# Significant figure round off when adding and subtracting

• n124122
In summary, when performing calculations, it is important to keep track of the error range throughout. For most scientific purposes, a statistical approach is taken, using the root-sum-square of independent error fractions. A compromise is to apply the rounding rules at each step, but this can lead to accumulation of errors. It is best to keep one or two more digits than expected in the final answer and be cautious of steps where the difference of two large numbers produces a significantly smaller number.

#### n124122

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So I know that you can't round off numbers in a longer calculation. (in physics)
but how is this with adding and subtracting. For example
52,32*13=680,16 -> 680,16/123=5,529...=5,5(because of the 13)

but what is correct when adding inbetween
530,014+0,01=530,02 -> 530,02*2=1060,04=1,1*1^3 (because of the *2)
or
530,014+0,01=530,024 -> 530,024*2=1060,048=1,1*1^3 (because of the *2)
It doesn't make any difference until now but it does in a longer calculation which is correct??

Basically your answer cannot be more certain than quantities given to you so your calculation should have as many significant digits as the number in the question which has the least no. of significant digits.

Mastermind01 said:
Basically your answer cannot be more certain than quantities given to you so your calculation should have as many significant digits as the number in the question which has the least no. of significant digits.
That doesn't work when there's a mix of additive steps and multiplicative ones.
E.g. (102.5+0.1)/100.0 = 1.026 is justified despite the 0.1. Similarly, you have to be very careful when additive steps result in substantial cancellation: 102-100.25 = 2, not 1.75.

Ideally, one keeps track of the error range throughout. How that is done depends on the risk associated with applying the result. For most scientific purposes, a statistical approach is taken, using the root-sum-square of independent error fractions. If you are designing equipment lives will depend on, it would be better to track the extremes.

A compromise is to apply the rounding rules at each step, as in your 530,014+,01=530,02 example, but that can lead to accumulation of errors. E.g. if we start with 0,1 and add 10 terms like 1,24, 0,33, 0,54, all happening to have a digit from 2 to 4 in the 0,0x position, then the answer would be too low by about 0,3.

Consider also that if 2 technically represents anything from 1,5 to 2,5, almost, then 530*2 can be from 795 to 1325.

haruspex said:
That doesn't work when there's a mix of additive steps and multiplicative ones.
E.g. (102.5+0.1)/100.0 = 1.026 is justified despite the 0.1. Similarly, you have to be very careful when additive steps result in substantial cancellation: 102-100.25 = 2, not 1.75.

Ideally, one keeps track of the error range throughout. How that is done depends on the risk associated with applying the result. For most scientific purposes, a statistical approach is taken, using the root-sum-square of independent error fractions. If you are designing equipment lives will depend on, it would be better to track the extremes.

A compromise is to apply the rounding rules at each step, as in your 530,014+,01=530,02 example, but that can lead to accumulation of errors. E.g. if we start with 0,1 and add 10 terms like 1,24, 0,33, 0,54, all happening to have a digit from 2 to 4 in the 0,0x position, then the answer would be too low by about 0,3.

Consider also that if 2 technically represents anything from 1,5 to 2,5, almost, then 530*2 can be from 795 to 1325.

I didn't know that. Why did that happen [the one in the bold] ?

haruspex said:
That doesn't work when there's a mix of additive steps and multiplicative ones.
E.g. (102.5+0.1)/100.0 = 1.026 is justified despite the 0.1. Similarly, you have to be very careful when additive steps result in substantial cancellation: 102-100.25 = 2, not 1.75.

Ideally, one keeps track of the error range throughout. How that is done depends on the risk associated with applying the result. For most scientific purposes, a statistical approach is taken, using the root-sum-square of independent error fractions. If you are designing equipment lives will depend on, it would be better to track the extremes.

A compromise is to apply the rounding rules at each step, as in your 530,014+,01=530,02 example, but that can lead to accumulation of errors. E.g. if we start with 0,1 and add 10 terms like 1,24, 0,33, 0,54, all happening to have a digit from 2 to 4 in the 0,0x position, then the answer would be too low by about 0,3.

Consider also that if 2 technically represents anything from 1,5 to 2,5, almost, then 530*2 can be from 795 to 1325.

So you shouldn't round anything (in a large calculation) till the endresult? (even 530,014+,01=530,015? when this is just one step in the middle of a long calculation)

n124122 said:
So you shouldn't round anything (in a large calculation) till the endresult? (even 530,014+,01=530,015? when this is just one step in the middle of a long calculation)
I tend to use spreadsheets, so very little rounding happens anyway. The same should apply to a calculator with enough store and retrieve slots. Otherwise, I would suggest keeping one or two more digits than you expect to quote in the final answer. And keep a wary eye on any step where the difference of two large numbers produces a number orders of magnitude smaller.

Mastermind01 said:
I didn't know that. Why did that happen [the one in the bold] ?
If we take the 102 as meaning anything from 101.5 to 102.5, then the answer is anything from 1.25 to 2.25. So even claiming one significant digit is a bit of a stretch.