To present the data, obtained experimentally or theoretically, some rules are followed for the sake of convenience

Since atoms and molecules have extremely low mass and are large in numbers. So, for dealing with such a large and small number scientific notation is used. It is also called exponential notation in which any number is represented as : eg : Andromeda galaxy (closest to milky way galaxy ) contains at least `2, 00 , 000, 000 , 000` stars. which is a large that causes problem so we write it in scientific notation `2.0xx10^(11)`.

`N xx 10^n`

where `n =` an exponent having positive or negative values.

`N =` any number between `1.000-9.999` is called digit term.

e.g. (i) `232.508` can be written as `2.32508xx10^2`.

(ii) `0.00016` can be written as `1.6xx10^(-4)`.

Both these operation follow the some rules i.e.

e.g. (i) `(5.6xx10^5)xx (6.9xx10^8) = (5.6xx6.9)(10^(5+8)) = (5.6xx6.9)xx10^(13) = 38.64xx10^(13)`

(ii) `(9.8xx10^(-2)) xx (2.5xx10^(-6)) = (9.8xx2.5)xx10^(-2+(-6)) = (9.8xx2.5) (10^(-2-6))` `= 24.50xx10^(-8)`

(iii) `(2.7xx10^(-3))/(5.5xx10^4) = (2.7 div 5.5 )(10^(-3-4)) = 0.4909 xx10^(-7)`

For both the operations, the numbers are written in the form which have some exponent. And after that coefficient are added or subtracted.

e.g. For Addition of `6.65xx10^4` and `8.95xx10^3`.

Step I : Make the exponent same. We can do it by changing the exponent of `6.65xx10^4` to `6.65xx10^3` or by changing the exponent of `8.95xx10^3` to `0.895xx10^2`.

Step II : Addition `(66.5xx10^3+8.95xx10^3) = 75.45xx10^3 = 7.545xx10^4`

or `(6.65xx10^4+0.895xx10^4) = 7.545xx10^4`

Similarly, subtraction of two numbers can be done in the same manner.

`2.5xx10^(-2) - 4.8xx10^(-3)`

`= (2.5 xx 10^(-2))-(0.48xx10^(-2))`

`= (2.5 - 0.48) xx 10^(-2) = 2.02xx 10^(-2)`

Precision : It is the closeness of various measurements for the same quantity.

Accuracy : It is the agreement of a particular value to the true value of the result

for example : value of g ( acceleration due to gravity `= 9.8065 m//s^2`) .
Now if the observed value are - `9.8714 , 9.8713 , 9.8712 , 9.8715` they are precise and if observed value are `9.81 , 9.79 , 9.82 , 9.77 => ` accurate.

`text(Importance of accuracy and percision)` :
(1) Accuracy comes in importance when you need to hit a target . Either you hit the target or don't hit it. It does no good to be precise if you miss all the shots .

(2). Precision gains importance in calculations when you use a measured value in calculation , you can be as precise as your least measurement from here arises the idea of significant figures ( which includes all the digits you know for certain plus the last digit , which contains come uncertainity).

For e.g., See Table 1.4.

Note : (i) The uncertainty in any value is indicated by mentioning the number of signification figures.

(ii) Significant figures are meaningful digits which have certainty.

(iii) Uncertainty is indicated by writing the certain digits and the last uncertain digit e.g. if there is a result written as `11.2` `mL`.

Formal Definition : The total number of digits in a number including the last digit whose value is uncertain is called

the number of significant figures.

Here, `11` is certain and `2` is uncertain. And uncertainty would be in the last digit. If uncertainty is not given then an uncertainty of `pm1` in the last digit is understood.

Defination : (1). A leading zero is any 0 digit that comes before the first nonzero digit

(2). trailing zeros are a sequence of 0's after which no other digit follows.

(i) All non-zero digits are significant.

(ii) Zeros preceding to first non-zero digit are not significant. These zeros only indicates the position of decimal point only.

(iii) Zeros between two non-zero digit are significant.

(iv) Zero at the end or right of a number are significant if they are on the right side of the decimal point.

e.g. `0.200g` has `3` significant figures.

If there is no decimal point then the terminal zeros are not significant.

e.g. `tt((text{Number} , text{no. of significant figures} ) , (100 , 1) , (100. , 3) , (100.0 , 4))`


(v) Objects which can be counted have infinite significant figure as these are exact numbers. And can be represented as writing infinite number of zero after placing a decimal.

e.g. `2 = 2.000000` or `20 = 20.00000`

The result obtained can not have more digits to the right of the decimal point than either of the original numbers.

e .g.

`12.11`
`18.0`
`1.012`
` underline overline(31.122)`

(Since 18.0 has only one digit after decimal point . So, the answer should be reported as `31.1`.)

The result obtained should have the significant figures equal to the number which has minimum number of significant figures.

e.g. `2.5xx1.25 = 3.125`

So, answer should be reported as `3.1`.

(i) If the right most digit to be removed `> 5`, then preceding number is increased by one.

e.g. `1.386`

If we remove `6`, then it is rounded off to `1.39`.

(ii) If the right digit to be removed `< 5`, then the preceding number is not changed.

e.g. `4.334`

If we remove `4`, then it is rounded off to `4.33`.

(iii) If the right most digit to be removed `= 5`

Case I : If preceding number is even, then it is not changed.

Case II : If preceding number is odd, then it is increased by one.

For converting units from one system to other a method was accomplished which is called `text(factor label method)` or `text(unit factor method)` or `text(dimensional analysis)`.

Note : Units can be handled just like other numerical part. It can be cancelled, divided, multiplied, squared etc.