Units, Measurements, Dimensions and Error Analysis
significant figures
The significant figures are a measure of accuracy of a particular measurement of a physical quantity. Thus, significant figures indicate the precision of the measurement which depends on the least count of the measuring instrument.
Significant figures in a measurement are those digits in a physical quantity that are known reliably plus the one digit which is uncertain. For example, the thickness of any material measured = 6.45 cm. Among the digits appearing in the value 6.45, 6 & 4 are reliable or certain, while the digit 5 is uncertain.
Significant figures are the number of digits up to which we are sure about their accuracy.
Significant figures don't change if we measure a physical quantity in different units.
Significant figures retained after mathematical operations like addition, subtraction, multiplication and division should be equal to the minimum significant figures involved in any physical quantity in the govern operation.
Other Topics from this Chapter

Errors in Measurement, Error Analysis

Dimensions, Dimensional Formulae, Dimensional Equation

Dimensional Analysis and its Applications

Rules for Counting (Finding) Significant Figures
v All nonzero digits are significant. For example, X = 2.345 has four significant figures.
v The zeros appearing between two nonzero digits are counted in significant figures. For example, 1.023 has four significant figures.
v The zeros occurring to the left of the last nonzero digit are not significant (insignificant). For example, 0.00123 has three significant figures. The two zeros appearing in the left of 1 are not significant.
v In a number without decimal, zeros to the right of nonzero digit are not significant (insignificant). However if the same value is recorded on the basis of actual measurement the zeros to the right of nonzero digit becomes significant. For example, L = 20000 m has five significant figures but X = 20000 has only one significant figure.
v In a number containing decimal, the trailing zeros (present to the right) are significant. For example, 1.500 has four significant figures.
v The exact number of digits appearing in the mathematical formula of different physical quantities has infinite number of significant figures. For example, perimeter of a square is given by (4 x side). Here, 4 is an exact number and has infinite number of significant figures which can be as 4.0, 4.00, 4.000 etc. as per the requirement.
v The powers of ten are not counted as significant digits. For example, 1.5 X 10^{7 }has only two significant figures 1 & 5.
v Change in the units of measurement does not change the number of significant figures. For example, distance between points is 1234 m, which has four significant figures. The same can be expressed as 1.234 km or 1234 X 10^{5} cm both having four significant figures.
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Rounding Off a Digit
v If the insignificant digit to be dropped is less than 5, then the preceding digit is left unchanged. For Example, 5.82 is rounded off to 5.8 since 2 < 5.
v If the insignificant digit to be dropped is more than 5, then the preceding digit is raised by one. For Example, 5.86 can be rounded off to 5.9 since 6 > 5.
v If the insignificant digit to be dropped is 5 followed by digits other than zero, then the preceding digit is raised by one. For Example, 5.852 is rounded off to 5.9
v If the insignificant digit to be dropped is 5 or 5 followed by zeros, then preceding digit is left unchanged, if it is even. For Example, 5.250 is rounded off to 5.2
v If the insignificant digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is raised by one, if it is odd. For Example, 5.350 is rounded off to 5.4
v The exact numbers like 2, 3, 4, p appearing in the mathematical formulae of different physical quantities and are known to have infinite significant figures, are rounded off to a limited number of significant figures as per the requirement.
Rules for Algebraic Operations (Arithmetic Operations) with Significant Figures
Certain rules need to be followed while doing arithmetic operations such as addition, multiplication, subtraction with significant figures so that the accuracy can be maintained in the final result (as in the original values or inputs).
Addition and Subtraction
If two or more physical quantities are added or subtracted, there should be as many decimal places retained in the final result as are there in the number with the least decimal places. Suppose the numbers to be added or subtracted, the least number of significant digits after decimal is n. Then in the result i.e., after addition or subtraction the number of significant digits after decimal should be n.
Example: 1.1 + 4.54 + 16.084 = 21.724 ⇒ 21.7
Multiplication and Division
If two measured values are multiplied or divided, there should be as many significant figures retained in the final result, as are there in the original number with the least significant figures. For example say, in the measured values to be multiplied or divided the least number of significant digits be n, then in the product or quotient, the number of significant digits should also be n.
Example: 2.2 x 13.222 = 29.088 ⇒ 29
In the above example the least number of significant digits in the measured values are two so, the result when rounded off to two significant digits become 29.
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