Numerical data should always be recorded in an appropriate manner, without giving the impression of unfeasible accuracy. For example, when using a digital timing device, it is quite possible for the instrument to give a readout to several decimal places. However, given that the ability of a human to react to a stimulus is only of the order of 0.2 seconds at best, any data recorded beyond the first decimal place is unfeasibly 'accurate'.
Data should only be recorded in such a way as to reflect the degree of inaccuracy.
Scientific notation and standard form
Very small or very large numbers can be recorded using standard form. This takes the form of a number between 1 and 10 (with the appropriate number of decimal places) and multiplies it by ten raised to the power of an integer (a base 10 exponential) to give the number in question.
Example: 0.00026 becomes 1.6 x 10-4
Example: 327 becomes 3.27 x 102
This allows us to express unmmanageably large or small numbers in a convenient and easy to handle way.
- Avogadro's constant: 6.02 x 1023
- Planck's constant: 6.63 x 10-34 Js
The concept of significant figures is important to indicate the degree of accuracy (or inaccuracy) of data. On many occasions a calculator will churn out a calculation result to as many digits as can be displayed. This does not so much reflect the accuracy of the calculation, rather the stupidity of the calculator.
If a titration has been carried out we know that the closest we have been able to read the burette is to ± 0.05 ml.
If the titre value is 25.00 ml then we should record it as 25.0 ± 0.1 ml.
This tells me that I can only be accurate to the third figure of the value (and even with this there is an inaccuracy). It stands to reason that any 'final answer' for a calculation using this titre value can only be as accurate as the titre value itself.
We are using the third figure of a three digit number, so we say that there are 3 significant figures in use.
It may be that more digits appear in subsequent calculations, but these must also then be adjusted to 3 significant figures. This is carried out by inspection of the fourth digit. If the fourth digit is 5 or greater, then the third digit must be increased by one unit.
Example: 3.457 to three significant figures = 3.46
Example: 3.454 to three significant figures = 3.45
Notice that following noughts after a decimal point in a number less than 1, i.e. 0.000xxx, do not count as significant figures.
The easy way to ensure the correct format is to convert a number into standard form first and then apply the significant figures.
Example: 603005 = 6.03005 x 105 = 6.03 x 105 to three significant figures
Example: 0.0005067 = 5.067 x 105 = 6.03 x 10-4 to three significant figures
The digital readouts on calculators can occasionally cause problems with students. They have several ways that they can express standard form and each student is encouraged to get to know exactly how their computer expresses numbers.
I have seen standard form expressed as numbers raised above the line, or separated by a space or by use of the letter E, both before and after a number. Some calculators do not express exponential functions at all.
Example: 27800000 may be expressed as 1.78E7 or 1.78E+7 or 1.78_07 or even 1.7807
These last two are probably the most misleading.
Similarly it is important to know how to enter values as exponential functions into a calculator. With some calculators you actually have to enter the values as if you are writing them down, while with others this is not correct.
The right number entry method depends on each individual calculator and once again it is important to know exactly how to proceed well before you actually need to use it for calculations in an exam.
The easiest thing to do is practice with a number such as Avogadro's constant. To find out how many molecules there are in 0.1 mol of hydrogen, multiply 0.1 x 6.02 x 1023.
If the answer comes to 6.02 x 1022 (6.02E22) then things are OK.