Significant Figures

Significant Digits
Approximate calculations (order-of-magnitude estimates) always result in answers with only one or two significant digits.  The last digit is always uncertain. For example, the uncertain digit in 2.56 is 6.
 
When are Digits Significant?
Non-zero digits are always significant. Thus, 22 has two significant digits, and 22.3 has three significant digits.
 
 
There are four rules for numbers with zeros:

1) Zeros in the middle of number are always significant. For example, 101 m has 3 significant figures.

2) Zeros at the beginning of a number only hold places to the right of the decimal point they are not significant. So 0.00 101m has only three significant figures.
 
 
3) Zeros at the end of a number and after a decimal point are significant. There would be no reason to show them if they were not. 0.1 010 has four significant figures.
 
4) Zeros at the end of a number and before a decimal point may or may not be significant. Therefore 10100 m may have 3, 4 or 5 significant figures depending on whether or not the last zeros are part of the measurement or only placeholders.

 
Significant Digits in Multiplication, Division

In a calculation involving multiplication, division, etc., the number of significant digits in an answer should equal the least number of significant digits in any one of the numbers being multiplied, divided etc.
Note that whole numbers have essentially an unlimited number of significant digits.
 
 
 
 
 
 
 
 
 
 Significant Digits in Addition and Subtraction
 
Whenever we multiply or divide numbers with a finite number of significant figures the answer cannot have more significant figures than any of the original numbers. If our mouse is expected to gain 21% more weight in a month our calculator says 75.3g x .21= 15.813g is the weight gain. Since .21 only has two significant digits we must report an expected weight gain of 16 g by rounding to two significant figures. 






Rounding Rules
Rule # 1:If the digit to be dropped is greater than 5, then add "1" to the last digit to be retained and drop all digits farther to the right.
For example:
3.677 is rounded off to 3.68 if we need three significant figures in measurement.
3.677 is rounded off to 3.7 if we need two significant figures in measurement.


Rule # 2:If the digit to be dropped is less than 5, then simply drop it without adding any number to the last digit.
For example:
6.632 is rounded off to 6.63 if we need three significant figures in measurement.
6.632 is rounded off to 6.6 if we need two significant figures in measurement.

Rule # 3:If the digit to be dropped is exactly 5 then:
(A) If the digit to be retained is even, then just drop the "5".
For example:
6.65 is rounded off to 6.6 if we need two significant figures in measurement.
3.4665 is rounded off to 6.466 if we need four significant figures in measurement.
(B) If the digit to be retained is odd, then add "1" to it.
For example:
6.35 is rounded off to 6.4 if we need two significant figures in measurement.
3.4675 is rounded off to 6.468 if we need four significant figures in measurement.
Remember: Zero is an even number
3.05 is rounded off to 3.0 if we need two significant figures in measurement.


Here is a video about significant figures found on YouTuBe: