# Significant figures in textbooks

I know that there are a plenty of questions similiar to this one, but none of them was able to definitely give an answer, at least in my point of view.

In so many Physics textbooks (Tipler, Halliday, Alonso & Finn), they usually explain at chapter 1 about the significant figures and how one should face the problems throughout the book.

However, they usually don't obey the convention established in their own books, or maybe it is just a typo.

Anyway, my question is: Why the answers in the back of the book are usually expressed with wrong rounding or with more significant figures than it should?

For instance, in the problem 5.39 in the book by Alonso & Finn (vol. 1), it is given the radius ($$r = 3.0 \;\mathrm{m}$$) of a circular motion and its linear velocity ($$v = 4.0 \times 10^5 \;\mathrm{m/s}$$) and asks for the radial component of acceleration (I know that in a uniform circular motion, there is no tangent acceleration).

I've found: $$\frac{v^2}{r} = \frac{(4.0 \times 10^5\;\mathrm{m/s})^2}{3.0\;\mathrm{m}} \implies a_r = 5.333\ldots \times 10^{10}\mathrm{m/s^2}$$. Since the quantity with smallest significant figures (s.f.) have only two s.f., I rounded the answer as $$5.3\times 10^{10}\;\mathrm{m/s^2}$$. In the back of the book, the answer is $$5.3\mathbf{3}\times 10^{10}\;\mathrm{m/s^2}$$.

There is another example, when I was studying exactly this question in the book by John R. Taylor (An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements):

He states that 1.27 means that the measure lies between 1.265 and 1.275. It is ok, as a rule.

But then, before this statement, he presents a measure of a pencil by a ruler, and says that the 36 mm is the best estimate of length, which translates into the interval 35.5 mm and 36.5 mm. How? Shouldn't be between 35 mm and 37 mm (figure attached)? Or, if the interval is $$35.5\;\mathrm{mm} \lt l \lt 36.5 \;\mathrm{mm}$$, then the measure shouldn't be reported as $$36.\mathbf{0}\;\mathrm{mm}$$, instead of only $$36\;\mathrm{mm}$$?

I know that may sound silly, but I wouldn't ask this kind of question if I hadn't seen this issue constantly in books and basically no answer to this.

In a technical paper a measurement of a quantity which is known between 1.265 and 1.275 would be described using the concept of the standard uncertainty described above. Assuming that the measurement is uniformly distributed in that range, then the standard uncertainty would be 0.003. So the number would be described as "1.260 with a standard uncertainty of 0.003" or as "1.260 $$\pm$$ 0.003" or "1.260(3)" if the context is understood.