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This may be too trivial to ask. But when I searched, I didn't find anyone asking this question previously on this site.

When I consider 3.0 and 3.00, from a mathematical perspective, absolutely I can see both of them being equal. But some long time ago, I heard that those two are actually not equal in a scientific perspective, because they differ in number of significant digits after decimal point. Now, I wanted to know is this claim correct? Or just a hoax?

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  • $\begingroup$ they are all equal because $0$'s after the decimal point don't matter and that's because $0000\dots0 = 0$ $\endgroup$
    – p_square
    Jul 22, 2021 at 5:13
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    $\begingroup$ In a scientific or engineering context, the number of decimal places may indicate the accuracy of the measurement, so that $3.0$ means $3\pm.1$ and $3.00$ means $3\pm.01$ (or something of the sort) so they're not really the same. Similarly, if I say that something is one and a half meters long, it conveys something different than if I say it's $150$ centimeters long. Most people would take the second measurement to be more accurate, I think. $\endgroup$
    – saulspatz
    Jul 22, 2021 at 5:13
  • $\begingroup$ They are equal, because $$3.000...=3+\frac{0}{1000...}=3+0=3$$ $\endgroup$
    – lone student
    Jul 22, 2021 at 5:32
  • $\begingroup$ Draw a right triangle with $1$ on both sides. The length of the hypotenuse will vary depending on the precision of your gauge. $$1.4, 1.41, 1.414, 1.4142, ...$$ Mathematically the length is clear: $\sqrt 2.$ But in real life you'll never know.Maybe the length of what you are measuring is $3.000001$, but because the measuring instrument does not have enough precision, you will find the required length $3.000=3$. $\endgroup$
    – lone student
    Jul 22, 2021 at 7:25
  • $\begingroup$ @lonestudent quick question. Is $\sqrt 2$ equal to any number of the form 1.41... mathematically? All 1.41... to any precision is still an approximation of $\sqrt 2$. Am I right? $\endgroup$
    – Sourav Kannantha B
    Jul 22, 2021 at 7:54

2 Answers 2

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Mathematically, they are the same.

However, in real life, with concrete measurements, you never know exactly which number you actually have. You're uncertain. And it is considered good practice to keep track of how uncertain.

If I'm measuring the width of a nail with a ruler, and I get 3mm, then I have no idea whether it's actually 3.15 or 2.8 or something in-between. It's just impossible to tell. In that case, i would report it as 3mm, and not 3.00mm because that would be dishonest.

If I'm measuring the length of a piece of wood with a measuring tape, and as far as I can see it matches perfectly with the 3m mark, then I can tell that it isn't 3.01m and it isn't 2.99m. in that case I can confidently report it as 3.00m (maybe even 3.000m, but that would be pushing it). If I just said 3m, then someone listening might not appreciate that I have sub-centimeter precision, as opposed to, say, just looking at the plank and guessing the length from experience.

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Mathematically they are equal, but in real world situations like computer science, the representation can matter due to storage mechanisms. From the perspective of a computer, 3.0 is not the same thing at all as 3, as they have different storage algorithms.

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