What is precision of a number representation?

[Ethan Furman]

On 07/11/2016 02:51 PM, Chris Angelico wrote:
> On Tue, Jul 12, 2016 at 6:56 AM, Ben Finney wrote:

>> Precision is not a property of the number. It is a property of the
>> *representation* of that number.
>>
>> The representation “1×10²” has a precision of one digit.
>> The representation “100” has a precision of three digits.
>> The representation “00100” has a precision of five digits.
>> The representation “100.00” also has a precision of five digits.
>>
>> Those can all represent the same number; or maybe some of them represent
>> “one hundred” and others represent “one hundred and a millionth”.
>>
>
> Yep. Precision is also a property of a measurement, the same way that
> a unit is. If I pace out the length of the main corridor in my house,
> I might come up with a result of thirty meters. The number is "30";
> the unit is "meters", the precision is two significant digits, and the
> accuracy depends on how good I am at pacing distance.
>
> This is why it's important to be able to record precisions of
> arbitrary numbers. If I then measure the width of this corridor with a
> laser, I could get an extremely precise answer - say, 2,147
> millimeters, with a precision of four significant digits, and
> excellent accuracy. But if I multiply those numbers together to
> establish the floor area of the corridor, the result does NOT have
> four significant figures. It would be 64 square meters (not 64.41),
> and the accuracy would be pretty low (effectively, the *in*accuracies
> of both measurements get combined). But on the other hand, if you want
> to know whether your new fridge will fit, you could measure it with
> the same laser and come up with a figure of 1,973 mm (four sig fig),
> which would mean your clearance is 174mm (four sig fig). How do you
> record this? Is it 174.0? 0174? "174 with four significant figures"?

174.0, because those last tenths of a millimeter could be very 
important, while knowledge that there are no thousands of millimeters is 
already present.

So, so far there is no explanation of why leading zeroes make a number 
more precise.

--
~Ethan~