Significant Figures

Accuracy of Measurement

​The accuracy of any measurements depends upon the

(a) accuracy of the measuring device used

(b) the skill of its operator.

If the average value of different measurements is close to the correct value ,the measurement is said to be accurate.

If the values of different measurements are close to each other and hence close to the average value, the measurement is said to be precise .

A measurement can have a good accuracy but poor precision because different measurements may give a correct average.

Errors in measurements when the same mistake is made repeatedly are called systematic errors. They do not affect the Precision but they often affect the accuracy of a measurement.

Significant Figures

The total number of digits in a number including the last digit whose value is uncertain is called the number of significant figures.

Rules for determining the number of significant figures

(1) All non zero digits as well as the zeros between the non zero digit are significant.

576 cm has three significant figures

5004 has 4 significant figures

0.48 has two significant figures

2.05 three significant figures

(2) Zeros to the left of the first non zero digit in a number are not significant.

0.05 m has 1 significant figure

0.0045 has two significant figures

(3) If a number ends in zeros but these zeros are to the right of the decimal point then these zeros are significant.

5.0 metre has two significant figures

2.500 g has 4 significant figures

0.0200 has 3 significant figures

(4) If a number ends in zero but these zeros are not to the right of a decimal point, these zeros may or may not be significant .

1.05 * 10³ has three significant figures

1.050 * 10³ has four significant figures

1.0500 * 10³ has five significant figures

Mathematical operation on numbers expressed in scientific notation

1) 4683.507 will be written as 4.683507 * 10³

Decimal is moved three places towards left so that only one non-zero digit is left and number of places moved is the exponent of 10 in scientific notation.

2) 0.000256 will be written as 2.56 * 10^-4

Decimal is moved four places towards right so that there is only one non-zero digit before the decimal point and the exponent of 10 is -4.

Calculations involving multiplication and division

In multiplication the coefficients i.e. the numbers before the factor 10ⁿ  are multiplied and the exponents of 10 are added up.

For Example: ( 5.7 ×  10⁶) × ( 4.2 × 10⁵)

( 5.7 × 4.2 ) (10 ⁶⁺⁵ )

23.94 × 10¹¹

In division the factor N are divided and exponents are subtracted.

For Example: (5.7 × 10⁶ ) ÷ ( 4.2 × 10³ )

( 5.7 ÷ 4.2 ) × (10^6-3 )

1.357 × 10³

In addition and subtraction, first the numbers are written in such a way that they have the same exponents. Taking out 10ⁿ common, the coefficients are added or subtracted.

For Example: 4.56 × 10³ + 2.62 × 10²

45.6 × 10² + 2.62 × 10²

( 45.6 + 2.62 ) × 10²

58.22 × 10²

For Example: 4.5 × 10^-3  – 2.6 × 10^-4

4.5 × 10^-3 -0.26 ×10^-3

4.24 × 10^-3

Rules for determining the number of significant figures 

Rule 1 The result of an addition or subtraction should be reported to the same number of decimal place as that of the term with least number of decimal places.

For Example: 4.523 + 2.3 + 6.24

Actual sum= 13.063

Reported sum = 13.1

4.523 has 3 decimal places

2.3 has 1 decimal places

6.24 has 2 decimal places

Answer should be reported only upto one decimal place.

Example 2 : 18.4215 – 6.01

Actual difference = 12.4115

Reported difference= 12.41

As the second number has 2 decimal places only the answer is reported upto 2 decimal places.

Rule 2 : The result of a multiplication or division should be reported to the same number of significant figures as is possessed by the least precise term used in calculation.

Example 1: 4.327 × 2.8

Actual product =12.1156

Reported product= 12

The first number has 4 significant figure while the second has 2.The actual product has been rounded off to give a reported product containing two significant figures.

Example 2: 0.46 ÷ 15.734

Actual quotient = 0.029236

Reported quotient = 0.029

It should contain only 2 significant figures because the least precise term in calculation has only 2 significant figure.

Rule 3 If a calculation involves a number of steps, the result should contain the same number of significant figures as that of the least precise number involved, other than the exact numbers.

(42.967 * 0.02435) ÷ (0.34 * 4)

=0.7692988

0.77