Significant Digits and Scientific Notation
Precision is the degree of exactness to which a measurement can be reproduced. The precision of a measuring device is limited by the finest division on its scale. For example, on a meter stick the finest division could be the millimeter. Any measurement finer than a millimeter can be only an estimate.
Because the precision of all measuring devices is limited, the number of digits that are valid for any measurement is limited also. The valid digits are called significant digits. Imagine that you want to measure a strip of metal with a meter stick. The smallest division on the meter stick is a millimeter. Using the meter stick, you would measure the strip to the nearest millimeter and then estimate to the nearest fraction of a millimeter. For example, suppose the strip were 37 mm in length and that the end of the strip was about seven tenths of the way between 37 and 38 mm. You would record your measurement as 37.7 mm. Remember that the last digit (in this case, 7) is an estimate - you did not actually measure it. Thus, we see that our measurement has three significant digits. Two were measured, 3 and 7, and one was estimated, 7. Had the end of the metal strip been exactly on the 38 mm mark, the length would be reported as 38.0 mm. Here, again, we have three significant digits. The last one (the zero) is an estimate.
There are a few rules to help you decide which digits are significant and which are not. They are a follows:
1. All non-zero digits are significant.
2. Zeroes to the right of the decimal point are significant.
3. Zeroes between non-zero digits are significant.
In other words, find the first non-zero digit, count it and everything after it.
Some confusion arises when we have to consider zeroes to the right of non-zero digits. For example, it is difficult to say how many significant digits are in the measurement 23 000 km. It is unclear whether the 3 or one of the zeroes was the estimated digit. To avoid this, numbers can be expressed in scientific notation. Before discussing scientific notation, let's look at simple mathematical operations using significant digits.
| e.g., | answer |
| a) 1.045 | 4 |
| b) 0.023 | 2 |
| c) 2.36 | 3 |
| d) 230.0 | 4 |
Mathematical Operations Using Significant Digits
The result of any calculation can never be more precise than the least precise value. For example, imagine that you have measured the volume of two quantities of water. One is 3.42 L and the other is 2.5 L. If these two were added together the answer would be 5.92 L. The 5 in 2.5 L is the least precise, which means that we do not know what the next digit is. Was it 2.52? Or maybe 2.57? Without knowing this, we cannot know what the answer is to the nearest hundredth of a liter.
| 3.42 L |
| +2.5? L |
| 5.9? L |
We must, therefore, round off our answer to the nearest tenth - the least precise digit of which we are sure. We can apply this rule when adding and subtracting measured values:
The answer is rounded off to the least precise value.
We use a different approach when multiplying or dividing. Suppose you are calculating the area of a piece of plastic. The width is 6.3 cm and the length is 8.43 cm. By multiplication, we find the area to be 53.109 cm2. Notice that our answer has 5 significant digits while our original measurements had only 2 and 3, respectively. The number of significant digits in our answer should be the same as the measurement with the least number of significant digits. In our example, the answer would be rounded off to 53 cm2 (2 significant digits as in 6.3 cm).
Scientific Notation
Scientific notation is used when working with very large or very small numbers. It can also be used to simplify the reporting of significant digits. To express a number in scientific notation, we write the value as a number between 1 and 10, multiplied by a whole number power of 10. Specifically,
M x 10n, where 1 £ M < 10, and n Î I
For example, 5 000 000 000 would be written as 5 x 109. Likewise, 0.000 000 564 would be written as 5.64 x 10-7.
The decimal is moved to the right or left until there is only one non-zero digit to the left. Then, you count the number of spaces the decimal was moved and use that number as the power of ten. If the decimal was moved to the left, the exponent is positive; if to the right, the exponent is negative.