Significant Digits

An Illustrative Story

Here's a little story that may help you understand the idea behind significant digits:
Some tourists at the Museum of Natural History are marveling at the dinosaur bones. One of them asks the guard, "Can you tell me how old these bonds are?"
The guard replies, "They're three million, four years and six months old."
"That's an awfully exact number," says the tourist. "How do you know their age so precisely?"
The guard answers, "Well, the dinosaur bones were three million years old when I started working here, and that was four and a half years ago."

What's the "Significance?"

When multiplying and dividing numbers it is important to know how many significant digits are in each measurement you have. Let’s pretend we are calculating the area of a rectangular room with the dimensions 15.778 m by 9.332 m. We find area by multiplying length times width, and the result our calculator gives is 147.240296 m2. Now let’s think about this for a second. What is the real likelihood that you can determine the square area to the millionths place from two original measurements that only went to the thousandths place? The answer is you can’t! The number 147.240296 m2 would be correct if we assumed that 15.778 m and 9.332 m were followed by an infinite number of zeroes. That is not likely the case!

If there were a more precise meterstick than the first one that was used to measure the room and the dimensions were found to be 15.7781 m by 9.3325 m, wouldn’t the value of the area now change too? Of course it would! When we perform multiplication and division, there is a simple rule we must follow in order to maintain a reliable answer:

The number of significant digits in your answer must have the same number of digits as the measurement that has the fewest.

What this means is that our answer 147.240296 m2 must be rounded to 147.2 m2 because the two original measurements (15.778 m and 9.332 m) have five and four significant digits respectively. This means that our answer must have only four significant digits (the fewer of five and four).

When you divide by an exact number (as you would in an average), that number has infinite significant digits, and therefore will never be the fewest. For example, when averaging five numbers, after summing the numbers, the division by five does not reduce your answer to one s.d. - five is an exact number here - it has infinite zeroes after it!

The rules are different when adding or subtracting numbers. Here, you are allowed to keep any digit in your answer so long as each measurement has a number in that same decimal place.