- Thread starter Indranil
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It will help if you state your specific confusion. What possibilities do you see for the answers? What about it are you unsure of?How many significant digits are in 1040. and 1040.0? I am confused

Do 1040. has four significant digits and 1040.0 has five significant digits?It will help if you state your specific confusion. What possibilities do you see for the answers? What about it are you unsure of?

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That's right. You aren't really confused, just perhaps uncertain.Do 1040. has four significant digits and 1040.0 has five significant digits?

When there is a decimal point present, all digits from the first

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And 1040 (without decimal point) has three significant digits.Do 1040. has four significant digits and 1040.0 has five significant digits?

Could you explain why without decimal point, 1040 has three significant digits? and what is the difference between 1040 and 1040.? because I knew 1040 and 1040. are the same.And 1040 (without decimal point) has three significant digits.

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When you write 1040 - it can be 1036 (min) to 1044 (max)Could you explain why without decimal point, 1040 has three significant digits? and what is the difference between 1040 and 1040.? because I knew 1040 and 1040. are the same.

When you write 1040. - it can be 1039.6 to 1040.4

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Of course you are right that 1040 and 1040. represent the sameCould you explain why without decimal point, 1040 has three significant digits? and what is the difference between 1040 and 1040.? because I knew 1040 and 1040. are the same.

I would say that 1040 is somewhat ambiguous, because you can't be sure whether the final zero is there because that digit is actually zero, or only because the digit is needed in order to have the right place values. It might really have either 3 or 4 significant digits - that is, it might have been rounded to the nearest ten or to the nearest unit, and be written the same. Many people assume the least possible accuracy. Ideally, significant digits should be read on in scientific notation, where this never happens (because there is only one digit before the decimal point).

On the other hand, with the decimal point, it is clear that the writer intends to stop after the decimal point, and the zero is significant.

According to the conventions of significant figures, 1040 and 1040. mean different things. That is because the whole idea of significant figures involves applied mathematics. In pure mathematicsWhen you write 1040 - it can be 1036 (min) to 1044 (max)

When you write 1040. - it can be 1039.6 to 1040.4

\(\displaystyle 1040. \equiv 1040\) as you recognize, but the concept of significant figures is not relevant to pure mathematics.

See https://en.m.wikipedia.org/wiki/Significant_figures

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I don't understand. Could you get your point easier, please?When you write 1040 - it can be 1036 (min) to 1044 (max)

When you write 1040. - it can be 1039.6 to 1040.4

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I think what others of us said make the point more clearly, showing the reason.I don't understand. Could you get your point easier, please?

Please read the Wikipedia article that was quoted: https://en.wikipedia.org/wiki/Significant_figures .

Take a more interesting example: 104000. I might write that as an estimate, or rounded number, whether the real number was exactly 104000, or 104001 and rounded to the nearest ten, or 104010 and rounded to the nearest hundred, or 104100 and rounded to the nearest thousand. There is no way to distinguish those situations by the way we write it, unless we use one of the (relatively rare) conventions Wikipedia suggests. As they say, "The significance of trailing zeros in a number not containing a decimal point can be ambiguous. For example, it may not always be clear if a number like 1300 is precise to the nearest unit (and just happens coincidentally to be an exact multiple of a hundred) or if it is only shown to the nearest hundred due to rounding or uncertainty."

One way to reduce the ambiguity when the number is accurate to the nearest unit is to include a decimal point and write it as "104000.". Since this can be done, some people tend to assume that if there is no decimal point, the number of significant digits should be assumed to be as few as possible, in this case three, with all the trailing zeros insignificant.

Back to the original, if a number was rounded to the nearest unit and the result was 1040, then the number could be anything from 1039.5 to just under 1040.5. (What Khan wrote was not quite right.) Any of those numbers, such as 1039.51 or 1040.49, would round to 1040.

If the number had been rounded to the nearest ten, then the original could have been anything from 1035 to just under 1045. (I should add that whether you would round 1035 to 1040 depends on the precise convention you are following for rounding.)

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