Understanding Significant Digits

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

  1. Non-zero digits are always significant
    • 47 has two significant digits
    • 47.3 has three significant digits
  2. 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
  3. Zero's placed between non-zero digits are always significant 
    • 1009 has four significant digits
    • 0.01002 has four significant digits
  4. 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.
  5. Zero's after non-zero digits but without a decimal place are ambigious
    1. 130 has at least two significant digits, possibly three.
Rule number 5 can be a little confusing, so lets use an example to illustrate the point, let's say you were counting the minutes until something happened, but you only had a clock with an hour and minuet hand--no second hand. You watch the minuet hand move three times and then the event happens. So you know if was longer than three minuets but less than four minuets. No suppose you wanted to calculate the number of second that transpired.
3x60 = 180
In this example, the zero is not a significant digit because you can't say for certain that the event happened at 180 seconds. One way to eliminate the ambiguity from this is to use scientific notation, by writing 180 seconds as:
1.8 x 102 
it is now clear that the measurement only has two significant digits. 

Significant Digits In Calculations

Since we can now identify the number of significant digits in a number, we will now learn how to determine the number of significant digits in a calculated number. There are two simple rules to remember:
  1. 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.
  2. 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 
Since the minimum number of decimal places in the input is 1 (3.2), the answer should have 1 decimal place also. We will round the value to 6.9

Subtraction Example

12.54 - 1.23 - 1 = 10.31 => 10
The input with the minimum number of decimal places in this example is 1 (0 decimal places), the answer should have zero decimal places.

Multiplication Example

47.3  * 12.23 = 578.479 => 578

Division Example

Let use our original example of trying to find our average lap time:
59+61+58 = 178
178/3 = 59.33333333333333 => 60
Since 3 only has 1 significant digit, out answer must only have 1 significant digit and we round up to 60.




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