Why does the rule for multiplication/division take into consideration the no. of significant figures?

Question

I've learned that the rule for multiplication, when taking into account the significant figures, is as follows:

The final result should retain as many significant figures as there are in the original number with the least significant figures.

So, by that definition, if I has the mass of an object = 4.237g and its volume = 2.51 cm^3, then the density would be 1.69 g cm^-3. However, let's take an example in which we have the side of a cube given to be 25.6 which has 3 significant figures. Now let's find out its volume: 16777.216. Now if I wanna report my result with the least significant figures (3), I would get an answer 168 which is plain wrong.

I could use scientific notation, but even there, I would have the significant figures which are significant.

Could someone help me out by explaining me this rule's implication? Thanks.

Answer 1

Now let's find out its volume: 16777.216. Now if I wanna report my result with the least significant figures (3), I would get an answer 168 which is plain wrong.

Your problem is that you seem to be truncating the value! When employing the significant figure rule, you turn all other values to zero. Trailing zeros are placeholders, so they can be eliminated from the reported value when using scientific notation. If you are not using scientific notation, you must record those zero's: $$ 16777.216 \Rightarrow 16800\neq168 $$ As compared to the scientific notation form, $$ 1.6777216\times10^4 \Rightarrow 1.6800000\times10^4\equiv1.68\times10^4 $$