# Do you round off insignificant digits in the middle of a calculation?

I have a question... Do you round with significant digits during each subcalculation of a problem or only when the entire problem is complete?

Example:

multiply the following number:

$$1.8 \times 2.01 \times 1.542$$

saving rounding until the end:

$$(1.8 \times 2.10) \times (1.542) = (3.78)\times(1.542) = (5.82876) \to 5.8$$

rounding at each sub-calculation:

$$(1.8 \times 2.10) \times (1.542) = (3.8)\times(1.542) = (5.8596) \to 5.9$$

I also have the strong feeling that if you round at each sub-calculation then multiplication is no longer commutative (although after experiencing matrices that no longer seems to be too much of a problem)

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Where is mechanics? –  Mr.ØØ7 May 5 '13 at 2:46
i was looking for a subject like introductory physics or elementary physics where this question would be appropriate but couldn't find one... so i went with the closest i could find (No i'm not bashing mechanics! i'm just saying there may be more beginners studying the beginner flavor of it than not) –  frogeyedpeas May 5 '13 at 2:47
Never round intermediate results. That introduces additional errors that propagate through the calculation. If you perform the error analysis you will see this is the case. For example en.wikipedia.org/wiki/Propagation_of_uncertainty –  Brandon Enright May 5 '13 at 4:00
You are also totally right that intermediate rounding destroys commutativity of multiplication. That can be a real problem in computer programming where you can't prevent intermediate rounding of floating points... –  Lagerbaer May 5 '13 at 4:05
@frogeyedpeas when you can't find a tag to fit your question, it usually means the question is off topic and you shouldn't ask it. Now, that wasn't the case here; it turns out the appropriate tag exists, you just didn't find it. But in the future, don't put unrelated tags on a question. You can ask for advice in Physics Chat if you can't find any appropriate tags for your question. –  David Z May 5 '13 at 4:11

Significant digits is a convention that only affects how you write numbers, not what the numbers actually are. So you only round when you are asked to drop down to a given number of significant digits - that is, at the end.

Think of it like this: there's a difference between a number, which is an abstract idea, and a written representation of a number. Some numbers have exact written representations; all numbers have approximate written representations, which represent another, nearby number. For example, the notation $5.82876$ is an exact representation of a particular number, and $5.8$ is an approximate written representation, to two significant figures, of the same number. $5.8$ is also an approximate written representation (to two significant digits) of many other numbers, such as $5.810394$ and $5.79928129$. This is the idea behind uncertainty, and significant digits: if you are given the written representation $5.8$, you don't know which actual number it represents - it could be anything between $5.75$ and $5.85$. The only exception is if you are told that $5.8$ is an exact representation, which uniquely specifies which number you are supposed to take it to mean.

When you calculate the product $1.8\times 2.01\times 1.542$, you start with three written representations which you are supposed to assume are exact. Then you multiply the first two of them, and get a number which is exactly represented by the notation $3.78$. Now, it's true that $3.8$ is an approximate written representation of that number. But does that fact change what the number is? No. If you do the intermediate rounding, you're effectively deciding to replace one number, the one which is exactly represented by $3.78$, which another number, the one which is exactly represented by $3.8$. And the operation "replace one number with another number" is not part of the mathematical expression you are supposed to simplify. So don't do it.

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It's unnecessary and impossible to avoid any rounding at all. E.g., if you calculate $\sqrt{17}$ on a handheld calculator, you're going to have to round. –  Ben Crowell May 5 '13 at 13:47
Sometimes people will say not to round at all at intermediate steps. This is wrong and in fact usually impossible. A calculator only has a finite number of digits of precision. If you calculate $\sqrt{17}$ at some intermediate step, you have to round it, because it can't be expressed as an exact decimal. It's also ridiculous to write down intermediate results on a piece of paper with 8 or 10 sig figs when you're doing a 2-sig-fig problem. It's a waste of time, because almost all of those digits are illusory precision.