Compare and contrast the multiplication/division significant figure rule to the significant figure rule applied for addition/subtraction mathematical operations. Give examples.
The Correct Answer and Explanation is :
When performing mathematical operations, the rules for significant figures (sig figs) differ between multiplication/division and addition/subtraction. These rules help determine how many digits in the result are meaningful based on the precision of the input numbers.
Multiplication and Division Significant Figure Rule:
The rule for multiplication and division is based on the number with the least significant figures. The result should have the same number of significant figures as the number with the least amount of significant figures in the calculation.
Example:
If you multiply ( 3.24 ) (3 significant figures) by ( 2.1 ) (2 significant figures), the result should have 2 significant figures because 2.1 has the fewest significant figures.
[
3.24 \times 2.1 = 6.804 \quad \text{(rounded to 2 sig figs)} \quad \Rightarrow 6.8
]
Addition and Subtraction Significant Figure Rule:
For addition and subtraction, the rule focuses on the decimal places. The result should have the same number of decimal places as the number with the fewest decimal places in the operation.
Example:
If you add ( 4.56 ) (2 decimal places) and ( 3.2 ) (1 decimal place), the result should have only 1 decimal place because 3.2 has the fewest decimal places.
[
4.56 + 3.2 = 7.76 \quad \text{(rounded to 1 decimal place)} \quad \Rightarrow 7.8
]
Key Differences:
- Multiplication/Division: Focuses on the total number of significant figures in the numbers involved.
- Addition/Subtraction: Focuses on the precision (decimal places) of the numbers.
These rules ensure that the result reflects the level of precision inherent in the input data. For instance, if an operation involves measurements with different levels of precision, the answer can’t be more precise than the least precise measurement used in the calculation.
In summary:
- Multiplication/Division: The result has as many significant figures as the input number with the least significant figures.
- Addition/Subtraction: The result has as many decimal places as the input number with the fewest decimal places.