Significant Figures & Precision

Learning Objective: to understand the meaning of significant figures, to be able to calculate numbers to the correct significance, and to understand how significant figures relate to the precision of a measurement.

Precision: I asked my students how old they were and their answers ranged from 15 to 16. One student said 15 and a half, another said “I’m almost 16”. Much to my surprise, no one answered “I’m 15 years, 2 months, 4 days, 17 hours, 18 minutes and 43 – no wait – 44 seconds old”!  The question to ask is how large or small a quantity do you need to properly represent something. In most conversations, saying you are 15 or 16 is good enough. We could say that you approximate your age to the last full year.  So the degree of precision is 1 year. You’re either 15 or 16 – that’s it. To be more precise, you might say that you are 15 and 1/2. Then the degree of precision would be 1/2 year. You can keep getting more and more precise to months, days, hours and seconds or even to 1/1000ths of a second.  This would be a very high degree of precision to be sure.

How does precision affect mathematical calculations?

The least precise measurement or number determines the overall precision of a calculation.

Consider the number 3. Is 3 really the same as 3.0?  The answer is that you don’t have enough information to tell for sure.  In fact the number 3 actually represents any number from 2.5 to 3.4999… which has been rounded to the nearest integer. 2.5 rounds up to 3 as does 2.6, 2.7, 2.8, and 2.9.  3.1 rounds down to 3 as does 3.2, 3.3 etc. right up through 3.4.

Now for a more difficult problem.  Is 3×3 the same as 9?

The smallest possible value for 3 could be 2.5.                                      2.5×2.5 = 6.25 rounds down to 6.                                                                       The maximum value of 3 could be 3.4999.                                                             3.4999 x3.4999  =  12.                                                                                             Take the average of these results (6 + 12) / 2 = 18/2 =9.