1. Introduction to Significant Figures &
Scientific Notation

2. Significant Figures
Scientist use significant figures to determine how precise a measurement is
Significant digits in a measurement include all of the known digits plus one estimated digit

3. For example… Look at the ruler below Each line is 0.1cm
You can read that the arrow is on 13.3 cm
However, using significant figures, you must estimate the next digit
That would give you cm

4. Let’s try this one Look at the ruler below
What can you read before you estimate?
12.8 cm
Now estimate the next digit…
12.85 cm

5. The same rules apply with all instruments
Read to the last digit that you know
Estimate the final digit

6. Let’s try graduated cylinders
Look at the graduated cylinder below
What can you read with confidence?
56 ml
Now estimate the last digit
56.0 ml

7. One more graduated cylinder
Look at the cylinder below…
What is the measurement?
53.5 ml

8. Rules for Significant figures Rule #1
All non zero digits are ALWAYS significant
How many significant digits are in the following numbers?
3 Significant Figures
5 Significant Digits
4 Significant Figures
274
25.632
8.987

9. Rule #2 All zeros between significant digits are ALWAYS significant
How many significant digits are in the following numbers?
3 Significant Figures
5 Significant Digits
4 Significant Figures
504
60002
9.077

10. Rule #3 All FINAL zeros to the right of the decimal ARE significant
How many significant digits are in the following numbers?
32.0
19.000
3 Significant Figures
5 Significant Digits
7 Significant Figures

11. Rule #4 All zeros that act as place holders are NOT significant
Another way to say this is: zeros are only significant if they are between significant digits OR are the very final thing at the end of a decimal

12. For example
How many significant digits are in the following numbers?
1 Significant Digit 3 Significant Digits 6 Significant Digits 2 Significant Digits x

13. Rule #5
All counting numbers and constants have an infinite number of significant digits
For example:
1 hour = 60 minutes
12 inches = 1 foot
24 hours = 1 day

14. How many significant digits are in the following numbers?x
2 Significant Digits 6 Significant Digits 3 Significant Digits 4 Significant Digits 1 Significant Digit

15. Rules Rounding Significant Digits Rule #1
If the digit to the immediate right of the last significant digit is less that 5, do not round up the last significant digit.
For example, let’s say you have the number and you want 3 significant digits
The last number that you want is the 8 – 43.82
The number to the right of the 8 is a 2
Therefore, you would not round up & the number would be 43.8

16. Rounding Rule #2
If the digit to the immediate right of the last significant digit is greater that a 5, you round up the last significant figure
Let’s say you have the number and you want 4 significant digits
– The last number you want is the 8 and the number to the right is a 7
Therefore, you would round up & get 234.9

17. Rounding Rule #3
If the number to the immediate right of the last significant is a 5, and that 5 is followed by a non zero digit, round up
(you want 3 significant digits)
The number you want is the 6
The 6 is followed by a 5 and the 5 is followed by a non zero number
Therefore, you round up
78.7

18. Rounding Rule #4
If the number to the immediate right of the last significant is a 5, and that 5 is followed by a zero, you look at the last significant digit and make it even.
(want 3 significant digits)
The number to the right of the digit you want is a 5 followed by a 0
Therefore you want the final digit to be even
2.54

19. Say you have this number
(want 3 significant digits)
The number to the right of the digit you want is a 5 followed by a 0
Therefore you want the final digit to be even and it already is
2.52

20. Let’s try these examples…
(want 3 SF) (want 2 SF) (want 3 SF) (want 1 SF) (want 2 SF)

21. Scientific Notation
Scientific notation is used to express very large or very small numbers
I consists of a number between 1 & 10 followed by x 10 to an exponent
The exponent can be determined by the number of decimal places you have to move to get only 1 number in front of the decimal

22. Large Numbers
If the number you start with is greater than 1, the exponent will be positive
Write the number in scientific notation
First move the decimal until 1 number is in front –
Now at x 10 – x 10
Now count the number of decimal places that you moved (4)
Since the number you started with was greater than 1, the exponent will be positive
x 10 4

23. Small Numbers
If the number you start with is less than 1, the exponent will be negative
Write the number in scientific notation
First move the decimal until 1 number is in front – 5.2
Now at x 10 – 5.2 x 10
Now count the number of decimal places that you moved (3)
Since the number you started with was less than 1, the exponent will be negative
5.2 x 10 -3

24. Scientific Notation Examples
Place the following numbers in scientific notation:
x x x x x 104

25. Going from Scientific Notation to Ordinary Notation
You start with the number and move the decimal the same number of spaces as the exponent.
If the exponent is positive, the number will be greater than 1
If the exponent is negative, the number will be less than 1

26. Going to Ordinary Notation Examples
Place the following numbers in ordinary notation:
3 x x x x x 103

27. UNITS (Systéme Internationale)
Dimension
SI (mks) Unit
Definition
Length
meters (m)
Distance traveled by light in 1/(299,792,458) s
Mass
kilogram (kg)
Mass of a specific platinum-iridium allow cylinder kept by Intl. Bureau of Weights and Measures at Sèvres, France
Time
seconds (s)
9,192,631,700 oscillations of cesium atom

28. Standard Kilogram at Sèvres

29. Dimensional Analysis Variable from Eq. x m t v=(xf-xi)/t a=(vf-vi)/t
Dimensions & units can be treated algebraically.
Variable from Eq. x m t
v=(xf-xi)/t
a=(vf-vi)/t
dimension L M T
L/T
L/T2

30. Dimensional Analysis Checking equations with dimensional analysis:
(L/T2)T2=L L
(L/T)T=L
Each term must have same dimension
Two variables can not be added if dimensions are different
Multiplying variables is always fine
Numbers (e.g. 1/2 or p) are dimensionless

31. Example 1.1 Check the equation for dimensional consistency:
Here, m is a mass, g is an acceleration, c is a velocity, h is a length

32. Example 1.2 Consider the equation:
Where m and M are masses, r is a radius and v is a velocity.
What are the dimensions of G ?
L3/(MT2)

33. Example 1.3
Given “x” has dimensions of distance, “u” has dimensions of velocity, “m” has dimensions of mass and “g” has dimensions of acceleration.
Is this equation dimensionally valid?
Yes
Is this equation dimensionally valid? No

34. Units vs. Dimensions Dimensions: L, T, M, L/T …
Units: m, mm, cm, kg, g, mg, s, hr, years …
When equation is all algebra: check dimensions
When numbers are inserted: check units
Units obey same rules as dimensions: Never add terms with different units
Angles are dimensionless but have units (degrees or radians)
In physics sin(Y) or cos(Y) never occur unless Y is dimensionless

35. Example 1.3 Grandma traveled 27 minutes at 44 m/s.
How many miles did Grandma travel?
44.3 miles

36. Prefixes
In addition to mks units, standard prefixes can be used, e.g., cm, mm, mm, nm

37. Example 1.4a
The above expression yields:
40.11 m
4011 cm
A or B
Impossible to evaluate (dimensionally invalid)

38. Example 1.4b
The above expression yields:
4.5 m kg
4.5 g km
A or B
Impossible to evaluate (dimensionally invalid)

39. Example 1.4b
The above expression yields:
-1.5 m
-1.5 kg m2
-1.5 kg
Impossible to evaluate (dimensionally invalid)

40. MARS CLIMATE ORBITER

43. Dec. 11, 1998

44. Sept. 23, 1999

45. The loss of the the Mars Climate Orbiter on September 23, 1999, was a most unfortunate and highly avoidable event.
The cause of the mishap has been traced to a mix-up over units. Preliminary findings indicated that one team used English units (e.g., inches, feet and pounds) while the other used metric units for maneuvers required to place the spacecraft in the proper Mars orbit.
The 'root cause' of the loss of the spacecraft was the failed translation of English units into metric units.
For nearly three centuries, engineers and scientists have been struggling with English units.