Why Significant Digits Matter in Multiplication & Division

[Diane_]

A thread here asking about significant digits got me to thinking. I understand why they work the way they do with addition and subtraction, but multiplication and division are so easy that I've never bothered to think about it.

So - can someone explain why a three s.d. number times a two s.d. number gives a two s.d. number?

 

[HallsofIvy]

Garbage In-Garbage out! A chain is no stronger than its weakest link. The result of a calculation is no more accurate than its least accurate input!

Enough with the platitudes!

Suppose your three s.d number is 1.43 x 10³ (in other words 1430 but writte so it is clear that we intend only 3 significant figures) and your 2 s.d. number is 4.8 x 10^-2 (i.e. 0.048). The first number has been measured to the nearest "ten" and the second to the nearest "thousandth". What that means is that the first number could be as high as 1.435 x 10³ or as low as 1.425 x 10³. Similarly, the second number could be as high as 4.85 x 10^-2 or as low as 4.75 x 10^-2.

Suppose we actually made the full error, high, in both cases. Then the actual value of the product is 1.435 x 10³*4.85 x 10^-2= 6.95975 x 10¹= 69.5975. Suppose we made the full error, low, in both cases. Then the actual value of the product is 1.425 x 10³*4.75 x 10^-2= 6.76875 x 10 ¹= 67.6875.

That's a difference of 1.91 or about 2. Clearly saying that we know the value correct to the nearest 0.1 would be untrue. It's actually pushing it to say we have 2 s.f. since we aren't really certain that it must be between 67.5 and 68.5 which what "68" or "6.8 x 10¹" would mean but it is certainly much closer.

 

[Diane_]

So what you're saying is that it's basically an approximation, taking into account the maximum error in each factor?