Rounding (Junior Cert Mathematics): Revision Notes
📚 Revision Notes
Rounding
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Learning intentions:
- Rounding to decimal places
- Rounding to significant figures
Rounding is a process used to reduce the number of digits in a number while keeping its value close to what it was.
Rounding can be done to:
- Decimal places or
- Significant figures.
Rounding to Decimal Places
Decimal places refer to the number of digits to the right of the decimal point in a number.
To round to a specific number of decimal places, follow these steps:
- Identify the digit at the place you are rounding to.
- Look at the next digit (the one immediately after the place you are rounding to).
- If this digit is 5 or more, round up the identified digit.
- If this digit is 4 or less, round down the identified digit.
- Remove all the digits after the place you are rounding to.
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Examples: 6. Round 4.768 to 2 decimal places.
- The number in the second decimal place is 6.
- The next number is 8 (which is more than 5), so we round 6 up to 7.
- The rounded number is 4.77.
- Round 3.2745 to 2 decimal places:
- The number in the second decimal place is 7.
- The next number is 4 (which is less than 5), so we keep 7 as it is.
- The rounded number is 3.27.
Rounding to Significant Figures
Significant figures are the digits that carry meaning contributing to the number's precision.
To round to a specific number of significant figures, follow these steps:
- Identify the digit at the place of the last significant figure you need.
- Look at the next digit (the one immediately after the last significant figure).
- If this digit is 5 or more, round up the identified digit.
- If this digit is 4 or less, round down the identified digit.
- Replace all digits after the last significant figure with zeros (if rounding whole numbers) or remove them (if rounding decimals).
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Examples: 8. Round 132,421 to 3 significant figures:
- The third significant figure is 2.
- The next digit is 4 (which is less than 5), so we keep 2 as it is.
- The rounded number is 132,000.
- Round 0.00472543 to 3 significant figures:
- The third significant figure is 2. (Zero is not counted as a significant figure. Therefore, the first significant figure is 4, the second is 7 and the third is 2.)
- The next digit is 5 (which is 5 or more), so we round 2 up to 3.
- The rounded number is 0.00473.
Rounding to significant figures can be tricky, especially when dealing with numbers that include zeros. Understanding which digits count as significant figures is important.
Which Digits Are Significant?
- Non-zero digits (1-9):
- Always significant. These digits are important no matter where they appear in the number.
- Example: In 345, all three digits (3, 4, and 5) are significant.
- Zeros between non-zero digits:
- Always significant. These zeros are sandwiched between other significant digits and matter for precision.
- Example: In 305, the zero is significant because it is between 3 and 5. So, all three digits (3, 0, and 5) are significant.
- Zeros after a decimal point and after a non-zero digit:
- Significant. These zeros are important because they show precision in a measurement.
- Example: In 0.450, the digits 4, 5, and the zero after 5 are significant. So, all three digits (4, 5, and 0) count.
Which Digits Are Not Significant?
- Zeros before the first non-zero digit (in decimals):
- Not significant. These zeros are just placeholders to show where the decimal point is.
- Example: In 0.0034, the first three zeros just help position the decimal point and are not significant. The significant figures here are 3 and 4.
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Examples to Make It Clearer :) 10. Number: 0.00789
- Significant figures: The first non-zero digit is 7, so the significant figures are 7, 8, and 9.
- Zeros before 7 are not significant. They are just there to place the decimal point.
- Number: 5002
- Significant figures: All digits are significant because the zero is between non-zero digits (5 and 2). So, the significant figures are 5, 0, 0, and 2.
- Number: 0.0407
- Significant figures: The first non-zero digit is 4, so the significant figures are 4, 0, and 7.
- The zeros before 4 are not significant, but the zero between 4 and 7 is significant because it's between non-zero digits.
Recap
- Count all non-zero digits (1-9) as significant.
- Count zeros between non-zero digits or after a non-zero digit in a decimal number as significant.
- Ignore zeros that come before the first non-zero digit in a decimal number (they're just placeholders).
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Exam Tip: When in doubt, write down the number and underline the digits you think are significant. This can help you visually separate significant and non-significant digits before rounding!
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Try it out! 13. Round 0.005678 to 3 significant figures 14. Round 0.04053 to 2 significant figures 15. Round 30245 to 3 significant figures
Solutions:
- 0.005678 rounded to 3 significant figures:
- The first non-zero digit is 5, so the significant figures are 5, 6, and 7.
- To round to 3 significant figures, look at the fourth digit, 8.
- Since 8 is more than 5, round up the third digit, 7, to 8.
- The rounded number is 0.00568.
- 0.04053 rounded to 2 significant figures:
- The first non-zero digit is 4. The significant figures are 4 and 0 (since it comes after a non-zero digit and is between non-zero digits).
- The third digit, 5, determines the rounding. Since it's 5 or more, round up.
- The rounded number is 0.041.
- 30245 rounded to 3 significant figures:
- The significant figures are 3, 0, and 2.
- To round to 3 significant figures, consider the fourth digit, 4.
- Since 4 is less than 5, round down, leaving the third digit, 2, unchanged.
- The rounded number is 30200.