I have often thought that $3=3.0$ so one can round $3.4\approx 3.0$. But some math teachers says this is wrong. Which one is the correct answer? Why this might be wrong?
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2$\begingroup$ This is actually a math and science question, not a math education question. You might find good answers to it at math.stackexchange.com, But math educators might have something particularly useful to say about why we think about it the ways that we do. $\endgroup$– Sue VanHattum ♦Commented Apr 24 at 20:16
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$\begingroup$ @SueVanHattum I suspect that at math.stackexchange, people would say that $3.4\approx 3$, $3.4\approx 3.0$, $3\approx 3.00$ or whatever number of zeros you want are exactly the same thing (because $3=3.0=3.00=\cdots $) and therefore it is not math question but a question for educators. $\endgroup$– PedroCommented Apr 25 at 0:41
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2$\begingroup$ I doubt that, Pedro. This is a part of mathematics (although more discussed in science), and mathematicians do understand this. $\endgroup$– Sue VanHattum ♦Commented Apr 25 at 1:22
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1$\begingroup$ As a mathematician, considering the usual meanings of $3$, equality, decimal expansion and $\approx$, I can tell you that from the logical point of view (which is what matters in the context of "mathematics") the statements $3.4\approx 3$ and $3.4\approx 3.0$ are equivalent. It is not possible to explain "$3\neq 3.0$" (which is the point of the question) outside of a very specific teaching context. $\endgroup$– PedroCommented Apr 25 at 9:39
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$\begingroup$ While 970 may contain either two significant figures (973) or three significant figures (970.3), 3.0 is widely understood by engineers to contain two significant figures. Based on this convention, 3.0 couldn't have been the result of rounding 3.4 to one significant figure. $\endgroup$– ryangCommented May 18 at 17:33
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4 Answers
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Suppose you measured a line segment with a metric ruler, but the ruler only had marks at whole numbers of centimeters:
With this lack of precision in your measuring instrument, the best you could say is "8cm". You wouldn't say "8.0cm", since you can't read that from the ruler. You can confidently report your answer to the nearest mark on the ruler.
If you later measured it with a ruler having marks at tenths of a centimeter...
...you could now say "8.0cm", since it is closer to the 0-tenths mark than 7.9 or 8.1. You are free to report the answer to the closest mark on your ruler. You would not say "8.00cm", though, because you can't see hundredth marks.
Keep going with a more precise instrument, and you might get this:
Now you could say "8.02cm". Etc.
But none of these is the exact length. Get an even better instrument, and it might turn out to be closer to 8.0193442 cm.
So, is 8 equal to 8.0? Not necessarily (in the strict sense of "equals"), since "8" might really be representing 8.2166254 and "8.0" might be representing 8.0133245.
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3.0 means you know something about the tenths place (which you don't know for 3). It means that your actual number is between 2.95 and 3.05. And 4.17 would mean that your number is between 4.165 and 4.175. You are saying that your last digit is the best choice for that digit.
Now for rounding. 3.4 can be rounded to 3, but not to 3.0, because 3.0 is wrong it its last digit. It you measure something as 2.3478kg, you can report that as is, or as 2.348kg, or as 2.35kg, or as 2.3kg. Why would you round? Perhaps your other measurements in a list were less accurate, and you want to report them all to the same accuracy? Or perhaps you've averaged 7 numbers that all had two places after the decimal. Your calculator gives you something like 10 digits after the decimal. It wouldn't reflect your measurements to give all that. You round it back to two places after the decimal.
answered Apr 24 at 20:14
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- It is true that $3=3.0$.
- It is also true that $3.4$ is approximately equal to $3.0$.
- Mathematically, it doesn't make any difference whether you use the approximation $3.4\approx 3$ or $3.4\approx 3.0$ in calculations.
However, in the rounding method, the final number is expected to have fewer decimal places than the rounded number. Therefore, as $3.4$ has one decimal place, it is expected that it gets rounded to a number with zero decimal places (in this case, $3$ according to the usual rounding rules).
Probably, this is why some teachers will not accept $3.4\approx 3.0$ as a result of rounding. Possibly, in the context in which this answer is considered wrong, they want to know if you know the rounding rules and this answer doesn't reflect explicitly this knowledge.
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Are you aware there's a difference between the sciences "mathematics" and "physics"?
In mathematics: $3 = 3.0$.
In physics: $3 \ne 3.0$.
As explained in some other answers, the "difference" is explained as the fact that $3.0$ means that you are dealing with a measured value, of which you are sure of two digits, not more. (In fact, the value $3.0$ can mean $3.0$, $3.01$, $3.02$, $3.04$, it can even mean $3.0123456789$ :-) ).
In fact:
In mathematics, the sentences $x=3$ and $x=3.0$ both mean that $x$ is the value you get, starting by $0$ and adding $1$ three times.
In physics, however, the sentence $x=3$ means $x \in [\frac{5}{2}, \frac{7}{2}[$,
while the sentence $x=3.0$ means $x \in [\frac{59}{20},\frac{61}{20}[$.
