# 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?