Significant figures

[Mojo]

When two or more numbers are multiplied or divided together we usually write the result to the least number of significant figures. But, why do we do this? What's the purpose? I mean I can understand if the number is an irrational number but why is it done when the result is a rational number?

For example:

21/4= 5.25

But, to the least number of significant figures it is:

21/4= 5

Doesn't this give a more inaccurate answer, since 5*4, in fact equals 20?
Some people say, "we must always round to the least number of significant figures to eliminate all traces of possible error". But is this true? My math textbook show answers to both, the least number significants and not. So, I am totally confused? How should I answer under test conditions.

[bigtick]

you need to understand that the use of significant figures is to indicate the accuracy of measured and calculated physical quantities, not the accuracy of calculations.

[Glockmeister]

It might be easier if we use an example.

Say we have a thermometer, and you look at between 20 and 21 and see 10 lines. If the mercury or whatever hits those lines, we know that it is 20.1 degrees. But if it is between those lines, we cant say for sure. It could be 20.11 or 20.12 or 20.13, we can't tell for certain. That is the rationale for sig figs.

[bigtick]

Useful reference http://itute.com/mathline/math_lesson_decpl_sigfig.doc

[shinny]

Well in maths, you'll tend to just use exact form anyhow (to prevent calculator abuse; especially CAS). Otherwise, the question will always indicate the number of decimals places to which you need to answer and VERY rarely the concept of significant figures. I personally don't recall many situations where sig figs have been used, and its unlikely to make much of an impact when it does come up, just look at the number of sig figs in the raw data you're given. Its more of a chem thing though really, as there isn't particularly much 'physical' data in maths...

[Mao]

I recall one instance in an exam question

the concept of significant figures isn't too hard, and it has widespread use in most scientific disciplines, so a useful tool to keep around

the basic principle is, your answer is only as accurate as the numbers you are given.

[shinny]

Typically in maths they'll state the actual number of sig figs you require (e.g. it says in the question, 'give your answer to three significant figures'), rather than letting you decide yourself based on the physical data given to you, so it's pretty obvious. But yeh, just learn it anyhow, you'll need it in sciences and such and its seriously not a very hard concept.