# How many significant digits are in 1040. and 1040.0?

#### Indranil

##### Junior Member
How many significant digits are in 1040. and 1040.0? I am confused

#### Dr.Peterson

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

#### Indranil

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

#### Dr.Peterson

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

When there is a decimal point present, all digits from the first non-zero digit on the left to the last written digit on the right are significant.

#### Subhotosh Khan

##### Super Moderator
Staff member
Do 1040. has four significant digits and 1040.0 has five significant digits?
And 1040 (without decimal point) has three significant digits.

#### Indranil

##### Junior Member
And 1040 (without decimal point) has three 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.

#### Subhotosh Khan

##### Super Moderator
Staff member
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 1036 (min) to 1044 (max)

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

#### Dr.Peterson

##### Elite Member
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.
Of course you are right that 1040 and 1040. represent the same number; it is only the convention about implied accuracy that is at issue.

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.

#### JeffM

##### Elite Member
When you write 1040 - it can be 1036 (min) to 1044 (max)

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

$$\displaystyle 1040. \equiv 1040$$ 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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#### Indranil

##### Junior Member
When you write 1040 - it can be 1036 (min) to 1044 (max)

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

#### Dr.Peterson

##### Elite Member
I think what others of us said make the point more clearly, showing the reason.

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.)

#### HallsofIvy

##### Elite Member
As has been said, the distinction between 1040 and 1040. is a convention. It has been agreed that if there is a decimal point after an integer then all digits in the number are "significant" and that if there is no decimal point then the significant digits are end with the rightmost non-zero digit.

#### Denis

##### Senior Member
...and if 1040. was at end of sentence, then you'd have 1040.. :cool: