Jerod has a homework problem that involves finding the area of a rectangle. He knows that the area of a rectangle equals its length times its width. The rectangle in question has a length of 6.9 m and a width of 6.5 m, so he multiplies the two numbers on his calculator. The answer he gets is 44.85 m2, which he records on his homework. To his surprise, his teacher marks this answer wrong. The reason? The answer has too many significant digits.
What Are Significant Digits?
In any measurement, the number of significant digits is the number of digits thought to be correct by the person doing the measuring. They are considered the "valid digits" in a measurement. It includes all digits that can be read directly from the measuring device plus one estimated digit.
Look at the sketch of a beaker below. How much blue liquid does the beaker contain? The top of the liquid falls between the mark for 40 mL and 50 mL, but it’s closer to 50 mL. A reasonable estimate is 47 mL. In this measurement, the first digit (4) is known for certain and the second digit (7) is an estimate, so the measurement has two significant digits.
Now look at the graduated cylinder sketched below. How much blue liquid does it contain? First, it’s important to note that you should read the amount of liquid at the bottom of its curved surface. This falls about half way between the mark for 36 mL and the mark for 37 mL, so a reasonable estimate would be 36.5 mL.
Q: How many significant digits does this measurement have?
A: There are three significant digits in this measurement. You know that the first two digits (3 and 6) are accurate. The third digit (5) is an estimate.
Rules for Counting Significant Digits
The examples above show that it’s easy to count the number of significant digits when you are making a measurement. But what if someone else has made the measurement? How do you know which digits are known for certain and which are estimated? How can you tell how many significant digits there are in the measurement? There are several rules for counting significant digits:
- Leading zeros are never significant. For example, in the number 006.1, only the 6 and 1 are significant.
- Any zeros between two significant digits are significant. For example, in the number 106.1, the zero is significant, so this number has four significant figures.
- Zeros that show only where the decimal point falls are not significant. For example, the number 470,000 has just two significant digits (4 and 7). The zeros just show that the 4 represents hundreds of thousands and the 7 represents tens of thousands. Therefore, these zeros are not significant.
- A final zero or trailing zeros in the decimal portion ONLY are significant.
Q: How many significant digits are there in each of these numbers: 20,080, 2.080, and 2000?
A: Both 20,080 and 2.080 contain four significant digits, but 2000 has just one significant digit.
Determining Significant Digits in Calculations
When measurements are used in a calculation, the answer cannot have more significant digits than the measurement with the fewest significant digits. This explains why the homework answer above is wrong. It has more significant digits than the measurement with the fewest significant digits. As another example, assume that you want to calculate the volume of the block of wood shown below.
The volume of the block is represented by the formula:
- Volume = length × width × height
Therefore, you would do the following calculation:
- Volume = 1.2 cm × 1.0 cm × 1 cm = 1.2 cm3
Q: Does this answer have the correct number of significant digits?
A: No, it has too many significant digits. The correct answer is 1 cm3. That’s because the height of the block has just one significant digit. Therefore, the answer can have only one significant digit.
Rules for Rounding
To get the correct answer in the volume calculation above, rounding was necessary. Rounding is done when one or more ending digits are dropped to get the correct number of significant digits. In this example, the answer was rounded down to a lower number (from 1.2 to 1). Sometimes the answer is rounded up to a higher number. How do you know which way to round? Follow these simple rules:
- If the digit to be rounded (dropped) is less than 5, then round down. For example, when rounding 2.344 to three significant digits, round down to 2.34.
- If the digit to be rounded is greater than 5, then round up. For example, when rounding 2.346 to three significant digits, round up to 2.35.
- If the digit to be rounded is 5, round up if the digit before 5 is odd, and round down if digit before 5 is even. For example, when rounding 2.345 to three significant digits, round down to 2.34. This rule may seem arbitrary, but in a series of many calculations, any rounding errors should cancel each other out.
Arithmetic with Significant Digits
Here are several basic rules when you have to make calculations or manipulate numbers:
- Multiplying and Dividing: When multiplying and dividing, an answer may only show as many significant digits as the multiplied or divided measurement showing the least number of significant digits. Example: 3.2 x 4.56=46.6944 but when you account for significant digits, the answer will be rounded to two significant digits. Therefore, the answer is 47.
- Adding and Subtracting: When adding and subtracting, an answer may only show as many decimal places as the measurement having the fewest number of decimal places. Example: 58.4 + 23.768=82.168 but when you take the rule into account, the answer will be rounded to three significant digits (with the decimal only extending to the tenths place). Therefore, the answer is 82.2.
- Averaging: The average cannot be more precise than any of the individual measurements. Use the adding rule in this case.
- Converting units: The final answer should have the same number of significant digits as the quantity being converted. Example: You are converting 8.5 grams to pounds. The answer should only have two significant digits since the measurement being converted only has two.
Helpful Mneumonic for Determining Significant Digits
The Atlantic Pacific Rule:
- Pacific: The "P" is for decimal point is present. If the decimal point is present, count significant digits starting with the first non-zero digit on the left (the Pacific side of the US).
- Atlantic: The "A" is for decimal point is absent. If there is no decimal point, start counting significant digits with the first non-zero digit on the right (the Atlantic side of the US).
- In any measurement, the number of significant digits is the number of digits thought to be correct by the person doing the measuring. It includes all digits that can be read directly from the measuring device plus one estimated digit.
- To determine the number of significant digits in a measurement that someone else has made, follow the rules for counting significant digits.
- When measurements are used in a calculation, the answer cannot have more significant digits than the measurement with the fewest significant digits.
- Rounding is done when one or more ending digits are dropped to get the correct number of significant digits. Simple rules state when to round up and when to round down.
- significant digits: Correct number of digits in an answer that is based on the least precise measurement used in the calculation.
Do the significant digits quiz at this URL. Be sure to check your answers.
- How do you determine the number of significant digits when you make a measurement?
- Measure the width of a sheet of standard-sized (8.5 in x 11.0 in) loose-leaf notebook paper. Make the measurement in centimeters and express the answer with the correct number of significant digits.
- How many significant digits do each of these measurements have?
- In this calculation, how many significant digits should there be in the answer? 1.0234 + 1.1 + 0.0056
- Round each of these numbers to three significant digits:
- It includes all the digits that can be read on the measuring device plus one estimated value.
- Remember multiplication rule for significant digits when answering
- Number of signifiant digits:
- 2, the addition rule states your answer should match the value with the least number of decimal places
- Rounded values: