Scientific Notation Anomaly Explained

[John Lee]

This question has to do with some annoying anomaly when expressing in scientific notation whole numbers that are multiples of powers of 10. For instance, 2000 is usually written in scientific notation as 2x10^3, implying that the original number has 1 significant digit. On the other hand, 2025 would become 2.025x10^3, with 4 significant digits.
Now, consider the scenario in which these two results came from a real estate appraiser measuring two rectangular houses, with exterior dimensions 50'x40' and 45'x45' respectively, using the same tape measure. How can we explain this (factually unjustified) disparity in presuming the number of significant digits? Could it be the case that we are generally suspect of trailing zeros, perhaps stemming from some subconscious probability estimation? Is there such thing as a "standards" document governing translation from decimal notation to scientific notation, or are we at the whim of the individual mathematician or scientist's authority we happen to run into? Please help clarify this phenomenon from the scientist's perspective.

 

[Danger]

You pretty much just use however many digits are necessary for the required accuracy of the answer.

 

[berkeman]

This question has to do with some annoying anomaly when expressing in scientific notation whole numbers that are multiples of powers of 10. For instance, 2000 is usually written in scientific notation as 2x10^3, implying that the original number has 1 significant digit. On the other hand, 2025 would become 2.025x10^3, with 4 significant digits.
Now, consider the scenario in which these two results came from a real estate appraiser measuring two rectangular houses, with exterior dimensions 50'x40' and 45'x45' respectively, using the same tape measure. How can we explain this (factually unjustified) disparity in presuming the number of significant digits? Could it be the case that we are generally suspect of trailing zeros, perhaps stemming from some subconscious probability estimation? Is there such thing as a "standards" document governing translation from decimal notation to scientific notation, or are we at the whim of the individual mathematician or scientist's authority we happen to run into? Please help clarify this phenomenon from the scientist's perspective.

Welcome to the PF. If they are significant digits with respect to the accuracy of the measurement, you write them out. Like, a 1% 1000 Ohm resistor is written as 1.00kOhm. Do not leave off zeros if they are significant to the accuracy of the measurement.

Note -- Excel may not cooperate when you try to do this, BTW. Thank BillG for that, or at least for propagating that.

 

[John Lee]

Welcome to the PF. If they are significant digits with respect to the accuracy of the measurement, you write them out. Like, a 1% 1000 Ohm resistor is written as 1.00kOhm. Do not leave off zeros if they are significant to the accuracy of the measurement.

Note -- Excel may not cooperate when you try to do this, BTW. Thank BillG for that, or at least for propagating that.

Thank you! May be I didn't present the question clearly enough. The point of the question is: Is there a reasoned method to decide how may significant digits to show when one is asked to convert a number from its decimal representation to scientific notation WITHOUT being given any further information, when the number is a whole number which happens to be a multiple of powers of 10? Or do we declare the request unsatisfiable because of the lack of accompanying information (e.g. context)?

 

[Tac-Tics]

WITHOUT being given any further information,

That kind of data is better left entirely to context. A number should not comment on its own accuracy. If you're doing an experiment, you mention how you measured (which will in turn suggest the precision). If you're using excell, you make an additional field which lists the precision of the first.

 

[jtbell]

Is there a reasoned method to decide how may significant digits to show when one is asked to convert a number from its decimal representation to scientific notation WITHOUT being given any further information, when the number is a whole number which happens to be a multiple of powers of 10?

No. There is no way, without further information, to decide whether 20000 should be written as $2 \times 10^4$ (one significant figure) or $2.0 \times 10^4$ (two significant figures) or $2.00 \times 10^4$ (three significant figures), etc.

Of course, if you got the number by reading from a scale of some kind, then the resolution of the scale should give you an idea of the number of significant figures.