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I've learned in scientific notation you have last number as probable. Meaning it could be anything...

so in $3$ significant digit number such as follows

$3.34$ has '$4$' which could be anything.

Now if had exact number and I wrote in scientific notation won't that make it inexact as it's implied that in scientific notation last digit is probably something else.

  • Exact number: $2650$
  • In s. Notation: $2.650$ x $10^3$

So now $0$ is probable and unsure number..?

J. M. ain't a mathematician
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Muhammad Umer
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    No, it is not. Don't think of these numbers behind the decimals as decimals numbers. They are not because of the 10-power. So that 0 at the end is still at unit position. With scientific notation we do not lose accuracy unless digits are deliberately ommitted. – imranfat Sep 11 '13 at 01:36
  • in s.n. its implied that last digit is probable. regardless of you get it. – Muhammad Umer Sep 11 '13 at 16:14
  • Generally, 2.65 would indicate that the number was accurate to the nearest hundredth and 2.650 would indicate that the answer was accurate to the nearext thousandth. – Steven Alexis Gregory Sep 27 '15 at 02:54
  • WolframAlpha uses triple dot (ellipsis) at the end of inexact numbers sometimes... – 4esn0k Jun 28 '20 at 13:27

4 Answers4

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Certainly I can write $10!=3,628,800=3.6288\cdot10^6$. This is exact, but if you just see $3.6288\cdot10^6$ you might wonder if it had been rounded. It may be more useful to write it this way, as it is more evident the magnitude of the number. There is nothing special about scientific notation. When you see $\pi \approx 3.1416$ it has been rounded and there is no power of $10$.

Ross Millikan
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  • so you can't write just a exact number in scientific notation without losing that last digit accuracy. – Muhammad Umer Sep 11 '13 at 16:15
  • I did write the exact number in scientific notation. The notation itself doesn't tell you whether it is rounded or not. Neither does regular decimal notation. There is no difference here. If I say the speed of light is 300,000,000 m/sec, that has been rounded to one significant place. – Ross Millikan Sep 11 '13 at 16:18
  • ok lets try different number... 341 this is exactly the number. Now if i write it in s.n. then it becomes 3.41 ⋅ 10^2. Now, digit 1 is implied to be doubtful due to how s.n. works. So i am asking is it not possible to write a exact number without introducing doubt. – Muhammad Umer Sep 12 '13 at 17:39
  • Even not in scientific notation 341 could be rounded from 340.6. If you believe it is exact without scientific notation, it is still exact outside it. If you automatically assume that a number presented in scientific notation is rounded while one presented without scientific notation is not, I can't help. That is what it sounds like you are saying. – Ross Millikan Sep 12 '13 at 18:21
  • i don't know i was told that such is the case...in s.n. last number is always probable and top of that probably was rounded. It's not rounding that i am talking about mainly though, it's the probable part. In which it says that last digit in s.n. is we are not sure of. i am so confused now. Again i want to know that this rule is correct last digit is probable. And can exact numbers be written as are in s.n. – Muhammad Umer Sep 12 '13 at 18:33
  • Yes, exact numbers can be written in scientific notation. I gave an example at the start. It is true that numbers presented in scientific notation are often approximate, but (depending on what is being talked about) so are numbers presented without powers of 10. You need to understand what sort of number you are looking at. In a recent post I calculated a probability as $\frac {323}{3289}$, which was exact, then said it was about $10%$. I could have said about $9.82%$, but neither is exact. No scientific notation here. – Ross Millikan Sep 12 '13 at 18:43
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Muhammad,

What you are missing here is a deeper understanding of significant digits. The general rule of sig. figs. is that the first number of the error (i.e. sigma / standard deviation) is the last significant digit of the measurement. This is were your idea of the last digit being "probable" comes into play.

For example, if you were measuring an unknown sample of 2-pentanone using a calibration curve and got a concentration of 0.71141 with standard deviation 0.0217... The concentration you would report would be 0.71 ± 0.02 Molar using the general rule of sig figs. The last number, "1" from 0.71 is probable because of the error.

Exact numbers are thought of to have an infinite number of sig figs. For example, the number of seconds in a minute (i.e. 60 seconds = 1 minute) or number of inches in a foot (i.e. 12 inches = 1 foot) do not have an error associated with it and so you can report them in scientific notation without losing information.

Miles
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  • Consider you see the number " 2.5x10³" written on a fence, how do you say if the number is exact or not? – 4esn0k Jun 27 '20 at 08:52
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Yes, exact numbers can absolutely be written in scientific notation. It's important to note the difference between scientific notation and the convention of significant figures.

Scientific notation is a standard used to simplify the expression of extremely large and extremely small numbers by expressing the figure as a product with $10^p$. Scientists routinely deal with numbers on both ends of the spectrum, and it's much easier to understand the numbers when they're expressed in this manner.

If we calculate $4 \cdot 10^6$, our product is always $4,000,000$. It doesn't become $3,000,000$ or $5,000,000$ simply because we expressed the value as a product.

$$ 4 \text{ million}= 4,000,000 = 4 \cdot 10^6 = 4.0 \cdot 10^6 = 4.00 \cdot 10^6 = \ ...$$

Since all of the expressions above equal the same thing, you don't have any loss of accuracy, mathematically speaking.

The concept of significant figures is part of a convention that allow any scientist in the world to immediately understand data that was measured by someone else. Significant figures are not mathematical law. This is why it's likely you were not introduced to this concept in a traditional math course, but rather in a science course.

The convention of significant figures in reporting measurements doesn't change the actual value of the number, it just confers information to anyone interpreting the data that there is a limit to which you believe this figure to be a wholly accurate representation of the quantity of unit being measured.

In the international scientific community, exact numbers are considered to have an unlimited number of significant figures. Exact numbers can come from only three sources:

  • Exact counting of discrete objects
  • Defined quantities
  • Numbers that are part of an equation

Convention for reporting scientific data dictates that the last digit of a reported measurement is the only estimated digit. But this doesn't only apply to values expressed in scientific notation. A reported measurement of $3.54 g$ of sodium chloride indicates the reporting scientist is absolutely certain of the digits '$3$' and '$5$', but the digit in the hundredths place is estimated. This doesn't change if the figure is expressed in scientific notation, where it would be represented as $3.54 \cdot 10^0g$.

This convention ensures that anyone analyzing your data — or you, analyzing someone else's data — will be aware of the degree of accuracy to which their measurements were taken. This is the meaning of "significant figures" in a measurement.

The example you provided is not enough to determine accuracy because we do not know what you are quantifying with your numbers or how you got your measurement. What you're addressing are two separate things. Any number can be expressed in scientific notation.

The idea is that human means of measuring things vary by method of measurement and the unit being measured. We can easily measure the number of pennies sitting in a bowl simply by counting. If I see 10 pennies, I can definitively say there are 10 pennies in the bowl with 100 percent accuracy, which is why the measurement would have infinite significant figures. But I can only estimate the number of atoms in the pennies, because I am limited by the accuracy of the instruments I am using to acquire my measurement.

See these resources for more information: http://www.chem.sc.edu/faculty/morgan/resources/sigfigs/ http://www.chem.tamu.edu/class/fyp/mathrev/mr-sigfg.html http://en.wikipedia.org/wiki/Scientific_notation#Significant_figures

MrCMedlin
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I would regard 3.34 as a three-significant-digit approximation to any number between (exact) 3.335 and 3.345, but not to, for example, 3.37. So I think "it could be anything" is misleading.

Andreas Blass
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