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The result of correcting $96$ to $1$ sig. fig. is obviously $100$. Yet, are the two zeros place holder?

In that case, I find the following question and answer given by the textbook interesting.

It is given that the approximated number of elastic in a box is $100$. If the number is correct to $1$ significant figure, what is the smallest possible number of elastic?

Answer: $100$

This example confuses my understanding of significant figure and place holder. Any clarification is appreciated.

Community
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Nighty
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1 Answers1

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With numbers like $0.0043$, the first few $0$s are just placeholders, so it has two significant figures. On the other hand, $2.400$ has four significant figures. You know the $0$s on the end are significant figures, otherwise you'd just write $2.4$.

However, it gets confusing with numbers like $100$. There are three possibilities: Either it is correct to one, two, or three significant figures. This depends entirely on the context. For example, if you round $101$ to two significant figures, you end up with $100$, so the first two digits are significant and the last one isn't.

In general you avoid this by writing in scientific notation. $100 = 1\times10^{2}$ has one significant figure, $100 = 1.0\times10^{2}$ has two significant figures, and $100= 1.00\times10^{2}$ has three significant figures.

Now let's think about your example. Suppose we have a number like $99.99$ and we round it up to $100$. At the moment, $99.99$ has four significant figures, so we want to see what happens if you round it to one significant figure. When we do that, you'll see that the digits in the 'hundreths', 'tenths', and 'ones' positions are insignificant, but the digit in the 'tens' position is significant. That means that when it rounds up to $100$, the digit in the 'tens' position is still significant. Therefore the "$100$" we end up with has two significant figures.

In other words, if "$100$" is correct to two significant figures, the true value may be less than $100$, but if it's correct to one significant figure the smallest it can be is $100$! Weird.

Harambe
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  • Thank you for the reply! In that case, if I correct $96$ to 1 significant figure, I will have $100$, which has 2 significant figures! Am I getting this right? – Nighty Aug 30 '17 at 13:39
  • Yeah exactly. It's just a technicality from the way significant figures work. If you end up using significant figures for something later on in life you'll realise it's a rare thing to have to deal with. – Harambe Aug 30 '17 at 20:04