# 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 1022      0.03
```

Why? Well, consider the first value of 2 million: all we know is that the value is smaller than 3 million, but larger than 1 million.

```

Q:  What is the percentage difference between
2,000,000  and  3,000,000?

Q:  What is the percentage difference between
2,000,000  and  1,000,000?

```
• Two significant digits corresponds roughly to an uncertainty of around 2 to 5 percent
```
2,100,000   8.9 x 1022      0.030
```

Why? Well, consider the first value of 2.1 million: we know that the value is smaller than 2.2 million, but larger than 2.0 million.

```

Q:  What is the percentage difference between
2,100,000  and  2,200,000?

Q:  What is the percentage difference between
2,100,000  and  2,000,000?

```

Is there a connection between +/- uncertainties and significant figures? Yes, indeed. For example, how many students at RIT are on campus today?

```

N  =  12,439  +/-   338

```

This number could be as big as 12,777, or as small as 12,101.

```

12,101   <   N    <    12,777

```

The first digit -- 1 -- doesn't change. The second digit -- 2 -- doesn't change. But the third digit could be 1, or 2, or 3, or 4, or 5, or 6, or 7. So, if we write the number N using significant digits, we should write

```

N  =  12,000

```

since these are the only digits we are SURE are correct.

Significant figures are like viruses: they infect any numbers they touch. If you combine two quantities, then the number of significant digits in the result is equal to the SMALLER of the number of significant figures in the input values.

For example, if you add two values, one with 5 significant figures and another with 1, then the result will have only 1 significant figure.

Can you write the answers to the following combinations?

```

83 meters/sec  *   1.459 seconds        =

0.04 kg        /   250 cubic meters     =

\$12 / hour     *   2.000 x 103 hours   =

```