Significant Figures

Every measurement involves error and this is why the result of measurement should be reported in a way that indicates the precision of measurement. In general, the reported result of measurement is a number that includes reliable digits plus the first digit that is uncertain. The reliable digits and the first uncertain digits are known as significant digits or significant figures.

The number of digits in a measurement that keeps meaning in a sense either being part of its magnitude or explaining the precision of the measurement is called a significant figure.

• Significant figure are used to express the order of accuracy of a measurement and significant figures are only defined for measurement.
• Pure numbers or unmeasured values do not have significant figure.
• Significant figures explain the accuracy of a measurement. Greater the number of significant figures in a measurement the more accurate the measurement.

Rules for Expressing the Significant Figures

1. All the noin-zero digits are significant.
   Eg. 23.456 cm has five significant figures.
   
2. The zeros existing to the left of extreme left non-zero digit are not counted as significant figures.
   Eg. 0.352 cm as three significant figure.
   
3. The zeros existing to the right of extreme right non-zero digit are not-counted as significant figures unless least count is not mentioned.
   Eg: 3500 cm has two significant figure.
   
4. The trailing zero(s) in a number with a decimal point are significant.
   Eg: 3.500 or 0.06900 have four significant figures in each.
   
5. The zeros existing between non zero digits are counted in significant figures.
   Eg: 1003 cm has four significant figure.
   
6. The position of decimal in a measurement does not affect the number of significant figures.
   Eg: 0.321 m and 32.1 cm both have three significant figures.
   
7. The change in the unit of measurement does not affect the number of significant figures.
   Eg: 0.321 m and 32.1 cm both have three significant figures.
   
8. All the zeros between two non zero digits are significant, no matter where the decimal point is.
   Eg: 2301.009 km has 7 significant figures.
   
9. If the number is less than 1, the zero(s) on the right of decimal point but to the left of first non-zero digit are not significant.
   Eg: 0.002308 has 4 significant figure.
   
10. Power of ten are not counted in significant figures.
    Eg: 21 x 10² cm and 3.4 x 10^-3 kg both have two significant figure.

Algebraic Operations with Significant figures

The result of a calculation involving approximate measured values of quantities must reflect the uncertainties in the original measured values. It cannot be more accurate than the original measured values. In general, the final result should not have more significant figures than the original data from which it was obtained.

1. Addition and Subtraction

In addition, and subtraction, the final result should retain as many decimal places as there are with a minimum number of decimal places in the given quantities.

In addition, or subtraction, care only the numbers right to the decimal point

For example: In addition

2.5 + 21.352
= 23.852 (normal addition)
= 23.8 (addition with correct significant figures after rounding off)

In subtraction

25.324 - 21.12
= 4.204 (normal subtraction)
= 4.20 (subtraction with correct significant figures)

2. Multiplication and division:

In multiplication or division, the final result should retain as retained as many significant figures as there are in the original number with the least significant figures.

In multiplication and division, care for all the numbers in the problem (not only after or before the decimal point)

Example: For multiplication

1.234 x 2.0
= 2.468 (normal multiplication)
= 2.5 (multiplication with correct significant figures)

3. For division

27.340 / 3.51
= 7.7891739 (normal division)
= 7.79 (division with correct significant figures)

Error in Measurement

The difference between the actual value and the measured value of a physical quantity is called error. Error in measurement is caused by the limit of accuracy of an instrument.

The least count of an instrument is the smallest quantity that can be measured by that instrument. The smaller the least count is more accurate the measurement will be.

1. Absolute Error
   Absolute erorr = Measured value - Actual Value
   
2. Relative Error
   \(Abolute\;error=\frac{Measured\;Value-Actual\;Value}{Actual\;value}\)
   
3. Percentage Error
   \(Percentage\;error=\frac{Measured\;Value-Actual\;Value}{Actual\;value}\times100\%\)

It is important to note that the absolute error has a unit whereas relative and percentage error is unitless.

1. The number of digits in a measurement that keeps meaning in a sense either being part of its magnitude or explaining the precision of the measurement is called a significant figure.
   
   
2. Significant figures explain the accuracy of a measurement.
   
   
3. The result of a calculation involving approximate measured values of quantities must reflect the uncertainties in the original measured values.
   
   
4. In addition, or subtraction, care only the numbers right to the decimal point.
   
   
5. In multiplication and division, care for all the numbers in the problem (not only after or before the decimal point).
   
   
6. It is important to note that the absolute error has a unit whereas relative and percentage error is unitless.