051% 365 days .066%

4 1/10,000 = .01% 3.142 .013% 365.2 days .012%

5 1/100,000 = .001% 3.1416 .00023% 365.24 days .0006%

Rules of Thumb for Problem Solving:

The answer to a problem cannot be more accurate than the accuracy of the numbers used to find the answer. The final answer is at best only as accurate as the accuracy of the least accurate value given.
Using the rounded output from one part of a problem as input to another part of a problem will normally generate error. You need to use at least two more SF for intermediate results than the accuracy you are endeavoring to obtain for the final answer.
When subtracting two numbers that are nearly equal, you will need more SF than normal if you wish your final answer to be accurate to the same number of SF as operations involving multiplication or division.
The best procedure to use to eliminate error propagation is to store all intermediate results internally in your calculator and to recall them when needed. In this way all the numbers you use will be as accurate as the precision of your calculator.

Units in Science

Quantity	Value	Number of Sig. Figs
Speed of light	299,792,458 m/s	9
Tropical Year	365.2422 days	7
Inch	.0254 m	3
Density of the Earth	5.515x10³ kg/m³	4
Day	8.6400x10⁴ sec	5
Mass of Proton	1.672x10^-27 kg	4

Zero's Location	Significance
Interior of a series of digits. 103, .1208	Counts as part of the number. It is Significant
To the Right of a series of digits that is before or after the decimal point. 10., 1.10	Counts as part of the number. It is Significant
To the Left of a series of digits .029, 0.598, 038	Not Significant

# SF	Approximate Accuracy	π	Actual % Error	Year	Actual % Error
1	1/10 = 10%	3	4.5%	4x10² days	9.5%
2	1/100 = 1%	3.1	1.3%	3.7x10² days	1.3%
3	1/1000 = .1%	3.14	.051%	365 days	.066%
4	1/10,000 = .01%	3.142	.013%	365.2 days	.012%
5	1/100,000 = .001%	3.1416	.00023%	365.24 days	.0006%