1. Significant Figures
- Definition: Significant figures in a number are the digits that carry meaningful contributions to its accuracy. They include all digits except:
- Leading zeros (zeros before non-zero digits)
- Trailing zeros in a number without a decimal point (unless they are specifically measured)
2. Rules for Identifying Significant Figures
Here are the rules to determine how many significant figures a number has:
a. Non-zero digits are always significant.
- Example: In the number 457, all three digits (4, 5, and 7) are significant, so the number has 3 significant figures.
b. Zeros between non-zero digits are significant.
- Example: In the number 405, all digits (4, 0, and 5) are significant because the zero is between non-zero digits, so the number has 3 significant figures.
c. Leading zeros (zeros before the first non-zero digit) are NOT significant.
- Example: In the number 0.0075, the leading zeros (0.00) are not significant, but the digits 7 and 5 are. So, the number has 2 significant figures.
d. Trailing zeros in a decimal number are significant.
- Example: In the number 3.500, all four digits (3, 5, 0, 0) are significant because the trailing zeros after the decimal point indicate precision in measurement. This number has 4 significant figures.
e. Trailing zeros in a whole number without a decimal are NOT significant (unless specified).
- Example: In the number 4000, only the digit 4 is significant unless there is a decimal point (e.g., 4000.0), in which case the trailing zeros become significant. Here, 4000 has 1 significant figure.
3. Rules for Calculations with Significant Figures
a. Addition and Subtraction:
- The result should have as many decimal places as the term with the fewest decimal places.
- Example: 12.52 + 349.0 + 8.24 = 369.76 → Round off to 369.8 (since 349.0 has only 1 decimal place).
b. Multiplication and Division:
- The result should have as many significant figures as the number with the fewest significant figures.
- Example: 4.56 × 1.4 = 6.384 → Round off to 6.4 (since 1.4 has 2 significant figures).
4. Rounding Off
Rounding off is the process of reducing the number of digits in a number while keeping it as close as possible to the original value. This is necessary when the result of a calculation gives more digits than are justified by the precision of the data.
a. Rules for Rounding Off:
- If the digit to be dropped is less than 5, the last retained digit remains unchanged.
- Example: 6.732 rounded to 3 significant figures becomes 6.73.
- If the digit to be dropped is 5 or more, the last retained digit is increased by 1.
- Example: 6.738 rounded to 3 significant figures becomes 6.74.
b. Special Cases:
- If rounding affects a digit followed by a 5 with no more digits or with zeros, and the preceding digit is even, it remains unchanged. If odd, it increases by 1. (This is called the round-half-to-even rule or banker’s rounding.)
- Example: 12.35 rounded to 2 decimal places becomes 12.4 (since 3 is odd).
- Example: 12.45 rounded to 2 decimal places remains 12.4 (since 4 is even).
c. Rounding Off in Whole Numbers:
- Example: If you have the number 4567 and need to round it to 2 significant figures, it becomes 4600 (the last two digits are replaced by zeros).
5. Practical Examples of Rounding Off:
- Rounding to Decimal Places:
- Number: 3.14159 → Rounded to 3 decimal places = 3.142
- Number: 9.876 → Rounded to 2 decimal places = 9.88
- Rounding to Significant Figures:
- Number: 0.004567 → Rounded to 3 significant figures = 0.00457
- Number: 23.876 → Rounded to 3 significant figures = 23.9
6. Why Significant Figures and Rounding Off Matter
Significant figures ensure that results in measurements and calculations are as precise as the equipment allows. Rounding off prevents overestimating the precision of the result, which can lead to false accuracy.
