Error Significant Figures Rounding Off

Edited By Vishal kumar | Updated on Sep 24, 2024 04:18 PM IST

Here, in this article, we will discuss round off round figures, how to round off numbers and decimals, significant figures etc...

What is Rounding Off?

We know that every measurement involves errors due to finite resolution of the instrument and several other factors. Results of scientific measurements are always written in a way that may indicate 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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Error Significant Figures Rounding Off

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Rounding significant figures

Let us suppose if 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 three significant digits in this measured value. Similarly if someone writes length of 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 and 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.

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Rules for rounding off significant figures

1. We consider all the non-zero digits to be significant. For example, 52.49 has four significant figures.

2. We consider all the zeros between two non-zero digits to be significant. Suppose we have a decimal point in the measured value even then this rule is not affected. For example, 2009 has four significant digits and 2050.08 has six significant figures.

3. If we have the measured value less than 1, then zeros to the right side of the decimal point up to the first non-zero digit are not considered as significant. For example, in 0.005038, the first three zeros are not significant and hence the total number of significant digits is four in this case.

4. If there is no decimal point in the measured value then all terminal or trailing zeros are not considered as significant digits. For example, in 32800, there are only three significant digits.

5. If there is a decimal point in the measured value then all trailing zeros are counted as significant figures. For example, 1.300 has four significant figures. Similarly, when we combine rule-3, we can understand that 0.06400 has four significant digits.

6. If we have some exact number which is not a measured value then it has an infinite number of significant digits. For example, p = 8.1415926.... has infinite significant digits.

7. Already we know that physical quantities are measured in standard units. If we change the units of measured value then it has no effect on the number of significant digits. For example, if length is equal to 3.207 cm, then we can understand that there are four significant figures in this measured value. We can write the same result as 0.03207 m, then also we can see that there are four significant figures. We can write the same result as 32.07 mm, then also we can see that there are four significant figures. Let us take an even smaller unit like µm then we can write the same result as 4070 µm, and the number of significant digits is still four because the last zero we know is not counted as a significant figure as per rule-4.

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Let us consider another example of length which is 6.500 m. Rule-5 can be used to understand that the number of significant digits measured in this case is four. Now let us write it in mm, then the measured value becomes 6,500 mm. Now rule-4 can mislead us to believe that there are only two significant digits, but we know that there are four significant digits in the measured value. We know that by changing units, the number of significant digits are not supposed to be affected. In order to avoid this kind of confusion in some cases, it is better to write the value in scientific notation.

In scientific notation we write the number as ‘a × 10^b’, here ‘a’ is a number between 1 and 10, and ‘b’ is any positive or negative power of 10. For example, the above-measured value can be written as follows:

NCERT Physics Notes:

2.500 m = 2.500 × 10² cm = 2.500 × 10³ mm

We can see that in all cases the number of significant figures is four. So, we conclude that the power of 10 does not affect the number of significant figures. But we need to note that all zeros in the base number of scientific notation are significant. Hence, we are clear with the trailing zeros in the base number because these zeros are always significant.

8. If the number is less than 1 then we conventionally put a zero to the left of the decimal point and this zero is never considered a significant figure, but zeros at the end of such numbers are considered as significant. For example, 0.1760 has four significant digits.

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