To keep track of error (uncertainty) during calculations, we usually follow simple rules when adding, subtracting, multiplying, and dividing numbers.
These rules include the following:
Rule 1:
- If a number consists of any combination of these digits –1,2,3,4,5,6,7,8, and 9. Then all the digits in the number are considered significant. As a result, the number 243 has three significant figures, the number 45.89 has four significant figures, and the number 89.678 has five significant figures.
Rule 2:
If a number consists of zero, we must consider the following before we can decide whether the zero in the number is significant or not
- If a zero appears between two non-zero digits, then it is considered significant. As a result, the number 405 has three significant figures, while 10.04 has four significant figures.
- If a zero appears to the right of a digit in a decimal greater than one, then it is considered significant. As a result, the number 46.0 has three significant figures, while 1.406 has 4 significant figures
- If a zero appears to the right of a significant digit in a decimal number less than one, then it is considered significant. As a result, the number 0.405 has three significant figures, while 0.3306 has four.
- If a zero appears only to fix the position of the decimal pointin a number less than one, then this zero is considered nonsignificant. As a result, the number 0.045 has only two significant figures, while 0.006080 has four significant figures.
- If zeros appear at the end of a number without a decimal point, then the zeros at the end are usually not significant. However, if the number is written in standard notation, then the zeros at the end can be significant. For instance, the number 6700 has only two significant figures, however, when written in standard notation as: 6.700 x 103, then it has four significant figures. If you think zeros at the end of number might cause confusion, it’s best to express the number in standard notation.
In addition to the above rules, you will also need to remember that:
- when adding or subtracting measured values, the number with the least number of decimal places determine the number of decimal places in the final answer. This is because the number with the least number of decimal places is less precise than the number with the greatest number of decimal places.
- When multiplying or dividing measured values, the number with the least number of significant figures determine the number of significant figures in the final answer. This is because the number with the least number of significant figures is less precise than the number with the greatest number of significant figures.
Rule 3
When rounding numbers, you must consider the following:
- if the number being rounded off is less than 5, drop the number
- if the number being rounded off is greater than 5, increase the last significant digit by 1.
- if the number being rounded off is exactly 5, the last significant digit increases by 1 when it is an odd number and remains unchanged when it is an even number.
For example, if we round the the number: 6.4355 to three significant figures, we will get: 6.44 –applying the odd number rule. If we round the number: 6.4655 to three significant figures, we will get: 6.46 — applying the even number rule. In summary, when number is odd, we round up. When number is even, we round down.
Rule 4
During calculations, reserve rounding until the end of your calculation or carry along one digit more than the allowed significant figures until the end of your calculation, and then round off to the required number of significant digits.
If you want to learn about errors in measurement, click here, and if you would want to learn about uncertainty in measurement, click here.