Rounding Off Significant Figures

Rounding off significant figures is an important part of scientific and engineering work, as it helps to ensure that data is reported accurately and consistently. 

• It is also important to round off significant figures correctly when performing calculations, as rounding off too early can lead to errors.
• To round off significant figures, we must remove one or more digits from the right side of the number until we reach the desired number of significant digits.
• To begin, we need to look at the digit on the right end of the number (to the right of the digit to which we want to round it off).
• If the digit is less than 5, the number is rounded down to the next whole number.
• If the digit is more than or equal to 5, the number is rounded up to the next whole number.

Key Terms: Significant figures, Whole numbers, Decimal numbers, Rounding off, Exact numbers, Even and odd numbers

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Rounding Off

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Rounding off implies simplifying a number by keeping its value but moving it closer to the next number. 

• It is done for whole numbers as well as decimals at various hundreds, tens, and tenth places. 
• Numbers are rounded off to preserve significant digits. 
• The number of significant figures in a result is simply the number of figures that are reliably known.
• There are three significant figures in the number 13.2. 
• Non-zero digits are always significant. 3.14159 contains six significant digits (all of the numbers are relevant). 
• Therefore, 67 has two significant numbers, but 67.3 has three.

Rounding off

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Rules for Rounding Off Whole Numbers

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The following are the rules that have to be followed while rounding off whole numbers

• Always use the smallest place value to get an accurate final result.
• Look for the next smaller place, which is to the right of the rounded-off number. When rounding off a number from the tens place, look for a digit in the one's place.
• If the insignificant digit to be dropped is more than 5, then the preceding digit is raised by 1. This is known as rounding up.
• If the insignificant digit to be dropped is less than 5, then the preceding digits are left unchanged. This is known as rounding down.
• If the insignificant digit to be dropped is 5, then the preceding digit is raised by 1 if it is odd and is left unchanged if it is even.
• When a complex multi-step calculation is involved, all the numbers occurring in the intermediate steps should retain a digit more than the significant digits present in them. The final answer at the end of the calculation, can then be rounded off to the appropriate significant figures.
• The exact numbers like π, 2, 3, 4, etc. that appear in the formulae and are known to have infinite significant figures, can be rounded off to a limited number of significant figures as per the requirements.

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Rules for Rounding Off Decimal Numbers

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The following are the rules that have to be followed while rounding off decimal numbers

• Determine the digit to be rounded off and look at the digit at the right side of the digit to be rounded off.
• If the digit on the right-hand side is less than 5, then consider it as equal to zero.
• If the right-hand side digit is greater than or equal to 5, then add +1 to that digit and consider all other digits as zero.

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Examples of How to Round Off

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Some examples of how to round off a number are given below

Round Off to Nearest Hundred

Let us take the number 4350. We have to round off this number to the nearest hundred places. The following steps should be followed

• First determine the digit present in the hundreds place: 3
• Next, determine the next smallest place in the number i.e tens place: 5
• Now if the number present in the smallest place is greater than or equal to 5, then the preceding number i.e. hundreds place digit is added by 1 i.e. 3 + 1 = 4. And the other digits on the right side become zero.
• So, we get the final answer as 4400

Round Off to Nearest Ten

Let us take the number 414.5. We have to round off this number to the nearest tens place. The following steps should be followed

• First determine the digit present in the tens place: 1
• Next, determine the next smallest place in the number: 4
• If the number present in the smallest place is greater than or equal to 5, then the preceding digit should be rounded up i.e. 1 is added to the preceding number.
• But in this case, the insignificant digit present in the one place is less than 5, therefore round down is followed, and hence the preceding digit remains unchanged.
• Also, all digits at the right side of the hundreds place become zero.
• So the final number is 410.

Let us consider another example of round-off to the nearest ten. Take the number 599. We have to round off this number to the nearest tens place. The following steps should be followed

• First determine the digit present in the tens place: 9
• Next, determine the next smallest place in the number: 9
• If the number present in the smallest place is greater than or equal to 5, then the preceding digit should be rounded up i.e. 1 is added to the preceding number.
• In this case, the insignificant digit present in one place is greater than 5, therefore the preceding digit is rounded up, and hence the preceding digit becomes 9 + 1 = 10 and the 1 is carried to the next place.
• As the digit in the one place is greater than 5, +1 has to be added.
• Therefore, 9+1=10, and the 1 is carried to the next place.
• So the final number is 600.

Round Off to Nearest Tenth

Let us consider an example of round-off to the nearest tenth. Take the number 0.84. We have to round off this number to the nearest tenth place. The following steps should be followed

• First determine the digit present in the tenth place: 8
• Next, determine the next smallest place in the number: 4
• If the number present in the smallest place is greater than or equal to 5, then the preceding digit should be rounded up i.e. 1 is added to the preceding number.
• But in this case, the insignificant digit present in the smallest place is less than 5, therefore round down is followed, and hence the preceding digit remains unchanged.
• So the final number is 0.8

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Things to Remember

• Significant figures (or significant digits) are the number of digits in a given value or a measurement, necessary to decide the accuracy and precision of measurement.
• Rounding off means simplifying a number by maintaining its value but moving it closer to the next number. 
• It is done for whole numbers as well as decimals at various hundreds, tens, and tenth places.
• If the digit to be rounded off is more than 5, then the preceding digit is raised by 1.
• If the digit to be rounded off is less than 5, then the preceding digit is left unchanged.
• If the digit to be rounded off is 5, then the preceding digit is raised by 1 if it is odd and remains unchanged if it is even.

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