# Does “exact” determine the precision or the accuracy of something?

When I say "the speed of light is exactly 300000000m/s", is that an incorrect or a correct statement?

That is, does "exactly" refer to the precision of an estimate (then my statement would be correct, I think, because I wouldn't have asserted that my statement doesn't deviate from the correct speed) or does it refer to the accuracy? (then my statement is incorrect, because the speed of light is slightly less than that).

## 10 Answers

Exactly in this case refers to the precision of the (presumed) measurement. It does not inherently imply any truth or falsehood about the statement.

If I were to tell you

I have exactly five apples.

it would mean that I want you to believe that I have five apples, no more and no less. But I could make that statement even if I had four or six or 150 apples. You would either have to take my word for the statement's accuracy or submit it to independent verification.

• I would say that then if you knew the answer. The sentence "I have exactly five apples" would then be an inaccurate one. – Ólafur Waage Jun 22 '11 at 15:22
• @Ólafur: That's what Robusto means. The sentence is precise but inaccurate (if he has 6). "I have five or six" apples is imprecise but accurate (if I have 6). The word "exactly" only serves to emphasis the precision. – RedGrittyBrick Jun 22 '11 at 15:42

Raw pedanticism.

It could clearly be either, depending on context.

It seems likely that, in a technical discussion, the term "exactly" implies that the measurement is accurate to the given amount of precision.

I would say it means both - it has to be a statement which has no error.

So, for example, "The speed of light is exactly 299,792,458 m/s" is true, but only because the meter is defined as the distance light travels in 1/299,792,458 s. When the meter was still defined otherwise, and this was only a measured value, "The speed of light is exactly X" would be a mistake no matter how precise X is.

Someone else in a comment said "All measurements are approximations to physical reality." I would say that therefore no measurements are exact. But the speed of light is not (anymore) a measurement in the sense that the speed of your car is.

• This is the right answer. In the sentence The speed of light is exactly 299,792,458 m/s, which is true, the word exactly indicates that the relationship is not a measurement, but a definition. It's true in the same sense that "A foot is exactly 12 inches" is true. If the speed of light were measured, the word exact would be inappropriate, and the question of accuracy vs. precision would not come into play. – coneslayer Nov 1 '11 at 23:25

Suppose we measure the length of a string to the nearest 0.1 cm and find it to be 14.3 cm. This could mean that the actual length is somewhere between 14.25 cm and 14.35 cm. In this case, we can say that the string is exactly 14.3 cm to mean that it is 14.3 cm and not 14.2 cm or 14.4 cm according to our chosen way of measuring.

The speed of light is exactly 299792458 m/s by definition, it is not exactly 300000000 m/s by definition, but it is exactly 300000000 m/s when measuring to the nearest 100000000 m/s say. It can be seen then that the meaning of exact depends on the context.

• if you're measuring to the nearest anything, you couln't say it's exact. you would have to denote it as "~14.3cm" because you don't know the exact measurment. – OghmaOsiris Jun 22 '11 at 15:46
• @OghmaOsiris, Robusto and RedGrittyBrick seem to disagree with you. – Johannes Schaub - litb Jun 22 '11 at 15:56
• In a scientific setting, you can't say something is exact if you're rounding off the measurments. I can't say I'm exactly 6' just because I rounded my height to the nearest foot. I would be approximately 6'. So technically, you can never be exact in the strictest sense to any sort of measurment unless its a predefined term like a foot is exactly 12 inches, etc. – OghmaOsiris Jun 22 '11 at 15:59
• @Jasper You still couldn't say something is exact to the nearest unit. You'd have to say it was accurate to the nearest unit. It would be equivilant to saying something is 'perfect except for 1 detail.' Since that '1 detail' exists, that something can't be 'perfect'. Same logic here. – OghmaOsiris Jun 22 '11 at 16:12
• 2+2=5 for sufficiently large values of 2 – mplungjan Jun 22 '11 at 16:19

IMO, the adverb "exactly" when referring to a measurement implies a high degree of both accuracy and precision. When you use "exactly", you infer that your number is accurate to the number of significant figures that are appropriate for some larger calculation. This in itself may not be "exact", but it's the best that can be done in the context of anything else you are measuring or talking about.

• Actually I think it would mean that precision and accuracy to a range aren't necessary for measurement. If you had five apples, you might have x lbs of apples +/- 2 oz, but you have exactly five apples. – Lee Louviere Jun 22 '11 at 15:39
• Yes, but stating that a mile is "exactly" 1.609344km means that the number is accurate for any calculation requiring 6 or fewer sig figs (which is plenty for most physics and engineering), while the corresponding term "about" as in a mile is "about" 1.6km implies a measurement that is good for rough work, but is not an "exact" conversion. – KeithS Jun 22 '11 at 15:46
• Yeah, at which point I'd consider the use of "exact" inappropriate. It's a misuse. You should just use the equivalency word "is". Is already implies an unspecified equivalency. Then you specify the accuracy. – Lee Louviere Jun 22 '11 at 16:00
• Well, in fact, a mile is exactly 1.609344 km because it is exactly 5280*12 = 63,360 inches, and an inch is exactly 25.4 mm. by definition. To contrast, a US survey mile is, by definition, 6336/3937 = ~1.609347 km (given that a survey inch is defined as exactly 1/39.37 m) – Random832 Jun 22 '11 at 18:12

Exact is regarding the full amount, no more, no less. I have five apples. (If I had six, this would also be true) I have exactly five apples. (If I had six, I would be lying).

• So if i measure the speed of light twice, and get 3Mm/s twice, then I think my measurement is exact? I'm not totally sure about the meaning of "precision" in a statement like "the car is blue" and in a statement like in my question. So maybe I should restate my question in terms of a repeated measurement? – Johannes Schaub - litb Jun 22 '11 at 15:31
• You don't repeatedly measure people and find out there's 5.2 people in the room. There are exactly 5 people in the room. According to use and definition, I don't believe exactly should be used for scientific measurement. – Lee Louviere Jun 22 '11 at 15:37
• I think people are missing the point. Exactly isn't a technical term, and shouldn't be used for technical measurements. It is a term that when measuring something you determine it to be exactly X. There are exactly 5 people, I have exactly 5 apples, there are exactly 50 states in America. It's not correct to say that the speed of the car was exactly 50 MPH.... Speed can't be exactly measured. – Lee Louviere Jun 22 '11 at 15:41

Exactly means

• without discrepancy
• in exact terms, without vagueness

My cousin met her girlfriend in 1994, and they got married exactly two years later.

• exactly two years later? To the Planck time (10^−43s)? ;) – Alain Jun 22 '11 at 16:59

Something can be stated precisely, but quite wrongly. To be accurate, a statement must be correct with reasonable precision. If it is exact, it is accurate to an ideal degree.

Exact would mean it has infinite significant figures. From http://www.splung.com/content/sid/1/page/sigfig:

Exact numbers have an infinite number of significant digits. For example, if there are two oranges on a table, then the number of oranges is 2.000... . Defined numbers are also like this. For example, the number of centimeters per inch (2.54) has an infinite number of significant digits, as does the speed of light (299792458 m/s)

You can essentially think of "exact" numbers as having infinite amount of zeros after the decimal point. It is not an approximating to a certain amount of significant figures, that IS the actual number. An example would be to say there are 2.5 cm in an inch (2 significant figures), but exactly 2.54 cm in an inch (infinite significant figures).

In my uses of the words, 'accuracy' and 'precision' are used with sets of things rather than just one thing.

If used with only one object, accurate and precice could be interchangeable because there is nothing to relate them to other than the fact at hand.

Saying something is exact would say that it is equal to what is true and known. So it would be both precise and accurate (assuming the claim itself is true).

So saying "c = exactly 300000000 m/s" is not true. So it wouldn't be precise, nor accurate to the truth.

• Sorry, I could also say that the speed of light is 123456789.123456 m/s which is extremely precise, and also extremely inaccurate. – Fraser Orr Nov 1 '11 at 23:02