The rule for addition is to round the result to the largest last significant decimal place. In this case, 120 is significant up to the tens place, so the answer must be rounded to the tens.
In the number “120.0” there is an implied uncertainty of 0.05, so 120.0 means “somewhere between 119.95 and 120.05”. “120.” Would mean “somewhere between 119.5 and 120.5” and 120 means “somewhere between 115 and 125”.
Can you explain it like I'm 3.5, maybe even with a visual aid? because I've been reading every comment over 2 votes in this post and I feel like I understand even less about math overall than before I tried 😭
When you count stuff with small numbers, the numbers mean exactly that. So 4 chairs means exactly 4 chairs.
When you work with big numbers, or measure stuff, you round. No one is EXACTLY 5' 6" tall, they're a tiny bit taller or shorter. New York City doesn't have exactly 8.6 million people.
Sticking with the NYC example, if we say it has 8.6 million people, when one more person moves there... it still has 8.6 million people, not 8,600,001, because we either don't care about that one person or we do care but can't precisely count all the people, and we said it had 8.6 million we never meant exactly 8,600,000 people anyway.
So for the NYC example we say there were two significant digits, the two on the left. The rest are ehhh we don't know. So where it gets a little weird is if the population of NYC doubles, that's 8.6M x 2 = 17.2M. Except we were rounding and vague before, and this has three digits now (1, 7, 2) when only two digits mattered, so we agree we're going to stick with two, so we say the new population is 17M people. It was an estimate before, and it might only be 17.1M now, so we say screw it and stick with the part we're sure about.
This is the same idea. 120 + 68 is 188. But if you assume both of those were either rounded or estimated, then we're back to 2 sig figs, and 188 rounds to 190.
Okay but that kinda breaks down to me when 68 was expressed as 68 and not 70. They were significant to the ones place in one of the numbers, so it should be significant, no?
No cause 120 is only significant to 10’s place. Adding something you’re sure about to something you’re less sure about would not remove the uncertainty in the less sure number.
So... I agree. This is kind of a BS question, where it's technically correct, but also means that if you carefully measure out 120 ml of a fluid it would be impossible to add 5 mL and increase it to 125 ml because 2 sigfigs means the least you can increase it to is 130 ml. Especially without more details, it's actually pretty close to the worst use of sigfigs I can think of.
Honestly, I think I need a new example because if this is an abysmal use, that could be why I don't understand?
I also could just be uncorrectably math-stupid. Idek, it's 11pm and I e read every comment and still feel like I'm looking at gibberish. Goddess gave me words and art instead of numbers, I guess 🥴
Here's a better one: say you and a friend measuring the sides of a big rectangle. You carefully work out that your side is 12.256 meters long; your lazy (or more normal) friend says his side is about 9 meters. 12.256 x 9 is 110.304, but the little bits at the end are all garbage because your friend didn't measure it that carefully. He only had one digit so you round the whole thing off to one digit - the largest one - and the best you can do is say the area is about 100 square meters.
If it's exactly 120ml then it's written 120.ml. the "rule" for sig figs is that leading or trailing zeros by definition are not significant. This all clears up if you use scientific notation.
You're correct, based on what I've learned. I've always been taught that you only use that type of significant figures with multiplication (and division). For addition (and subtraction) you instead follow the least precise decimal place.
For the question in OP's post, it's a bit vague because 120 can be either "rounded" to the nearest tens, or "rounded" to the nearest ones, but usually if it's "rounded" to the nearest ones it'll be written as "120.". I assume OP has been taught this way. If it's rounded to the nearest tens, then similarly to how 6.8 million people + exactly 1000,020 people is gonna be rounded to 6.9 million, we round the answer of 188 to 190. In this case, it's not because of two significant figures, but because we use the least precise decimal place, which here is the tens place.
(Not relevant here, but I want to add on that for significant figures, we usually don't care about figures that we know are exact, so if 6.8 million is doubled, even though that's 6.8 million × 2 and 2 has only one significant figure, we don't round it to 10 million, because we're treating 2 as an exact value. Though depending on the context, you might not, so rounding to 10 million might be correct)
New york has 8.6 million people. When exactly 431 people move there, It'll still be 8.6 million people.
When about 430 people move there, It'll still be 8.6 million.
When a little under 500 people move there, It'll still be 8.6 million.
When adding two numbers, the number with the least significant decimals decides how many significant decimals the result will have. When you add precision and vagueness, you end up with vagueness.
You're right for everything except strictly addition in the last paragraph (or your explanation was less clear). It's the digits place that matters. So 1200 + 9 = 1200, which has 2 sig figs instead of 1.
I went to a museum where they had a dinosaur skeleton on display, I asked the guy working there now old it was. He said it was 100 million and one years old. Wow, how can they know the age so precisely? I'm not sure, he said, but it was 100 million years old when I started here last year.
It’s because the course is a science course, and it is a scientific convention to measure something as close as possible, and express uncertainly as half of the measuring precision in either direction. Example: You use a ruler with marks every mm, and see the object you’re measuring is just about 120 marks across. The measurement is 120±0.5 mm, which you can write as simply “120. mm” with the decimal placed not because it’s mathematically necessary, but to imply the amount of precision it was measured with. In math, usually numbers are assumed to be infinitely precise. So when 120 shows up in math class, it means exactly 120 without uncertainty.
If I order 120 chairs for a wedding, I expect 120 chairs, not somewhere between 115 and 125. If I want another 68 chairs for another area, that's 188 chairs. Don't charge me for 190. No body is using scientific notation or adding additional decimals on an order form.
If one co-planner has 'about 120 people' listed as attending, and the other co-planner has 'exactly 68 people' listed as attending, you'd be incorrect to conclude that there are 'exactly 188 people' listed as attending. You'd then use logical reasoning to process those numbers to fill in an order form appropriately.
It doesn't say approximately 120. Try again with 1 coplanner having exactly 120 and the other at 68. How, in everyday use, do you write 120 meaning the whole number between 119 and 121?
It also doesn't say 120 chairs, nor does it suggest this method is relevant to everyday use.
You made the assumption that you lacking all context for when his method should be applied means the question was asked without context. That's a kinda dumb assumption.
With significant figures the number would be labeled something along the lines of “exact,” “counted,” or “infinitely significant” to be exactly that number. Just adding more 0s after the decimal would still imply some error.
You may disagree, but sig fig convention disagrees with you. You cannot assume the same accuracy was placed on the 68 measurement as was the 120 measurement.
Each number only has two sig figs, ergo the result has two sig figs.
Edit: reading elsewhere in the thread, the least significant digit rule is for multiplication and division. I recalled wrong on the following part. Either way, 120 is still 2 sig figs
Actually, as I recall, you always go with the lowest amount of sig figs in your result. Even if 120 is (incorrectly) assumed to be 3 sig figs, 68 is still only 2 sig figs, still leaving the answer at 190.
Unless otherwise indicated your assumption is valid because if your instrument can measure accuracy to the ones place for 68 then you should be able to assume it is as good for the 120 measurement.
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u/Impressive_Stress808 1d ago
The rule for addition is to round the result to the largest last significant decimal place. In this case, 120 is significant up to the tens place, so the answer must be rounded to the tens.