In absence of precise measurement we take many readings of the same measurement.

$a_{\text {mean }}=\frac{a_1+a_2 \cdots+a_n}{n}$

Absolute error: $\left|\Delta a_1\right|=\left|a_1-a_{\text {mean }}\right|$

Mean absolute error: $\Delta a_{\text {mean }}=\frac{\left(\left|\Delta a_1\right|+|\Delta Q| \cdots+\left|\Delta a_n\right|\right)}{n}$

$a_{\text {mean }}-\Delta a_{\text {mean }} \leqslant a \leqslant a+\Delta a_{\text {mean }}$

Relative error: $\frac{\Delta a_{\text {mean }}}{a_{\text {mean }}}$

% error $\delta a=\frac{\Delta a_{\text {mean }}}{a_{\text {mean }}} \times 100$ %

Sum or Difference

$A=A \pm \Delta A$

$B=B \pm \Delta B$

$Z=A+B$

Find error in Z

Given $\Delta A, \Delta B$

$Z \pm \Delta Z=(A \pm A A)+(B \pm \Delta B)$

$=A B(A+B) \pm(\Delta A)$

$\Delta Z=\Delta A+\Delta B \pm(A B)$

Subtraction:

$Z=A-B$

$Z \pm \Delta Z =(A \pm \Delta A)-(B \pm \Delta B)$

$=(A-B) \pm \Delta A \pm \Delta B$

$\Delta Z = \pm \Delta A \pm A B$

For maximum error

$\Delta Z = \Delta A+\Delta B$

So, $Z= A-B$

$\Delta Z= \Delta A+\Delta B$

Product or Quotient

$Z = \frac{A}{B} \text { or } A \times B$

$Z=A B$

$Z \pm \Delta Z=(A \pm \Delta A)(B \pm \Delta B)$

$Z \pm \Delta Z=(A B) \pm \Delta A B \pm \Delta B A \pm \Delta A\Delta B$

Now, divide above equation by Z.

$1 \pm \frac{\Delta Z}{Z}=1 \pm \frac{\Delta A}{A} \pm \frac{\Delta B}{B} \pm \frac{\Delta A \Delta B}{A B}$

a) Convert $\pm$ to "+" for maximum error.

b) $(\Delta A \Delta B) \rightarrow$ neglect product of 2 small quantity.

$\frac{\Delta Z}{Z}=\frac{\Delta A}{A}+\frac{\Delta B}{B}$

$Z =\frac{A}{B}$

$\frac{\Delta Z}{Z}=\frac{\Delta A}{A}+\frac{\Delta B}{B}$

If $Z=\frac{A^n}{B^m}$

$\frac{\Delta Z}{Z}= \pm\left(n \frac{\Delta A}{A}+m \frac{\Delta B}{B}\right)$

% error in Z = n(% error A)+ m(% error B)

If $H=I^2 R t$, where, $I$ is current, $R$ is resistance, $t$ is time, and $H$ Heat generate. Error in measurements of I, R and t are 2%, 3% and 1%. Find relative error in measurement of $H$ ?

$\frac{\Delta H}{H}=\frac{2 \Delta I}{I}+\frac{\Delta R}{R}+\frac{\Delta t}{t}$

$\frac{\Delta H}{H} \times 100$%$=2 \frac{\Delta I}{I} \times 100$%$+\frac{\Delta R}{R} \times 100$%

$=2 \times 2$%+3 %+1 % = 8 %

Error in $\Delta H(\text { in })$ % $= \pm 8$%

Whenever any measurement is reported, the uncertainty in the measurement is in the last digit.

For example: Time period $=1.62 {s}$.

1 and 6 are reliable digits.

$2 \rightarrow$ uncertainty.

$1.62{s} \rightarrow$ correct upto 3 significant digits.

$1.62 \rightarrow 3$ significant digits.

$287.5 \mathrm{cm} \rightarrow 4$ significant digits.

Significant digits indicate the precision of an instrument which defend on the lead count.

Choice of different units should nut affect the number of significant digits.

$2.308 \mathrm{cm} \rightarrow 4$ significant.

$2.308 \mathrm{cm}$ = $23.08 \mathrm{mm}$ = $0.02308 \mathrm{m}$.

Rules:

1) All non zero digits are significant.

2) Zeros between 2 non zero digits ore significant irrespective of decimal point.

3) If the number is less than 1, the zero(s) on the right of decimal point but to the

left of the first non-zero digit are not significant.

0.00238  3 significant digits.

4) The terminal or trailing zero(s) in a number without a decimal point are not

significant.

123 m = 12300 cm = 123000 mm has three significant figures, the trailing zero(s)

being not significant.

5) The trailing zero(s) in a number with a decimal point are significant.

The numbers 3.500 or 0.06900 have four significant figures each.

Measurement in scientific notation.

Notation: $a \times 10^{b}$.

b is exponent (can be positive or negative)

$a$ is between 1 and 10.

Order of magnitude:

$a \times 10^b$

If $1 \leqslant a \leqslant 5$

round it off to 1

$\approx 10^b \quad$ b is called order of magnitude.

$5 \leqslant a \leqslant 10 \rightarrow$ round of to the next digit

$10^{(b+1)} \rightarrow(b+1)$ is called the order of magnitude.

For example: $1.28 \times 10^7 \mathrm{m}$

Order of magnitude $\sim 10^7 \mathrm{m}$

Diameter of Hydrogen atom: $1.06 \times 10^{-10} \mathrm{m}$

Order of magnitude $\sim 10^{-10} \mathrm{m}$

Exact numbers can be considered to have an infinite number of significant figures.

For example :

Diameter $={2} \times$ radius

2 is exact number.

So it has $\infty$ significant digits.

Final result 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.

Example:

$\text { Density }=\frac{\text { mass }}{\text { volume }} =\frac{4.237 \mathrm{gms}}{2.51 \mathrm{cm}^3}$

$\text { Density } =1.68804780876 \frac{\mathrm{g}}{\mathrm{\textrm {cm } ^ { 3 }}}$

$m = 4.2379$, 4 significant digits.

$V = 2.51 \mathrm{cm}^3$, 3 significant digits.

Density should have three significant figures.

$\text { Density }=\frac{\text { mass }}{\text { volume }}$

$\quad$$\quad$$\quad$$=\frac{4.237 \mathrm{gms}}{2.51 \mathrm{~cm}^3}$

$\text { density } =1.68804780876 \frac{\mathrm{g}}{\mathrm{cm}^3}$

$=1.69 \mathrm{gm} / \mathrm{cm}^3$

In addition or subtraction, the final result should retain as many decimal places as are there in the number with the least decimal places.

For example, the sum of the numbers 436.32 g, 227.2 g and 0.301 g by mere arithmetic

addition, is 663.821 g. But the least precise measurement (227.2 g) is correct to only one decimal place. The final result should, therefore, be rounded off to 663.8 g.

Preceding digit is raised by 1 if the insignificant digit to be dropped (the

underlined digit in this case) is more than 5,

Is left unchanged if the latter is less than 5.

2.745 rounded off to three significant figures becomes 1.74.

The number 2.735 rounded off to three significant figures becomes 1.74 since

the preceding digit is odd.

$L=16.2 \mathrm{cm}$, and $B=10.1 \mathrm{cm}$

Area = $163.62 \mathrm{cm}^2$.

$L=16.2 \pm 0.1 \mathrm{cm}$, $\Delta L=0.1 \mathrm{cm}$

$B=10.1 \pm 0.1 \mathrm{~cm}$

$A=(16.2 \pm 0.1)(10.1 \pm 0.1) \mathrm{cm}^2$

$\frac{\Delta L}{L}=\frac{0.1}{16.2} \times 100$% $=0.6$ %

$\frac{\Delta B}{B}=\frac{0.1}{10.1} \times 100$%=1 %

$A={L B} ; \frac{\Delta A}{A}=\frac{\Delta L}{L}+\frac{\Delta B}{B}$

$(\frac{\Delta A}{A})$% = 0.6 %+1 % = 1.6 %

$\Delta A=\frac{1.6}{100} \times A$

$A=16.2 \times 10.1$

$A=163.62 \mathrm{cm}^2 \pm 1.6$%$\leftarrow \text{2.6cm}$

$=164 \mathrm{cm}^2 \pm 3 \mathrm{cm}^2$

For example, the accuracy is $1.029 \pm 0.019$.

Relative error $=\frac{0.01}{1.02} \times 100=1$%

Similarly, $9.89 \mathrm{~g} \pm 0.01 \mathrm{g}$

Relative error $=\frac{0.01}{9.89} \times 100$ =0.1 %

Intermediate results in a multi-step computation should be calculated to one more significant figure in every measurement than the number of digits in the least precise measurement.