# discrete mathematics – Thermometer 5°C increments: How does resolution relate to significant figures? How interpret \$x pm y\$ and translate to number of significant figures?

Given: The smallest increment on thermometer is 5°C.
When reading a temperature, should I note 15°C or 15.0°C?
I learnt from 2.2 Accuracy, Precision and Resolution, that the last sig fig in a number is always guessed and therefore uncertain.

They say that $$resolution=frac{smallest increment}2$$.
Here we would get $$resolution = 2.5°$$.
Therefore I would be able to write $$15.0°pm2.5°$$, wouldn’t I?
Would it be also correct to write $$18.0°pm2.5°$$? I guess no.
Because here Uncertainties – Physics A-level & GCSE they say, that it is not possible to use any value which is not an increment of the measuring scale.
So is the notation of $$xpm resolution$$ only valid for multiples of $$Delta x=2times resolution$$?

From the logical point of view I would interpret a notation of
A)
$$x pm y$$
as
$$x_{true} in (x – y; x + y)$$

Let’s say, that we want to round to the $$10^{-3}$$.

B)
$$x_{true}=1.23456$$
$$y=round(x_{true},3)=1.235$$
If we define
$$f(x)=round(x,3)=y$$
then $$f^{-1}(y)={x|f(x)=y}$$
In this case the notation of $$pm$$ would just mean:
if y is positive (the minus case):
$$1.235 pm 0.001$$
$$equiv$$
$$1.235 – 0.001 = 1.234$$
if y is negative (the plus case):
$$-1.235 pm 0.001$$
$$equiv$$
$$-1.235 + 0.001 = -1.234$$
Because in the set of $$f^{-1}(1.235)=(1.2345;1.2355($$ some of the elements are $$=….5$$ and some are $$=….4$$.
For the latter case, that is the reason for writing $$pm 0.001$$?

So which interpretation of $$x pm y$$ is correct?
A) or B)

What’s the relation of rounding and sigfigs.
If a number is rounded, does it change the certainty quality of the last digit into uncertain?

Please also have a short look at these video clips, to understand my confusion:

Which numbers are certain in each measurement:
In spite of 1 and 3 are certain in 130 pounds, it’s still possible that these sigfigs are affected by rounding ?
If a digital scale reads 133.6, does that mean that the 0.6 is estimated or is it the rounding result of the estimated digit in the $$10^{-2}$$ position?

Measuring with a metric ruler:
Possible readings 3.38 … 3.41 by having a scale with $$Delta = 0.01$$

Measure pencil with a ruler $$Delta 1mm$$
Is 1mm the resolution or is it $$0.5mm$$?
If the ruler increment is $$Delta 10 cm$$, are we free to claim the length any value in the range $$(0;10)$$?
Uncertainty $$frac 1 2 times resolution$$ vs $$uncertainty=resolution$$