Chapter 1. Significant Figures

Making a Measurement

Many numbers we work with in science are the result of measurement and are therefore known only within a degree of uncertainty. This uncertainty should be reflected in the number of digits used. For example, if you have a 1-meter-long rule with scale spacing of 1 cm, you know that you can measure the height of a box to within a fifth of a centimeter or so. Using this rule, you might find that the box height is 27.0 cm. If there is a scale with a spacing of 1 mm on your rule, you might perhaps measure the box height to be 27.03 cm. However, if there is a scale with a spacing of 1 mm on your rule, you might not be able to measure the height more accurately than 27.03 cm because the height might vary by 0.01 cm or so, depending on where you measure the height of the box.

When you write down that the height of the box is 27.03 cm, you are stating that your best estimate of the height is 27.03 cm, but you are not claiming that it is exactly 27.030000 . . . cm high. The four digits in 27.03 cm are called significant figures. Your measured length, 27.03 cm, has four significant digits. Significant figures are also called significant digits.

Try It Yourself 1: Making a Measurement

Although some of the numbers you will find in physics problems are known exactly (e.g. π or c (the speed of light in a vacuum), most are not. They are measured quantities and therefore can only be as accurate as the measuring device allows.

For example, let's say you use a meter rule to measure some wood for a home project. The rule is graduated in centimeters and millimeters.

Calculations with Significant Figures

The number of significant digits in an answer to a calculation will depend on the number of significant digits in the given data. When you work with numbers that have uncertainties, you should be careful not to include more digits than the certainty of measurement warrants. Approximate calculations (order-of-magnitude estimates) always result in answers that have only one significant digit or none.

When you multiply, divide, add, or subtract numbers, you must consider the accuracy of the results. Listed below are some rules that will help you determine the number of significant digits of your results.

1. When multiplying or dividing quantities, the number of significant digits in the final answer is no greater than that in the quantity with the fewest significant digits.
   • For example, 2.75/6.3 = 0.4365079, but 6.3 has only 2 significant figures so the answer should be written as 0.44.
2. When adding or subtracting quantities, the number of decimal places in the answer should match that of the term with the smallest number of decimal places.
   • For example, 101m + 2.3m = 103.3m, but the least accurate input is 101m so the answer should be written as 103m.
3. Exact values have an unlimited number of significant digits.
   • For example, a value determined by counting, such as 2 tables, has no uncertainty and is an exact value. In addition, the conversion factor 0.0254000 . . . m/in. is an exact value because 1.000 . . . inches is exactly equal to 0.0254000 . . . meters. (The yard is, by definition, equal to exactly 0.9144 m, and 0.9144 divided by 36 is exactly equal to 0.0254.)
4. Sometimes zeros are significant and sometimes they are not. If a zero is before a leading nonzero digit, then the zero is not significant.
   • For example, the number 0.00890 has three significant digits. The first three zeroes are not significant digits but are merely markers to locate the decimal point. Note that the zero after the nine is significant.
5. Zeros that are between nonzero digits are significant.
   • For example, 5603 has four significant digits.
6. The number of significant digits in numbers with trailing zeros and no decimal point is ambiguous.
   • For example, 31,000 could have as many as five significant Digits or as few as two significant digits. To prevent ambiguity, you should report numbers by using scientific notation or by using a decimal point.

Worked Example: Calculations Using Significant Figures

In most physics problems you will complete throughout the book you will need to write an numerical answer with the correct number of significant figures. This will involve deciding how many significant figures your answer should include.

For example, say you want to find the total mass of yourself and a cat. You use a bathroom scale accurate to the nearest 0.1lb to find your mass and get a value of 142.2lb. Then you use a vet’s scale accurate to the nearest 0.01lb to find the cat’s mass and get a value of 18.21lb. What is your total combined mass?

However, there is a problem here. The number quoted above for the total mass has too many decimal places and suggests that the measurement is accurate to the nearest 0.01lb. In fact your accuracy in finding the total mass is limited by your bathroom scale (the least accurate measuring device used) to the nearest 0.1lb.

The correct way to quote the answer to this problem would be:

P'Cast M-1: Finding the Average of Three Numbers

Try It Yourself: Calculations Using Significant Figures

Write the answers to the following calculations to the correct number of significant figures.