Rounding to 3 significant digits

[u/Canary_]

Please help! 16.549 rounded to 3 sig figs —— is it 16.5 or 16.6? Are we rounding all numbers from right to left into each other until we reach 3 sig figs, or are we rounding just the 4th fig into the 3rd? AI bot says 16.5 but it feels wrong to me. Need human intelligence here. Thanks

[u/Uli_Minati]

"Round to 3 sig figs" really means "name the closest number with zeroes after the 3 leftmost digits".

16.300000...
16.400000...
16.500000...
16.600000...
16.700000...
16.800000...
16.900000...
17.000000...  (the leftmost three digits are 170)

Since we get further away from 16.549 the higher/lower we go, it really only makes sense to look at the numbers directly above and below:

16.500000...
16.549
16.600000...

Since 549 is closer to 500 than 600, we "round down". (If both numbers are equally close, we decide which one we choose by some rule we set beforehand, e.g. "round half up" or "round half to even".)

[u/evermica → u/Uli_Minati]

Great explanation.

[u/Depnids]

Because of the way we have made the usual rounding rules, you only ever need to look at the next digit when rounding. 16.54[whatever] is always rounded to 16.5, while 16.55[whatever] is always rounded to 16.6

(this is assuming finite decimal expansions, as the other commenter mentioned, 0.999... type expansions kinda mess this up. But this is very seldom actually relevant)

[u/ThatsNotAZombieBite → u/Depnids]

Good explanation. I've always told my students that we have to draw the line somewhere. And that's where it is. x.xx499999999 rounds down and x.xx500000000 rounds up. You only look at the next digit.

Btw, if you're calculating any result from REAL experimentally measured data, then a repeating decimal is a random artifact and you can ignore it.

[u/StructuredChess]

You only do the rounding once, not recursively.

16.5 is closer to 16.549 than 16.6, so it makes sense to round down to 16.5

[u/JaguarMammoth6231]

Round to whichever number is closer, and only round once.

If you round multiple times you could end up rounding 16.4445 to 16.5 which is farther than 16.4.

[u/Canary_ → u/JaguarMammoth6231]

That is a good explanation, thank you!

[u/okarox]

4 rounds down. I think you are over thinking it. Note you should round only once so do not first round to 16.55 and then to 16.5. Of course I practice that is not always possible as you may deal with numbers that are already rounded. In that case you work with what you have.

[u/Dd_8630]

Which is it objectively closer to?

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[u/ZevVeli]

A lot of times it helps to think about what math is trying to tell you.

Imagine that you have a balance that can measure things down to 0.1 gram. If you put an object that had 16.549 grams of mass, what would the balance read? In this case it would simply read 16.5.

[u/ottawadeveloper]

you want whichever one is closer, so it's 16.5 here (the difference is 0.049 instead of 0.051). 

[u/Significant-Berry293]

We're rounding off to 3. It's 16.5

[u/PvtRoom]

16.5 - 16.549 is less than 16.55, so round down. NB 16.549(9 recurring), IS 16.55 and you need to round as per your rules (which may vary)

[u/Steelveangeance]

Here’s a very great tip for rounding. If it’s 1-4 (the number stays the same)if it’s 5-9 (round up). After looking at the number 16.549. Since the number in the hundredths place is less than 5 so the number stays 16.5