SIGNIFICANT DIGITS

The number of significant digits in an answer to a calculation will depend on the number of significant digits in the given data, as discussed in the rules below. Approximate calculations (order-of-magnitude estimates) always result in answers with only one or two significant digits.

I. When are Digits Significant?

• Non-zero digits are always significant. Thus, 22 has two significant digits, and 22.3 has three significant digits.

• With zeroes, the situation is more complicated:
1. Zeroes placed before other digits are not significant; 0.046 has two significant digits.
2. Zeroes placed between other digits are always significant; 4009 kg has four significant digits.
3. Zeroes placed after other digits but behind a decimal point are significant; 7.90 has three significant digits.
4. Zeroes at the end of a number are significant only if they are behind a decimal point as in (c). Otherwise, it is impossible to tell if they are significant. For example, in the number 8200, it is not clear if the zeroes are significant or not. The number of significant digits in 8200 is at least two, but could be three or four. To avoid uncertainty, use scientific notation to place significant zeroes behind a decimal point:
5.  8.200 x 103 has four significant digits 8.20 x 103 has three significant digits 8.2 x103 has two significant digits

6. Zeros in Scientific notation are always significant.
 Never write down more digits than you can measure. Most of the time you can only measure 3 or 4 digits.

Examples:

 9,683 has four significant digits 15.60007 has seven significant digits 0.0005 has one significant digit 15.0000 has six significant digits 3.1560 has five significant digits

II. Rule #1: Significant Digits in Multiplication, Division, Trig. functions, etc.

In a calculation involving multiplication, division, trigonometric functions, etc., the number of significant digits in an answer should equal the least number of significant digits in any one of the numbers being multiplied, divided etc.

Thus in evaluating sin(kx), where k = 0.097 m-1 {two significant digits} and x = 4.73 m {three significant digits}, the answer should have two significant digits.

 Example: 1.23 * 4567.89 1.23 has three significant digits; 4567.89 has six significant digits. The result will have the smaller of these: three significant digits. Your calculator produces 5618.5047 as a result; round it to three significant digits and report = 5.62 x 103 (or 5620). Note: some people say that you should not report 5620 as your result as the last zero is ambiguous. I say that it is fine. We'll just assume the worst case and say that there are 3 significant digits.

Note that counted numbers have an unlimited number of significant digits.
As an example, if a hair dryer uses 1.20 kW of power, then 2 identical hairdryers use 2.40 kW:

1.20 kW {3 sig. dig.} x 2 {unlimited sig. dig.} = 2.40 kW {3 sig. dig.}

III. Rule #2: Significant Digits in Addition and Subtraction

When quantities are being added or subtracted, the number of decimal places (not significant digits) in the answer should be the same as the least number of decimal places in any of the numbers being added or subtracted.

Example:

``` 5.67   J (two decimal places)
1.1    J (one decimal place)
0.9378 J (four decimal places)
7.7    J (one decimal place)
```
Be especially careful with numbers which are given in scientific notation.

Example:

1.2 + (3.45 x 10-4)

The best way to solve this problem is to write the numbers in a column in ordinary notation:

```     1.2
+ 0.000345
```
Your calculator returns 1.200345, but only one digit right of the decimal is significant.
Report your result as 1.2.

You may also convert all numbers into scientific notation with the same exponent:

Example:

(1.23 x 105) + (4.56 x 106) + (7.89 x 107)
```     1.23 x 10^5
45.6  x 10^5
+ 789.   x 10^5
----------------```
The full answer would be 835.83 x 105, but the last two digits are not significant.
Report your result as 836. x 105 or even better 8.36 x 107.

 IMPORTANT: Keep One Extra Digit in Intermediate Answers When doing multi-step calculations, keep at least one more significant digit in intermediate results than needed in your final answer. For instance, if a final answer requires two significant digits, then carry at least three significant digits in calculations. If you round-off all your intermediate answers to only two digits, you are discarding the information contained in the third digit, and as a result the second digit in your final answer might be incorrect. (This phenomenon is known as "round-off error.")

The Two Greatest Sins Regarding Significant Digits

1. Writing more digits in an answer (intermediate or final) than justified by the number of digits in the data.
2. Rounding-off, say, to two digits in an intermediate answer, and then writing three digits in the final answer.
Practice Exercises:
1. ekt = ?, where k = 0.0189 yr-1, and t = 25 yr.
2. ab/c = ?, where a = 483 J, b = 73.67 J, and c = 15.67
3. x + y + z = ?, where x = 48.1, y = 77, and z = 65.789
4. m - n - p = ?, where m = 25.6, n = 21.1, and p = 2.43
5. 6.53 cm + 12.1 cm =
6. 405 m ? 3.90 s =
7. (1.2 V + 10.8 V) (0.350 A) = _______ W