# Significant Figures in Long Sequences of Calculation

• Lightfuzz
In summary, when dealing with significant figures in long sequences of calculations, intermediate answers should not be rounded to avoid accumulated errors. Instead, take a few more digits in the calculation process than required in the final answer and then round off at the end. The number of significant figures in the final answer should be based on the number with the least significant figures in the original data, regardless of whether it is a multiplication/division or addition/subtraction calculation. In the case of addition/subtraction, the number of decimal places should also be considered.
Lightfuzz
After losing marks in an exam due to significant figures, I have decided to clear up all my doubts about this concept. But since my teacher hasn't been very helpful, I've decided to post my question here.

I understand the rules for significant figures in both single-step multiplication/division and addition/subtraction calculations, but I am uncertain about what to do with them in long sequences of calculations, especially ones involving both types of arithmetic calculations. Based on my brief research on the Internet, I have found that intermediate answers should not be rounded as this would lead to accumulated error. Instead one should take a few more digits in the calculation process than required in the final answer and then round off at the end. But what I don't understand is, how do we decide how many significant figures to round off to at the end? Do I use the rule for multiplication/division (as many significant digits as the data with the least significant digits) or the rule for addition/subtraction (as many decimal places as the data with the least decimal places)?

Any help will be appreciated. Thank you.

Hi Lightfuzz!
Lightfuzz said:
… But what I don't understand is, how do we decide how many significant figures to round off to at the end? Do I use the rule for multiplication/division (as many significant digits as the data with the least significant digits) or the rule for addition/subtraction (as many decimal places as the data with the least decimal places)?

Not really following you

what is wrong with using the multiplication/division rule for multiplication/division and the addition/subtraction rule for addition/subtraction?

(If it's eg (A+B)C, then use the addition rule for A+B to find the number of decimal places, convert that into significant figures, then use that result in the multiplication rule for the final result)

Thanks for the response.

So would the following example be correct?

(1.5 + 2.03) x 1.1
= 3.53 x 1.1
= 3.883

But according to the addition rule, the addition intermediate step yields a result with 1 decimal place (3.5), which contains 2 significant figures. Therefore when this intermediate result is multiplied to a number with 2 significant figures, the answer should contain 2 significant figures (3.9).

Last edited:
Hi Lightfuzz!
Lightfuzz said:
So would the following example be correct?

(1.5 + 2.03) x 1.1
= 3.53 x 1.1
= 3.883

But according to the addition rule, the addition intermediate step yields a result with 1 decimal place (3.5), which contains 2 significant figures. Therefore when this intermediate result is multiplied to a number with 2 significant figures, the answer should contain 2 significant figures (3.9).

Yes, that's exactly correct

Though it would be 2 significant figures anyway, since that is the smallest number of sig figs in the original

a better example would be
(11.5 - 2.72) x 2.22
= 8.78 x 2.22
= 19.4916

which should be written as 19 (to 2 sig figs), even though all the original terms had 3 sig figs

(and which, if you round off too early, gives you 8.8 x 2.22 = 19.536, which would be 20 )

You don't really need separate "rules" for addition or multiplication. The basic rule is "the result of a calculation has the same number of significant figures as the number, in the original data, with the least number of significant figures.

Here, both 1.1 and 1.5 have two significant figures while 2.03 has three. The smaller of those numbers is "2" so the result of ANY calculation with those numbers would have two significant figures.

HallsofIvy said:
The basic rule is "the result of a calculation has the same number of significant figures as the number, in the original data, with the least number of significant figures.

no, i think for addition or subtraction it has to be based on decimal places

eg in 3.16 - 3.15 + 0.111, all three terms have 3 significant figures,

but the correct answer is 0.12 (2 decimal places, and only 2 sig figs)

## What are significant figures?

Significant figures are the digits in a number that contribute to its precision. They indicate the level of accuracy in a measurement or calculation, and are determined by the precision of the measuring tool or the number of known digits in the original data.

## How do you determine the number of significant figures in a long sequence of calculation?

In a long sequence of calculation, the number of significant figures is determined by the number with the least amount of significant figures. This is because in a calculation, the result cannot be more precise than the least precise number used in the calculation.

## What is the purpose of using significant figures in calculations?

The use of significant figures helps to maintain and communicate the level of precision in a calculation. It also helps to avoid giving a false sense of accuracy in the result.

## What are the rules for rounding off numbers when dealing with significant figures?

When rounding off numbers, the general rule is to round to the same number of significant figures as the least precise number in the calculation. If the digit to be dropped is less than 5, the preceding digit stays the same. If the digit to be dropped is 5 or greater, the preceding digit will be increased by one.

## What are some common mistakes to avoid when dealing with significant figures?

Some common mistakes to avoid when dealing with significant figures include rounding off too early in a calculation, using more significant figures than the original data, and forgetting to include zeros that are placeholders in a number. It is important to pay attention to the rules of significant figures and apply them correctly to ensure accuracy in calculations.

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