Significant Figures: Multiplying and Dividing

[nebbione]

Hi everyone, i have a doubt about significant figures, on a book I read this:

"When multiplying several quantities, the number of significant figures in the final
answer is the same as the number of significant figures in the quantity having the
lowest number of significant figures. The same rule applies to division."

My question is: if i have to multiply 5 x 7 the answer would be 35 but if i apply the law the result is 3 x 10^1 is that right ? because i have to use the same nbumber of significant figures of the number in the operation that has less significant figures, in my case '5' and '7' both have 1 significant figure.

Is that correct ? if not, how should i think about it?

 

[jbriggs444]

You 3 x 10¹ is a correct answer. So is 4 x 10¹ depending on how you round.

This is a quick and easy approximation for a more correct result that could be obtained by other means.

5 plus or minus 0.5 multiplied by 7 plus or minus 0.5 is a number somewhere in the range between 4.5 times 6.5 and 5.5 times 7.5.

That's 35 nominally, with 29.25 on the low end and 41.75 on the high end. One significant figure.

If you look, you can note the the percentage errors are:

+/- 10% on the 5
+/- 7% on the 7
-16% low through +19% high on the 35.

Roughly speaking, percentage errors add when you multiply or divide.

A factor of 2 increase in percentage error is not a whole significant digit. It's a little less than 1/3 of a significant digit. The simple rules for computing significant figures do not require you to account for this.

 

[nebbione]

Sorry i didn't understand very well the second part; I have 3 questions :

1) If I have to compute some calculations to solve a problem, should i round every time? or only for the final result ? For example lets' say that i have to multiply 5*7.25 then divide by 6.234 then add 5.6 then multiply by arctan(pi/4).
do i have to round for each operation that i make? o i have to keep all the digits and round only for the final result ?

2) I didn't understand how uncertainty work, for example if i take a measure, the uncertainty is determined by me or by the tool I'm using?

3) How did you doi to say that on 5 the error is 10% and on 7 is 7% ? Can you explain me ?

Thank you for the patience! And sorry for my ignorance

 

[jtbell]

1) If I have to compute some calculations to solve a problem, should i round every time? or only for the final result ?

Round only when you get to the end. This happens naturally when you do the calculation in a single chain of steps on your calculator.

 

[jbriggs444]

3) How did you doi to say that on 5 the error is 10% and on 7 is 7% ? Can you explain me ?

"5" is stated with one significant digit. So the implicit error bound is 5 plus or minus 0.5.
"7" is stated with one significant digit. So the implicit error bound is 7 plus or minus 0.5.

Expressed as a percentage, 0.5 is 10% of 5.
Expressed as a percentage, 0.5 is 7% of 7.

 

[nebbione]

Ahhh i got ya! Thank you very much! I understand now!
So i f for example a measure is given to me like 5 meters, i know that by default the uncertainty is ± 0.5... i understand now, and about question 2 i heard that the uncertainty of a measure is given by the tool normally, for example if i have a balance that has a precision of 100 grams i know that if i put on it an object and i read a value of 7.5 kg i have to specify that the mass of the object is 7.5 kg ±(100/2)grams is that right ?