Tag: units and measurement: error analysis

Given :  $R = 7\times 10^8 \ m$     and    $M = 2\times 10^{30} \ kg$
Volume of sun  $V = \dfrac{4}{3}\pi R^3 = \dfrac{4}{3}\pi\times (7\times 10^8)^3 = 1.4\times 10^{27} \ m^3$
Density of sun   $\rho = \dfrac{M}{V} = \dfrac{2\times 10^{30}}{1.4\times 10^{27}} = 1.43\times 10^3 \ kg/m^3$
Thus order of magnitude of density of sun is $10^3 \ kg/m^3$.

When a number rounded to the nearest power of 10, it is called order of magnitude. Here a number which is less than 5 is taken as 1 and a number greater than 5 is taken as 10. 
As $6.4>10, $ so it would be takes as 10. 
The order of magnitude of earth's size $=10\times 10^6=10^7 m$  

Here, time taken $t=2.9$ billion years $=2.9\times 10^9$ years $ =2.9\times 10^9\times (365\times 24\times 3600) s=9.14\times 10^{16} s$
Distance $=$ velocity $\times $ time
or $d=(3\times 10^8)\times (9.14\times 10^{16})=2.74\times 10^{25} m$
As $2.74 <5$ so it will be taken as 1. 
Thus, the order of magnitude of distance $d=1\times 10^{25} m=10^{25} m$

Here column-I represents the some prefixes and column-II represents the power of ten corresponding prefixes. 

We know that  $1\mu s = 10^{-6} \ s$
Thus,  $5\times 10^{7}\mu s = 5\times 10^{7}\mu s\times \dfrac{10^{-6} \ s}{\mu s} = 50 \ s$

Order of magnitude of $379$ is $\dfrac { 379 }{ { 10 }^{ 2 } } =3.79<{ 5 }^{ -1 }$    i.e, $379\approx { 10 }^{ 2 }$

All non-zero numbers are always significant.All zeroes before a non zero number are insignificant. All zeroes which are simultaneously to the right of the decimal point and at the end of the number are significant. The only non significant digit here is the zero before the decimal point.
Hence, number of significant digits is 4.

The significant figures of a number are those digits that carry meaning contributing to its measurement resolution. This includes all digits except:
1. All leading zeros;
2. Trailing zeros when they are merely placeholders to indicate the scale of the number (exact rules are explained at identifying significant figures); 
3. Spurious digits introduced, for example, by calculations carried out to greater precision than that of the original data, or measurements reported to a greater precision than the equipment supports.

According to rules,
$6.032$ 
All the digits are significant. (no leading and trailing zero)
No. of significant digits $= 4$

According to the rules of significant figures $6.320$ has four significant figures.
$6.032$ has four significant figures.
$0.0006032$ has four significant figures.