The concept of Significant Digits arose from the need to keep uncertainty out of calculations. Let's consider a simple example to help understand why we need significant digits:

Suppose you ran a lap around the track every morning for 3 days and you recorded you times to be: 59s, 61s, and 58s. You made these measurements on your watch by keeping track of the where the second hand was when you started and where is was when it ended. Now suppose you wanted to know what your average time was--simply add the results and divide by the number of results:

59+61+58 = 178 178/3 = 59.33333333333333

I pasted the results of this calculation right from the computer's calculator, but clearly I wasn't able to measure that many decimal place from the second hand on my watch, so the calculation has added some uncertainty to my results. Below we'll learn more about how to use significant digits and see how are calculation will change.

### Identifying Significant Digits

- Non-zero digits are always significant
- 47 has two significant digits
- 47.3 has three significant digits
- Leading zero's are not significant, they simply are a product of the unit of measure you are using
- 0.00125 has three significant digits
- Zero's placed between non-zero digits are always significant
- 1009 has four significant digits
- 0.01002 has four significant digits
- Zero's after non-zero digits and after the decimal place are significant
- 1.20 has three significant digits
- 0.20 has two significant digits.
- Zero's after non-zero digits but without a decimal place are ambigious
- 130 has at least two significant digits, possibly three.

3x60 = 180

1.8 x 10^{2}

### Significant Digits In Calculations

- During addition and subtraction, the calculated value should have the same number of decimal places as the input number with the least number of decimal places.
- During multiplication and division, the calculated value should have the same number of significant digits and the input number with the least amount of significant digits.

#### Addition Example

3.2 + 1.23 + 2.453 = 6.883 => 6.9

#### Subtraction Example

12.54 - 1.23 - 1 = 10.31 => 10

#### Multiplication Example

47.3 * 12.23 = 578.479 => 578

#### Division Example

59+61+58 = 178 178/3 = 59.33333333333333 => 60Since 3 only has 1 significant digit, out answer must only have 1 significant digit and we round up to 60.