1-6 Significant Digits

-         When combining numbers in calculations, the uncertainty of the measurements must be considered in the final answer- this is done by keeping tract of significant figures

-         Significant digits: The certain digits and the uncertain digit make up the significant digits

Non- Significant Zeros:

-         As a place keeper- not significant:

o       If the zero is not a known digit, or an estimated digit, the number is not significant

-         Zeros to the Right of a number without a decimal point:

1040 – 3 significant figures

1040. – 4 significant figures

104 – 3 significant figures

1000 – 1 significant figure

1000. – 4 significant figures

1000 – 2 significant figures (the line tells that that zero is the uncertain digit.

-         The Atlantic Pacific rule???

More Information on Zeros:

Zeros to the right of the decimal place before a whole number are not significant.  .001 has only one significant figure. .0000001 also only has one significant figure.

Zeros between two whole numbers are significant.  1.01, now has 3 significant figures.  .0101 also has 3 significant figures.

Numbers following a whole number:

If a decimal place is not present, the zeros to the right of the number are not significant unless marked with a bar.  So, 1000 has only one significant figure.

If a decimal place is present, the zeros to the right of the number are significant.  So 1000. has 4 significant figures.

Zeros following a whole number after a decimal are significant. So, in the number 14.000 the three zeros are significant so there are 5 significant figures.  .0002012300 has 7 significant figures.  Remember, those first three zeros are not significant, but the ones following the three are significant.

Significant Digits in Calculations:

            Rule for exact numbers

When an exact number appears in a calculation, it does not effect the significant digits of the calculations

                        **All Defined Value Unit Factors are Exact Numbers**

            Rule for addition and subtraction of measurements:

Find the measurement with the most uncertainty (fewest number of decimal places) and round the answer to that uncertainty.

Ex.  4.5 g + 3.221 g + 4.3232 g = ?

            4.5 has an uncertainty of .1

            3.221 has an uncertainty of .001

            4.3232 has an uncertainty of .0001

So, the number with the most uncertainty is 4.5 with an uncertainty of .1, therefore, the answer must be rounded off to that uncertainty (in this case, one decimal place)

The Math: 

12.0442 is the answer so far, but remember to correct for uncertainty the answer may only have one decimal place.  In this case, 0 should be the last significant figure.  Look to the right of zero, notice that it is a four.  Since it is a four, we need to round that four down.  The zero does not change, so the answer to the question would become 12.0 +/- .1 g.

            Rule for multiplication or division of measurements:

Find the measurement with the fewest number of significant figures and round the answer to that number of significant figures.

                        Ex. 5.32 g x .01 g

5.32 has 3 significant figures and .01 has 1 significant figure, so the answer to the problem must have only one significant figure.

The answer is .0532.  Remember, now you need to correct for significant figures.  Since the answer can only have one significant figure, anything after the 5 must be rounded off.  Since 3 is less than five, it is rounded down, and the five does not change.  The answer to this problem then is .05 (one significant figure).

Scientific notation:

Scientific measurements usually appear in the form of scientific notation, written in the general form as:

t.tt x 10ⁿ  where t is any integer and n is the some whole number of times that 10 has been multiplied to reach a given answer.

Rule # 1

Find the first non-zero significant figure in a number and place the decimal place immediately to the right of that number.

                        Rule # 2

IF the decimal was moved to the left, then the power of 10 is a positive number

IF the decimal was moved to the right, then the power of 10 is a negative number

Remove all non-significant numbers (place holding zeros)

Example:

Write .00923 in scientific notation:

Ok, since the 2 zeros are place holders, they will not appear in the final answer.

The fist non-zero sig fig is the 9, so place the decimal after the 9

            9.23

Now, since the decimal was moved to the right 3 places, the power of ten is a – number

                                    9.23 x 10^-3

            Write 5600 in scientific notation:

Ok, the first sig fig is the 5, so place the decimal immediately to the right of the 5

                                                            5.600

Since the two zeros are not significant, they should not appear in the final answer:

            5.6

Since the decimal was moved to the left three place, the power of 10 is a positive number:

5.6 x 10³ (to check to see if you have done this correctly simply revert to the ordinary number)

Since 10³ is 10 x 10 x 10, 5.6 is being multiplied by 1000.  when 5.6 is multiplied by 1000, the answer becomes 5600