First of all.. Sig figs is an e**stimate **of precision. Estimate being the key word here. Sometimes it works well. other times it doesn't work so well. It is an estimate!

Next, the conversion between C and F is

°F = 1.8*°C + 32

where

1.8 is exact and 32 is exact.

notes..

(1) in multi / div. the result is limited to the same # of sig figs as the

factor with the least number of sig figs

(2) in add / sub, the result is limited to the same precision as the addend

with the lowest precision.

(3) in this conversion, there is both a multiplication AND an addition step.

So we do this in 2 steps.

example #1

5°C = 1.8*5 + 32 = 9 + 32 = 41

in this case, 5 has 1 sig fig. When we multipy it by 1.8 we get a number with 1 sig fig.. "9"

when we add the number "9" which is precise to the 1's column to 32, we get 41 also

precise to the 1's column

example #2

7°C = 1.8*7 + 32 = 12.6 (rounded to 1 sig fig) + 32 = 10 + 32 = 40

in this case, 7 has 1 sig fig. When we multipy it by 1.8 we round to 1 sig fig.. "10"

when we add the number "10" which is precise to the 10's column to 32, we get 40 also

precise to the 10's column

and as you can see, sig figs breaks down.

But let's take a second and think about those numbers. In sig figs, we assume the precision in the rightmost sig fig is ± 1 So our results really mean

5°C = 41°F ± 1°F

7°C = 40°F ± 10°F.

Both of those numbers are accurate, but the precision is drastically different.. right?

IF we were to apply formal propagation of errors, we would calculate these values

(5 ± 1)(1.8 ± 0) = (5 x 1.8 ) ± (5 x 1.8 )*sqrt((1 / 5)^{2}+(0 / 1.8 )^{2}) = 9 ± 1.8

then

(9 ± 1.8 ) + (32 ± 0 ) = (9 + 32) ± sqrt(1.8^{2} + 0^{2}) = 41 ± 1.8

likewise

(7 ± 1)(1.8 ± 0) + (32 ± 0 ) = 44.6 ± 1.3

so our "sig figs" results gives us a fair estimate for 5°C but terrible for 7°C.

Why? because sig figs is an ESTIMATE. Sometimes it works well, other times not so well.

*******

let's try these examples.. 5.1°C and 7.1°C ( hint.. read 5.1 as 5.1 ± 0.1)

via sig figs 5.1°C = 41.2°F ± 0.1°F

via FPE 5.1°C = 41.2°F ± 0.2°F

and

via sig figs 7.1°C = 45°F ± 1°F

via FPE 7.1°C = 44.8°F ± 0.2°F

and we can see sig figs still is accurate and now the precision is a more reasonable.

some numbers for YOU to try on your own.

-5.1°C and -7.2°C

25°C and 27°C

52°C and 57°C

are they all still accurate? does the precision estimate improve if you have more digits (temps in the 20's vs in the single digits).

*********

back to your question. how to handle this

(1) I would suggest you consider using FPE for temp conversion IF precision is important to the story.

(2) OR read your thermometer to 0.1°C and report °F to no more precision than 0.1°

(3) OR at an absolute minimum.. don't report °F to a higher precision than °C

the reverse conversion is similar. You get to think through that.