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Is there any difference between saying that say $\pi \approx 3.14$ and $\pi = 3.14(2dp)$?

A Chinese student has asked me this. They claim that the latter version doesn't exist in Chinese. They always use the $\approx$ notation.

Can anyone explain whether these two notations have any difference and if so what is it?

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    $\begingroup$ I've never seen the "2dp" notation either and I have no idea what notation exists in Chinese. I can guess what it means because I'm a native English speaker, but that suggests that the use of this notation is at least limited to English speaking places. $\endgroup$ – Dan Fox Oct 14 '17 at 19:37
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    $\begingroup$ Why not $\left|\pi-3.14\right|<10^{-2}$. $\endgroup$ – user5402 Oct 14 '17 at 19:43
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    $\begingroup$ @DanFox Well for high school maths in Ireland we express the number of decimal places as say $\pi = 3.142(3dp)$ as above and the number of significant figures as say $\pi = 3.142(4sf)$. How would you express these? $\endgroup$ – Kantura Oct 14 '17 at 19:53
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    $\begingroup$ @Derek: I'm not judging the merits of the notation (it seems quite reasonable), simply saying that as someone educated in the US I've never seen the notation (nor any standard notation specific for this purpose), and that the notation is surely not used any place that does not speak English. $\endgroup$ – Dan Fox Oct 14 '17 at 20:05
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    $\begingroup$ When a notation is not known to your reader, don't use it. Instead use words: $\pi$ to $2$ decimal places is $3.14$. And if you are in Ireland and schools in Ireland use the "2dp" notation, go ahead with it. A pedant may say that the "dp" should be in roman not italic, since it is not a variable: $\pi = 3.14\;(2\mathrm{dp})$. $\endgroup$ – Gerald Edgar Oct 15 '17 at 11:31
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As a U.S. person, I have never seen this second notation. My first instinct was that it was 3.14 multiplied by some error factor(s).

In my experience, there tends to be a difference in notation between pure math courses and other scientific or statistical fields. In math we would tend to write $\pi \approx 3.14$. In a statistics or programming text I tend to see $\pi = 3.14$, where the significance is implied by the number of decimal digits, or in narrative text, or in the context of a certain section.

For example, I think it's legitimate to write "Rounded to the hundredths place, $\pi = 3.14$", or, "The values in this table are accurate to the 4th decimal place" or somesuch.

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    $\begingroup$ @Michael I can't see why you would call it complicated. How would you express the same idea? I haven't seen a less complicated method here. $\endgroup$ – Kantura Oct 15 '17 at 22:23
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    $\begingroup$ @Derek: In the U.S. we would normally just write $3.14$. Is there any point in the parenthetical? Would you ever write $3.14159(2\mathrm{dp})$? If the only added value of the notation is to indicate the value is truncated, then $\approx$ does the job just as well. So does a convention that any numeral with a decimal point should be treated as a truncation rather than an exact value. $\endgroup$ – user797 Oct 15 '17 at 23:31
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    $\begingroup$ @Michael Yes there is point in the parenthetical. It state concisely that this number 1: is not complete and 2: has been rounded to the stated number of d.p. $\pi$ absolutely does not equal 3.14 so that statement is not complete and misleading. $\pi$ is also approximately equal to say 3.123456 but has not been rounded correctly. If the time it took for an event was 3.12345 s, that to me says that $t=3.12345 \pm 0.000005$ or more concisely $t=3.12345(5dp)$. If you stated that $t \approx 3.12345$ it is not clear to what degree of accuracy this number has. That's my understanding anyway. $\endgroup$ – Kantura Oct 16 '17 at 11:19
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The difference in my mind is that the former can be used more widely yet is less precise.

For instance one might also write $\pi \approx 22/7$ to indicate that $\pi$ and $22/7$ are approximately equal, where there is no precise meaning to "approximately."

Or, I would also consider it as correct to say that $\pi \approx 3.15$ as the quantities are approximately equal.

Others might not as possibly they only use the notation (in this context) if all the digits written out are 'correct.' But I would not be aware of this being a universal convention.

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My understand is as follows. When using the $\approx$ notation one can state $\pi \approx 3.14$, $\pi \approx 3.142$ or even $\pi \approx 3.0$ the latter approximation is useful for doing calculations in ones head. The $\approx$ notation does not need to be rounded correctly. For example $\pi \approx \frac{22}{7} = 3.142857(6dp)$ but $\pi \ne 3.142857(6dp)$ because $\pi = 3.141593(6dp)$. The $dp$ notation must be rounded correctly. Finally, sure, one can use words to express $dp$ but to write $3.14(2dp)$ as 3.14 rounded to two decimal places seems to me to be similar to writing $1+1=2$ as one plus one equals two.

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Many answers are opining on the 2dp notation, which is not the subject of the question.

FWIW, no the two are not equivalent. Approximately 3.14 does not mean +/- .005. It could be a higher (or slightly lower error). Consider science papers listing an approximate value.

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    $\begingroup$ Can you clarify this? When I see a number, say, 2.14, I don't know if it was rounded up from 2.135 or down from 2.144. Of course, it might be truncated from 2.140000. $\endgroup$ – JTP - Apologise to Monica Apr 1 '19 at 0:18
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    $\begingroup$ That is an unsophisticated view of error analysis (sig figs only). If you do experiments to estimate a value, you will get some mean and standard deviation. There is no requirement for the SD to be exactly half of a next decimal. $\endgroup$ – guest Apr 1 '19 at 3:03
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    $\begingroup$ That may be. The question is for high school student education. Your answer may be far more sophisticated than my own thinking on this as well. $\endgroup$ – JTP - Apologise to Monica Apr 1 '19 at 9:37

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