Significant figures: a problem.

Question

After going all through web and posts I can't get a complete idea of significant figures. I'll try to explain the problem.

The definition that seems more frequent is:

significant figures: number of figures carrying on precision.

It is easy to see that in the number $1000$ zeros are non-significant figures unless we specify it as $1000.$. It is also clear that $0010.$ is two significant figures.

Here it comes when definition shows unuseful (at least to me), because leading zeros as well as non zero numbers talk about precision. For example $0.0017$ and $0.1217$ are same precision. They indicate the measuring instrument can detect variations in the ten-thousands. If not, please explain how. That's the specific problem. I beg you answer with concrete examples.

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The most interesting site I've read is this , and I understand how significant figures work, but the previous problem remains.

Answer 1

I've understood it by this way:

Suppose we have sand grains of $0.0005$ grams each one, weighed with a 0.0001 precision balance. Of course, the numbers before $5$ have a meaning: they indicate magnitude. But they are not significant figures. It can be seen in this way:

somebody estimate grains' number in $63566$. If we want to estimate the total weight, we should multiply. But there is 'no difference' between $0.0005$ and $0.0006$. Let's see how it modifies the multiplication:

$ 0.0005\times 63566=31783$

$ 0.0006\times 63566=38396$

So we can't even trust the first number, as if only the five $0.0005$ were meaningful. We can see it easily if we write it as $5\times 10^{-4}$

Once understood, $31783$ should be written as $3\times 10^{4}$

Answer 2

Let's try an actual example. Suppose you have a pile of sand that weighs $0.1217$ on your scale. The sand is a mixture of white and red grains. Your boss asks you to determine what percentage (by weight) of the sand is red.

About how accurate will your answer be? Will you be able to measure the proportion of red sand in tenths of a percent?

The next sample of sand weighs only $0.0017$ on your scale. Again it has white and red grains and your boss wants to know the percentage of red grains by weight.

Now how accurate will your answer be?

It is sometimes true in practice that it doesn't matter that a particular measurement has few digits of precision, because the only use for that measurement is to add it to a much larger measurement. Then only the absolute precision of both measurements matters. But the reason significant figures are so often used is that we are very often multiplying or dividing numbers.

If you write every number in standard scientific notation, even the ordinary numbers such as $10$ or $0.1$ that do not seem to need this notation, then you will get a better idea of significant digits. In scientific notation, all the digits in the mantissa are significant, because we never write leading zeros and we never write more digits than the accuracy of measurement justifies. For example, $0.1$ is $1 \times 10^{-1}.$ The mantissa then is $1$ -- one significant digit.