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I was wondering:

Why does

$1.30 \times 10^3$ have $3$ significant figures

while $1300$ has $2$ significant figures

(they are both the same number)

Why is that distinction ?

When should I use each ?

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up vote 1 down vote accepted

I found the following sentence in

"In particular, the potential ambiguity about the significance of trailing zeros is eliminated. For example, 1300 to four significant figures is written as 1.300×10^3, while 1300 to two significant figures is written as 1.3×10^3".

This is the problem of trailing zeros. In Wiki page, the expalnations are quite clear on this topic.

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But if I see merely 1300 - should i conclude it has 2 SF ? – Royi Namir Nov 14 '13 at 7:40
If your teacher says yes, then yes, at least until the next teacher. Truth is that it is not universally agreed on. – André Nicolas Nov 14 '13 at 7:42

If you write $1.30\times 10^2$ instead of $1.3\times10^2$, some people construe that to mean you're claiming it's accurate to the nearest $0.01\times 10^2$.

I.e. "$1.30\times10^2$ means $(1.30\pm\text{half of }0.01)\times10^2$, whereas $1.3\times10^2$ means $(1.3\pm\text{half of }0.1)\times 10^2$.

In one case you're rounding to the nearest $0.01\times10^2$, and in the other to the nearest $0.1\times10^2$.

It's just a matter of how much accuracy is being claimed.

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That is what I was taught, but I don't think it is universal. The difference is in approximate numbers. $1.30 \times 10^3$ represents $(1.30 \pm 0.0005) \times 10^3$, while $1300$ could represent $1300\pm 50$ as any number in that range would round (using 2 SF) to $1300$. In the case of $1.30 \times 10^3$ the writer went to the extra trouble to write the trailing zero. In the case of $1300$ we don't know how many of the zeros are significant. I was also taught that when I cared and had 3 SF, I should write it $13\underline 00$, but I haven't seen that since. If you really care, you should use $\pm$ the error bound.

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