Q18PE - College Physics (Urone)

Question

(a) How many significant figures are in the numbers 99 and  100 ? (b) If the uncertainty in each number is  1 , what is the percent uncertainty in each? (c) Which is a more meaningful way to express the accuracy of these two numbers, significant figures or percent uncertainties? 

Step-by-Step Solution

Answer

1. The percent uncertainty in  99  is  1.00% .
2. The percent uncertainty in  100  is  1.00% .
3. The better way to express the accuracy of  99  and  100  is percent uncertainty.

Step 1: Defining concept of uncertainty

Uncertainty is the doubt that exists about the result of a measurement which means how much the values deviate from each other. The precision of the measuring by significant figures. According to the rule, a significant figure in result must be equal to the quantity with the least number of significant figures.

Consider the given data as below.

Numbers:  99 and  100

Step 2: (a) Defining the significant figures in the numbers and

Since 99  is the exact value, therefore all the number in 99  are significant, while in the case of  100 , the zeros at the end of  100  may or may not be significant, they could be placeholders or they could mean the number is known to the last digit, therefore  100  could have either  1,2,  or  3  significant figures.

Hence, the number of significant figures in  99  is two and  100  could three significant figures at most.

Step 3: (b) Defining the percent uncertainty in each, if the uncertainty in each number is

If the uncertainty in each number is  1, then using uncertainty percent uncertainty in each can be calculated as follows.

Percent uncertainty in  99  can be calculated as follows:

$$\begin{gathered} \mathrm{Percentage} \mathrm{uncertainty}=\frac{\mathrm{\delta A}}{\mathrm{A}}\times 100\% \\ =\frac{1}{99}\times 100\% \\ \square 1.0\% \end{gathered}$$

Percent uncertainty in  100  can be calculated as follows:

$$\begin{gathered} \mathrm{Percentage} \mathrm{uncertainty}=\frac{\mathrm{\delta A}}{\mathrm{A}}\times 100\% \\ =\frac{1}{100}\times 100\% \\ =1.00\% \left(\mathrm{If} \mathrm{all} \mathrm{the} \mathrm{zeros} \mathrm{are} \mathrm{significant}\right) \end{gathered}$$

Hence, the percent uncertainty in  99  and  100  is  1.00% .

Step 4: (c) Calculating significant figures or percent uncertainties

The precision of the measuring tool used to measure a value is indicated by significant figures while present uncertainty from one another in percentages.

Therefore, the better way to express the accuracy of  99  and  100  is percent uncertainty.