Error Significant Figures Rounding Off

Understanding the ideas of errors, significant figures, and rounding off is important in academic life as well as in real life. From the measurement of different ingredients of a recipe to the calculation of the exact distance between pieces of construction, errors along with approximations affect the accuracy of the results. Significant figures ensure that the measurements are valid and relevant; but rounding off brings a complicated solution to simpler terms, without causing too much loss of accuracy. It enables us to make better decisions in any scientific experiment or in solving everyday problems.

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What are Rounding Off numbers?

We know that every measurement involves errors due to the finite resolution of the instrument and several other factors. Results of scientific measurements are always written in a way that may indicate the accuracy involved in the measurement. When we measure a physical entity then some of the digits in the measured value are reliably correct. When we take measured value we should include all digits which are reliably correct plus one digit which is uncertain. All reliable digits plus the first uncertain digit, in the measured value, are known as significant digits or significant figures. If we are including more uncertain digits while taking the measured value then it will give us a false impression about the precision of measurement. So we need to include only one uncertain digit.

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Rounding Significant Figures

Let us suppose we have measured the time interval of a certain phenomenon as 3.57 s then here 3 and 5 are reliable digits whereas 7 is the first uncertain digit. So, here there are a total of three significant digits in this measured value. Similarly, if someone writes the length of an object as 256.7 cm then here 2, 5 and 6 are reliable digits but 7 is an uncertain digit and there are a total of four significant digits in the measured value. Finally, we know that the significant figures in measured value indicate the precision of measurement which in turn depends on the least count of the measuring instrument.

We can refer to the following rules to determine the number of significant digits in a measured value of measurement.

Rounding Rules for Whole Numbers

• To get an accurate final result, always choose the smaller place value.
• Look for the next smaller place which is towards the right of the number that is being rounded off. For example, if you are round figure of a digit from the tens place, look for a digit in the one's place.
• If the digit in the smallest place is less than 5, then the digit is left untouched. Any number of digits after that number becomes zero and this is known as rounding down.
• If the digit in the smallest place is greater than or equal to 5, then the digit is added with +1. Any digits after that number become zero and this is known as rounding up.

Rounding Rules for Decimal Numbers

• Determine the rounding digit and look at its righthand side.
• If the digits on the right-hand side are less than 5, consider them as equal to zero.
• If the digits on the right-hand side are greater than or equal to 5, then add +1 to that digit and consider all other digits as zero.

Examples of Rounding Off

Rounding to the Nearest Hundred

Consider the number 3450. To round off to the nearest significant number, consider the hundreds place and follow the steps below:
Identify the digit present in the hundreds place: 4
Identify the next smallest place in the number: 5
If the smallest place digit is greater than or equal to 5, then round up the digit.
Now add 1 to the digit in the hundreds place. $4+1=5$. Therefore, the other digits become zero.
So the final number is 3500.

Rounding to the Nearest Ten

Consider the number 327.5. To round off to the nearest significant number, consider the tens place and follow the steps below:

• Identify the digit present in the tens place: 2
• Identify the next smallest place in the number: 7
• If the smallest place digit is greater than or equal to 5, then round up the digit.
• Since the digit in the smallest place is greater than 5, a round-up has to be done and the digit increases by 1.
• Every other digit becomes zero.
• So the final number is 330.

Rounding to the Nearest Ten

• Consider the number 489. To round off to the nearest significant number, consider the tens place and follow the steps below:
• Identify the digit present in the tens place: 8
• Identify the next smallest place in the number: 9
• If the smallest place digit is greater than or equal to 5, then round up the digit.
• As the digit in the one place is greater than 5, add 1.
• Therefore, $8+1=9$ and the 1 is carried to the next place.
• So the final number is 490.

Rounding to the Nearest Tenth

• Consider the number 0.68. To round off to the nearest significant number, consider the tenth place and follow the steps below:
• Identify the digit present in the tenth place: 6
• Identify the next smallest place in the number: 8
• If the smallest place digit is greater than or equal to 5, then round up the digit.
• As the digit in the smallest place is greater than 5, the digit gets rounded up.
• So the final number is 0.7.