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Scientific Notation And Significant Figures

Scientific Notation

Scientific notation consists of two parts:
A number between 1 and 9 A power of 10 N x 10x

* Examples * Given: 289,800,000 Given: 0.000567
Use: (moved 8 places) Answer: x 108 Given: Use: 5.67 (moved 4 places) Answer: 5.67 x 10-4

Learning Check Express these numbers in Scientific Notation: 405789
x 105 3.872 x 10-3 3 x 109 2 x 10-8 x 10-1

Significant Figures

Significant Figures AKA (also know as): Sig Figs Significant Digits

Definition: A method of rounding off calculated measurements
No answer can be more precise than the least precise measurement

Rounding rules Look at the number behind the one you’re rounding.
If it is 0 to 4 don’t change it If it is 5 to 9 make it one bigger

Rounding 5.87192 7.9237439 Round 2 digits Round 3 digits
5.9 5.87 5.872 8 7.9 7.924 7.9237

Significant Figures 2 1 3 4 5 How many numbers mean anything
When we measure something, we can (and do) always estimate between the smallest marks. 2 1 3 4 5

Significant Figures The more marks the better we can estimate. Scientist always understand that the last number measured is actually an estimate 1 2 3 4 5

Significant Figures 1 2 3 4 5 How do we read the ruler? 4.5515 cm?
We needed a set of rules to decide 1 2 3 4 5

Rules for Working with Significant Figures:
Leading zeros are never significant. Imbedded zeros are always significant. Trailing zeros are significant only if the decimal point is specified. Hint: Change the number to scientific notation. It is easier to see. Addition or Subtraction: The last digit retained is set by the first doubtful digit. Multiplication or Division: The answer contains no more significant figures than the least accurately known number.

Significant Figure Rules
Rule #1: All real numbers (1, 2, 3, 4, etc.) count as significant figures. Therefore, you only have to be concerned with the 0 Whether a 0 is significant or not depends on the location of that 0 in the number

Which zeros count? Rule #2: Zeros at the end of a number without a decimal point don’t count 12400 g (3 sig figs) Rule #3: Zeros after a decimal without a number in front are not significant. 0.045 g (2 sig figs)

Which zeros count? Rule #4: Zeros between other sig figs do count.
1002 g (4 sig figs) Rule #5: Zeroes at the end of a number after the decimal point do count g (6 sig figs)

Other Information about Sig Figs
Only measurements have sig figs. A a piece of paper is measured 11.0 inches tall. Counted numbers are exact A dozen is exactly 12 Being able to locate, and count significant figures is an important skill.

Learning Check A. Which answers contain 3 significant figures? 1) cm 2) cm 3) 4760 cm B. All the zeros are significant in 1) mL 2) mL 3) x 103 mL C. 534,675 g rounded to 3 significant figures is 1) 535 g 2) 535,000 g 3) 5.35 x 105 g

Learning Check 1) 22.0 and 22.00 2) 400.0 and 40
In which set(s) do both numbers contain the same number of significant figures? 1) and 22.00 2) and 40 3) and 150,000 4) 63,000 and 2.1 5) and 144 6) and 2000 NO NO YES- 2 YES- 2 NO YES-1

Learning Check How many sig figs in the following measurements? 458 g g 4850 g g g g 3 4 3 3 4 8

Next we learn the rules for calculations
Unfortunately, there are different rules for addition and subtraction and for multiplication and division

Rules for Addition and Subtraction

Your answer must have the same number of digits to the right of the decimal point as there are in the measurement having the fewest digits to the right of the decimal point in the problem. WOW!!! What does that mean?

Example: 2.515 cm cm cm cm Answer stops here

Another Example 27.96 mL + 6.6 mL 34.56 mL 34.6 mL
If mL of NaOH is added to 6.6 mL of HCL, what is the total volume of your solution? First line up the decimal places Then do the adding Find the estimated numbers in the problem This answer must be rounded to the tenths place 27.96 mL mL 34.56 mL mL

Rules for Multiplication and Division

What the heck does that mean?
Your answer MUST have the same number of sig figs as the least number of sig figs in the numbers from the problem. What the heck does that mean?

Round off the answer to 4300 cm3 which is 2 sig figs.
Example: 135 cm x 32 cm = 4320 cm2 3 S.F. 2 S.F. 2 S.F. Round off the answer to 4300 cm3 which is 2 sig figs.

What is the correct answer?
Another Example 610 m x 6.20 m = 3782 m2 2 S.F. 2 S.F. 3 S.F. What is the correct answer? 3800 m2

Learning Check A) 9 m2 B) 9.2 m2 C) 9.198 m2 2. 4.311 cm2 ÷ 0.07 cm =
m X 4.2 m = A) 9 m2 B) 9.2 m2 C) m2 cm2 ÷ cm = A) cm B) 62 cm C) 60 cm (2.54 mL X mL) = mL X mL A.) mL B)11 mL C) 0.041mL

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