1. Scientific Notation Significant figures And Dimensional Analysis

2. Scientific Notation
An alternate way of writing numbers – especially those that are very LARGE or very SMALL
The only measures you CAN’T use it with is money or temperature!

3. 4.1 x 105 Scientific Notation Two Parts to the notation
Mantissa – the base 10 number
Characteristic – the exponent to which the base 10 number (mantissa) is raised
Correct scientific notation MUST HAVE BOTH PARTS even if the characteristic is zero!
4.1 x 105
Characteristic
Mantissa

4. Scientific Notation
Positive characteristic means the number is very LARGE
3.0 x 1010 = 30,000,000,000
If the original number is > 1 then the characteristic will be positive
Negative characteristic means the number is very SMALL
3.0 x =
If the original number is < 1 then the characteristic will be negative

5. Examples
17600
10.2
-67.30
4.76
301.0

6. Scientific Notation to Normal Notation
To expand scientific notation to regular or normal notation  look at the characteristic.
If the characteristic is POSITIVE move the decimal that many places to the RIGHT
4.96 x 102 = 496.
If the characteristic is NEGATIVE move the decimal that many places to the LEFT
5.61 x 10-3 =

7. Examples
36.7 x 101
x 10-3
75.4 x 10-1
-14.5 x 10-2
0.123 x 104
97723 x 102
851.6 x 10-3
94.2 x 10-4

8. Scientific Notation Multiplying
MULTIPLY the mantissa as you would any other multiplication
ADD the characteristics
CONVERT the results to the correct notation if necessary
(2x104)(3x10-3)
2x3 = 6
4 + (-3) = 1
6 x 101 final answer

9. Examples
(5x10-3) (4x10-4)
(6x104) (-7x10-6)
(-4.5x10-2) (2x10-7)

10. Dividing Scientific Notation
To divide
DIVIDE the mantissa as you normally would
SUBTRACT the characteristic of the denominator (bottom) from the characteristic of the numerator (top)
CONVERT the result if necessary
8x10-5
2x10-3
8/2 = 4 ; (-5) – (-3) =-2
4x10-2 final answer

11. Examples 4x103/8x10-3 6x10-7/3x10-8 4.5x104/9.0x10-12

12. Adding and Subtracting
In order to ADD and SUBTRACT scientific notation the characteristic MUST BE THE SAME
First correct or “incorrect” the mantissa so that the characteristics of the numbers are the same
Then add or subtract the manitssas as you normally would
Write the characteristic as you have it for both numbers DO NOT ADD the characteristics!!

13. Examples
1.64 x 10-2
+3.78x10-3
4.77x10-2
+2.13x10-4
9.88x101
+6.37x103

14. Significant Figures Two basic rules
All non-zero digits are ALWAYS significant
Any number 1-9
Zeros have three rules:
Zeroes at the beginning of a number are NEVER significant – they are only place holders (0.0001)
Zeroes between NON-zero digits are ALWAYS significant (101)
Zeros a the END of numbers are significant only if the number contains a decimal point (100 vs 100.)

15. Significant figures Zeros that come BEFORE significant numbers
are NEVER significant
But just like paparazzi you can’t get rid of them!!

16. Significant Figures ALWAYS significant
Zeros BETWEEN two significant figures are
ALWAYS significant

17. Significant figures
Zeros at the END are ALWAYS significant

18. Examples
114.0
733.02
9.080
4.50 x 103
-23.40
6.0040

19. Multiplying & dividing sig figs
Count the significant figures in the given numbers.
The answer must be rounded to the LEAST number of digits in the given numbers
4.51 x 3.4 = = 15
3 digits
2 digits
Final answer = 2 digits

20. Adding & subtracting sig figs
Count the number of DECIMALS PLACES in the given numbers.
The answer must be rounded to the LEAST number of DECIMALS PLACES in the given numbers

21. Examples
( ) – 2.9 –

22. If the problem contains both addition/ subtraction AND multiplication/division you must take the problem apart and deal with each part appropriately
78.4 x

23. Metric Units
System International units or SI are the units used in science.
English units are the COMMON unit used in the United States.
BOTH systems are both accurate
There are 88 English units
There are only 3 major SI units and 9 prefixes

24. Metric Units What are the English units for length?
The metric units for length is the meter (m)
What are the English units for volume
The metric unit for volume is the liter (L)
What are the English units of mass (weight)?
The metric unit for mass is the gram (g)

25. Metri Prefixes Giga G 1,000,000,000 (billion) 1x109
Mega M 1,000,000 (million) 1x106
Kilo k x103
Deci d 1/ x10-1
Centi c 1/ x10-2
Milli m 1/ x10-3
Micro µ 1/ x10-6
Nano n 1/1,000,000,000 1x10-9
Pico p 1/1,000,000,000,000 1x10-12
No prefix is ever used alone, it is always attached to a unit!

26. Examples Give the metric abbreviation for each One-millionth of a gram
One billion liters
On-hundredth of a liter
On thousand grams
On-thousandth of a meter
On million grams

27. Dimensional Analysis Organized method of unit conversion
What is 3 hours in minutes?
How did you do that?
How about hours?
Not so easy in your head…
So HOW do you do it?

28. Dimensional Analysis
There are two relationships which exist between hours and minutes
60 minutes “per” 1 hour OR minutes
1hour
And 1 hour “per” 60 minutes OR 1hour
60 minutes

29. Dimensional Analysis
To convert 3 hours to minutes your brain took the value of 3 hours and multiplied it by one of the two fractions
3 hours x (1 hour) OR 3 hours x (60 minutes)
60 minutes hour
Only ONE of these will cancel out the units of hours when multiplied. That’s the one you want!

30. Dimensional analysis
Dimensional analysis is a way of arranging the labels (units) so they will cancel out.
Instead of writing fractions in parentheses we are going to place them on a straight line dimensional analysis grid.
3.179 hours 60 min = min = min
1 hour

31. Dimensional analysis Draw the dimensional analysis grid like so
Put what is GIVEN in the UPPER LEFT corner with the unit label
GIVEN UNIT LABEL

32. Dimensional Analysis
Put the SAME label that you put in the upper left in the LOWER RIGHT corner
Put the label of what you are converting to in the UPPER RIGHT
Put the SAME label that you put in the upper left in the LOWER RIGHT corner
Put the label of what you are converting to in the UPPER RIGHT
GIVEN UNIT LABEL
MATCH THE GIVEN UNIT LABEL
GIVEN UNIT LABEL
UNIT LABEL
YOU WANT
MATCH THE GIVEN UNIT LABEL

33. Dimensional Analysis
Fill in the correct numerical relationship that exists between the two labels i.e. 60 seconds = 1 minutes
Cancel out any labels that appear in BOTH the numerator and denominator.
Multiply everything together that is above the grid line; Divide everything that is below
UNIT LABEL
YOU WANT
GIVEN UNIT LABEL
3.179 hours
60 seconds
seconds
MATCH THE GIVEN UNIT LABEL
1 hour

34. Examples How many eggs are there in 10.25 dozen?
Metric system unit conversions
Convert cm to meters
Convert 4.32 x10-4 g to micrograms
How many picoliters are in 4.56x10-7 liters?
Convert 88.1 km to meters

35. Multiple Prefixes
How do you know where to put the unit labels for a problem with multiple prefixes?
Use one dimension to convert from one prefix to the “plain” unit
Then convert from the “plain” unit to the other prefix
231 mm 1x10-3 m 1 nm = 231 x = 2.31x108nm
1 mm x 10-9 m x 10-9

36. Examples Convert 5.43 GL to dL Convert 6.99 x 108 pg to cg
How many micrometers are there in 45.2 kilometers?

37. English Conversions
The following are conversions you are EXPECTED to KNOW!
1 ton = _________ pounds 1 day = _______ hours
1 pound = _________ounces 1 hours = ______ minutes
1 mile = __________ feet 1 minute = _____seconds
1 foot = __________ inches 1 year = ______days
1 yard = __________ feet 1 month = ____ days
1 pint = ___________ cups
1 quart =__________ pints
1 gallon = _________quarts

38. English to metric unit conversions
2.54 cm = 1 inch (length)
1.06 quarts = 1 liter (volume)
454 grams = 1 pound (mass)

39. Examples Convert 3.92 yards to picometers
Convert 1.75 x 106 grams to ounces
Convert 25.6 tons to mg

40. Dimensional analysis of Double units
Put the number on the TOP label in the upper left corner of the grid
Put the number on the bottom label in the lower left corner of the grid
You may begin working with either label - just be sure to arrange the labels diagonally so they will cancel
Work until you get ONE of the labels in the unit you want THEN work on the other label.

41. Example Convert 35 miles/hour to km/sec
35 mi 1 hr 1 min ft 12 in cm 1x10-2 m 1 km
1 hr min 60 s 1 mi 1 ft 1 in cm x103m

42. Examples Convert 45.8 Pounds/gallon to ng/microliter
Convert 1.023X107 pm/day to inches/sec

43. Cubic units
Write what is given in the problem in the upper left part of the grid with label
STOP AND THINK! You do not know one SINGLE conversion which utilizes cubic units …so transfer just the “plain” unit in the lower right corner and the other “plain” unit in the upper right
Now go ahead and fill in the known relationship (i.e. the numbers)
BUT!!! Since the label that you started with is CUBIC you MUST CUBE THE ENTIRE DIMENSION both NUMBER AND LABEL!!
Sometimes it is necessary to cube more than one of the dimensions to get to the label you want…don’t panic!

44. Example Convert 70.6 cm3 to m3 70.6 cm3 1 x 10-2m 7.06 x 10-5 m3 1 cm

45. Examples Convert 35 km3 to cubic feet
Convert cubic yards to cubic millimeters

46. Final Notes on Dimensional Analysis
Read the problem and write down ALL “double label” conversions
Begin your dimensional analysis with anything in the problem which is NOT a double label conversion
Use each double label conversion factor you have written down only ONCE and arrange the conversion factors on the dimensional analysis grid so that the labels cancel out.

47. Precision versus Accuracy
if your data is accurate it is close to true value
if your data is precise it is consistent from trial to trial
How close a set of measurements are to EACH OTHER

48. Precision versus Accuracy
This is precise and accurate

49. Precision versus Accuracy
This is precise but not accurate

50. Precision versus accuracy
Student weighs a beaker and records its mass as g, 47.23, and 47.19, The actual mass of the beaker is 48.44g is this student accurate? Precise?
What is the accuracy and/or precision of these basketball free throw shooters?
99 out of 100 shots made
99 out of 100 hit the front of the rim and bounced off
33 out of 100 shots are made and the rest miss the rim completely or rebound off the backboard

51. Percent Error
Sometimes you need to know the percent ERROR in your measurements whether it is too high or too low.
This is the formula for % error – MEMORIZE IT!!
% error = experimental or observed value – actual or true value x 100
actual or true value

52. Examples
A student weighs a beaker and records its mass as g. The actual mass of the beaker is 47.93g. What is the student’s % error?
% error = 47.2 g g x = %
47.93 g
HINT: a negative sign means your answer is that percentage TOO LOW
a positive sign means your answer is that percentage TOO HIGH