What is significance of (.) in significant figures? [closed]

I read in my textbook (NCERT class 11 chemistry) that

• 100 has 1 significant figure
• 100.0 has 4 significant figures
• 100. has 3 significant figures

Including the dot increases the number of significant figures.

So what is the significance of this dot to significant figures?

I really don’t know whether this question should be posted to Physics or Mathematics Stack Exchange. As I think this is an elementary measurement based issue and in the purview of mathematics, I posted this in Mathematics Stack Exchange.

Solution 1:

This is a notational issue (and I agree with @SpineFeast's comment above). The book is using the way the number is written to give you information about how accurate the value is.

• Writing $100$ means that our value could have been rounded. In other words, we did some calculation and got an answer between $50$ and $150$ and rounded our answer to the nearest hundred. In scientific notation, this might be written as $1.\times 10^2$ to indicate the number of significant digits. If one were to write $1.0\times 10^2$ this means that the we rounded to the first two digits, so the actual value is between $95$ and $105$.

• Writing $100.$ means that we've put in a decimal point. The decimal point is unnecessary for the number $100$, but by putting in the extra work of the dot, we're indicating that everything to the left of the decimal point has not been rounded. This means that the value that we actually have is between $99.5$ and $100.5$. This case corresponds to $1.00\times 10^2$ in scientific notation.

• Writing $100.0$ means that we're even more certain about the value. We've put in two extra things, the decimal point and the $0$ to the right of the decimal point. By writing extra, we're indicating how much more we know about the value. In this case, we might write, in scientific notation, $1.000\times 10^2$ to indicate that the actual value is between $99.95$ and $100.05$.

Why does all of this matter? The problem lies in that we sometimes round before being done with our calculations. Perhaps we want to compute $$ 100+30 $$ What does this mean, well, in a scientific situation, the $100$ and $30$ could both have been rounded before you received them, so the $100$ represents a number between $50$ and $150$ while the $30$ represents a number between $25$ and $35$. So, the actual sum could be anywhere between $75$ and $185$, a rather large range. The difficulty here is that the two numbers have been rounded to their leading digit but because the leading digit is in a different place value, the errors in each number get complicated.

If, on the other hand, one were to compute $$ 100.+30. $$ then the decimal points mean that you're adding a number between $99.5$ and $100.5$ to a number between $29.5$ to $30.5$. The result is between $129$ and $131$, a much smaller range. So, the decimal point tells you that the values are more accurate, and, since they're more accurate, you can get a better range of possible values after operations.

Solution 2:

The dot is a decimal point. The idea is that when you see $100$ the zeros may just be there to show which place the $1$ belongs in. You could have measured $66$ and rounded it to the nearest hundred, getting $100$. When you write $100.0$ you are claiming that the real value is $100.0 \pm 0.05$, which is why we say it has four significant figures. By convention, $100.$ is representing $100 \pm 0.5$, so both zeros are significant and you have three significant digits. The original $100$ could represent any number in the range $100 \pm 50$ so only the leading $1$ is significant. I have also seen an underline to show significant figures, so $1000 \pm 5$ would be written as $10\underline00$. When the figures are not zeros they are considered to be obviously significant, but the measurement could be accurately $100$ and we need to know that.