Significant figures and uncertainty

When asked to estimate the number of steps required to walk the length of the Appalachian Trail,

Joe said
```
            2,589,842
  
```
Pamela said
```
            2,100,000
  
```

Which student's answer is more sensible? Why?

Scientists pay careful attention to the way they write numerical values. They use only as many digits as they can justify, a practice we describe as providing only the significant digits. There is a connection between the number of significant digits in a written quantity and the uncertainty in that quantity.

One significant digit corresponds roughly to an uncertainty of around 20 to 50 percent
```
            2,000,000     8 x 10^(22)      0.03
  
```
Two significant digits corresponds roughly to an uncertainty of around 2 to 5 percent
```
            2,100,000   8.9 x 10^(22)      0.030
  
```

Can you explain the mathematical basis for this connection?

Some examples

What is the proper way to write each of the following quantities?



     blue whale has mass                   M  =  56.35  +/-  3.3   tonnes


     Joe "The Nose" Scarlatti erased       N  =  123 +/- 4   mobsters


     my measurements of pi yields         pi  =  3.1287356 +/- 0.0793234

What happens when you combine values?

Significant figures are like viruses: they infect any numbers they touch. If you combine two quantities, one with 5 significant figures and another with 1, then the result will have only 1 significant figure.

Try determining the result of the following calculations.

A door has height H = 73.3 inches and width W = 40 inches. What is its area?

Paris Hilton goes to Rodeo Drive and spends $56,000 in just 2.8 minutes. How much money did she spend each second?

Bob drives from NY to LA (3637 miles) in just 4 days. What was his average speed in miles per hour?

You can read more about significant figures in Section 1.5 of your textbook.