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I saw another question which says the speed of light is "$3\times 10^8 \:\rm m/s$", and I know that the speed of light is $299,792,458\ \rm m/s$.

My chemistry teacher taught me that $3.0$ means $3.0 \pm 0.05$, so it is a range from $2.95$ to $3.05$. My chemistry teacher also taught me that $3$ means "exactly $3$ without any range".

How does this work with a number like $3\times 10^8$?

Does the lack of a decimal point mean that the number is exactly $300,000,000$ with no range?

Is it accurate to say that the speed of light is $3\times 10^8\:\rm m/s$?

My question is about precision, not about the speed of light.

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    $\begingroup$ This seems like a legitimate question about how to specify precision, even though the discussion about speed of light is (as noted in the question) beside the point since speed of light is exact. Not sure the downvotes are appropriate. $\endgroup$ – Brick Sep 10 '18 at 14:14
  • $\begingroup$ To one significant figure, yes. I disagree with your chemistry teacher re: 3 without a radix. That may be how chemists interpret numbers, but different contexts infer different interpretations. Sometimes 3 might be exact, sometimes not. Always consider context. $\endgroup$ – Bill N Sep 10 '18 at 21:15
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Is it accurate to say that the speed of light is $3\times 10^8\ \rm m\, s^{-1}$?

For most purposes, it's fine.

To within 7 parts in 10,000 the value of $3\times 10^8$ is fine. Unless your calculations require more accuracy then it's a perfectly reasonable statement to say the speed of light is $3\times 10^8\ \rm m\,s^{-1}$.

You could do most physics with this value with negligible error.

How does this work with a number like $3\times 10^8$?

This is exactly 3 with eight zeros after the decimal place.

There is no implied error range.

I know that the speed of light is $299\,792\,458\ \rm m\,s^{-1}$.

Normally a physical value like this would have an experimental error range. However the speed of light is defined as having this value in modern standards, and we derive other values from this definition. Before that definition was made the most accurate experimental value was (from Wikipedia) $299\,792\,456.2\pm1.1\ \rm m\,s^{-1}$. At that time the meter was defined in a different way.

From Wikipedia :

It is exact because by international agreement a metre is defined to be the length of the path traveled by light in vacuum during a time interval of 1/299792458 second

The speed of photons versus the speed of light.

There is a slight complication because although we know light is made up of particles called photons, which we think are massless and should therefore travel at the speed of light, we cannot measure the speed of photons exactly.

So the speed of light is defined by international standard as $299\,792\,458\ \rm m\,s^{-1}$ but the most accurate measurements of the speed of photons has a finite (but very, very small) error range - you cannot avoid some margin of error in experiment. This is connected with experiments to determine if the photon has a mass (which would be a surprise to physics but we still check). A photon with a non-zero mass could not travel at the speed of light (as defined).

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$3 \times 10^8$ as a number means the number 300 000 000.

However, when used in the context of a measurement, where that one come to expect inexactness, there may be an implication of the amount of precision, e.g. if I say "about 260,000" generally the latter zeros are taken not to be actually zeros for the true, exact quantity in question. It's a rounded approximation to make it simpler and also because we may not know really what the exact amount is (e.g. there may be dispute in figuring it or information we don't have, or our measurement devices aren't precise enough). The trouble with this is, of course, that what if we measured 260,000 but the measurement as actually good enough to say that first zero was in fact accurate - i.e. we really narrowed the true value to between 260,000 and 260,999 and not 260,000 and 269,999. This is an ambiguity, and where that is important, e.g. in scientific work, that is one of the reasons that notation and conventions have been invented to specify imprecision more precisely and when a measured quantity is specified like this, one way to do this is to use the scientific notation to specify the precision by specifying the number of digits in the significand, i.e.

$$2.6 \times 10^5$$

is considered to imply the first two digits are accurate, while

$$2.60 \times 10^5$$

to imply the third digit, a zero, is also accurate. Both expressions are mathematically equal to the approximating number 260,000, but this notation allows us to specify in a crude way the accuracy of that approximation. When you have a $\pm x$ on the end, that provides a more accurate way of specifying errors, by indicating the actual range around the given value the truth is at least likely to be, given the imprecisions in your measurements.

Another thing that is sometimes done is if one is writing out the full number, i.e. 260 000, to put a vinculum (bar) over the last accurate digit, i.e.

$$26\bar{0}\ 000\ \mathrm{widgets}$$

means that it is in fact accurate to that first zero, but not further.

The accuracy of $3 \times 10^8\ \mathrm{m/s}$ as an approximation for the speed of light is, as a mathematical number, on the order of a couple parts in a thousand. However, by these conventions, since we did not write $3.00 \times 10^8$, one would typically be saying that we are being more conservative and not assuming it to be that accurate (even though it is).

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First, in everyday life it is accurate to say that the speed of light, c= 3×10^8m/s. This is only true in vacuum, when measured locally.

Now nowadays, the speed of light, c is defined as a universal physical constant, it's exact value is 299,792,458 metres per second.

This is exact, because the meter is defined by light, traveling in 1/299,792,458 seconds.

Now this needs too, that the second be defined, and nowadays, the second is defined as 9 192 631 770 cycles of a Caesium atomic clock.

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    $\begingroup$ My question is about units and precision $\endgroup$ – user202776 Sep 9 '18 at 23:51
  • $\begingroup$ @Gimmethe411 I think I explained the units in which the speed of light is defined. Now about precision, I mention too, that it is an exact value. There is no precision question, it is exact. $\endgroup$ – Árpád Szendrei Sep 10 '18 at 0:18
  • $\begingroup$ Do you mean 3×10^8m/s is exactly 299,792,458m/s? $\endgroup$ – user202776 Sep 10 '18 at 0:21
  • $\begingroup$ I mean that the speed of light, cis exactly that amount. It is an exact value. What you are mentioning, 3×10^8m/s is just for everyday purposes, and does not have to be precise and it does not have to be at a certain precision. $\endgroup$ – Árpád Szendrei Sep 10 '18 at 0:24
  • $\begingroup$ So you mean it's not 3×10^8m/s but it doesn't matter? $\endgroup$ – user202776 Sep 10 '18 at 0:32

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