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Physics 621
Math Review
SCIENTIFIC NOTATION
Scientific Notation is based on exponential notation (where decimal
places are expressed as a power of 10).
The numerical part of the measurement is expressed as a number
between 1 and 10 multiplied by a whole-number power of 10. M *
10n , 1≤ M < 10, where n is an integer (+ or - #).
Standard Notation 2000
Standard Notation
→ 2 * 103
180 g → 1.8 * 102 g or 1.8 * 10-1 kg
SIGNIFICANT FIGURES
Significant Figures - The number of digits is rough but useful
indication of a measurements precision.
Each digit obtained as a result of measurement is a significant
figure.
The last digit of each measured quantity is always estimated.
The zeros in a number warrant special attention. A zero that is the
result of a measurement is significant, but zeros that serve only to
mark a decimal point are not significant.
Example:
A)
65 ml
(2 sig figs)
B)
173.4
(4 sig figs)
C)
13.2 g
(3 sig figs)
D)
5 ml
(1 sig figs)
Rules for Significant Figures
1.
Non-zero digits are always significant.
Ex.
A) 234.7 L
2.
A zero between other SF is significant.
Ex.
A) 1.05
(4 sig figs)
(3 sig figs)
B) 1.921 kg
B) 2001 m
(4 sig figs)
(4 sig figs)
3. Final zeros to the right of the decimal point are significant
Ex.
A) 6.30 g
(3 sig figs)
B) 10.00 ml
(4 sig figs)
4. Initial zeros are not significant and serve only to show place of
decimal.
Ex.
A) 0.069 km (2 sig figs)
B) 0.0107
(3 sig figs)
5.Final zeros in numbers with no decimal point may or may not
be significant.
Example:
A) 20 marbles
*count, exact, infinite*
B) 2000 m
*1*
C) 20 lbs
*1*
precise from 15 - 24.99....
D) 2.0 x 10¹ lbs
*2*
precise from 19.5 - 20.499....
E) 2.00 x 10¹ lbs
*3*
precise from 19.95 - 20.0499...
F) 1 km = 1000 m
*definition, infinite*
Interpretations of Significant Figures
200 lbs
*1* significant figures ... better written 2 * 102 lbs
200 lbs
*2* significant figures ... better written 2.0 * 102 lbs
200 lbs
*3* significant figures ... better written 2.00 * 102 lbs
COUNTS, CONSTANTS, DEFINITIONS
All have an infinite number of significant figures.(∞)
COUNT - Ex. 10 marbles, 3 people .... Exact
CONSTANTS - Ex. Consider a + 2b = c .
The number 2 is a constant.
DEFINITIONS - Ex. 1 km = 1000 m, 12 = 1 dozen
SIGNIFICANT FIGURE CALCULATIONS
The result of any mathematical calculation involving
measurements cannot be more precise than the least precise
measurement. (Assume all the numbers below are from
measuring)
Addition and Subtraction
When adding or subtracting measured quantities, the answer
should be expressed to the same number of decimal places as
the least precise quantity used in the calculation. ( If needed use
a LINE OF SIGNIFICANCE to aid in solving these.)
Example:
A)
B)
C)
D)
E)
F)
94.02 + 61.1 + 3.1416 = 158.2616 = 158.3
4.01 - 2.30642
= 1.70357 =
1.70
6500 + 730
= 7230
= 7230
98 + 9
= 107
1107 - 107
= 1000
= 1000
1100 -51
= 1049
= 1049
Multiplication, Division, and Square Root
When multiplying, dividing, or finding the square root of measured
quantities, the answer should have the same number of significant
digits as the least precise quantity used in the calculation.
Example:
A)
5.6432 * 0.020
= 0.112864 = 0.11
B)
2500
*2
= 5000
= 5*103
C)
26.3
* 35
= 920.5
= 920
ROUNDING
When completing calculations, do not round any of the
intermediate answers on your way to finding a solution to a
problem. The only rounding that should occur is in the final
answer that is being reported.
Example:
16.75 2.4   40.2  1.2078601......
 3.698 9  33.282
Now, at the very end round to 1 sig fig
Answer is “1”
INVERSELY AND DIRECTLY PROPORTIONAL
When considering what effect changing one or more variables
has on another variable in mathematical relationships
inversely and directly proportional relationships are used.
AB
C
D
C ∝ A,
C ↑ A↑, or C↓ A↓, if C doubles then A doubles
C ∝ 1/D
D ↑ C↓, or D↓ C↑ , if D doubles then C becomes half
Directly Proportional Quantities
Quantities that are directly proportional to one another
increase or decrease by the same factor.
Quantities that are directly proportional to one another occupy
the same position on opposite sides of the proportion sign
(either both located in the numerator position or both in the
denominator position).
AX = ZP
Z & X are directly proportional to each other. ( Z α X )
Inversely Proportional Quantities
Quantities that are inversely proportional to one another change
by the reciprocal of one another (or 1/x of one another).
In a proportion, quantities that are inversely proportional to one
another occupy opposite positions on opposite sides of the
proportion sign.
Z
N

X M
Z & M are inversely proportional to each other. ( Z α 1/M)
Example: Given the following formula, what would happen to
v if r is doubled and T is tripled?
2 r
v
T
Answer: v would change by a factor of,
2
v
3
Example: Given the following formula, what would happen to
mc if T is changed by a factor of 2 and G by a factor of ½?
r 3 Gmc

2
2
T
4
Answer: v would change by a factor of,
1
mc 
2
Unit Analysis
Often we need to change the units in which a physical quantity
is expressed. For example we may need to change seconds and
minutes, hours, days or even years, to do this we use conversion
factors.
Example:
60 sec
1
1min
and
1min
1
60 sec
When a quantity is multiplied by conversion factor it does not
change the amount of quantity just the units the quantity is
measured in.
When a conversion factor is evaluated its value is equal to 1.
Any number multiplied by 1 remains unchanged.
Example:
180sec1 
1min
180sec

60sec
Second are cancelled which leaves units of minutes
Answer: (3 min)
Example: How many centimetres are in 1 km?
1000m 100cm
1km 

?
1km
1m
1*105 cm
Example: How many seconds are in one day?
24hr 60 min 60sec
1day 


?
1day
1hr
1min
86400 sec
Example: Convert 2.4 km/hr to m/s
km
1hr
1min 1000m
2.4



?
hr 60 min 60sec 1km
0.666… m/s
The metric system
Prefix
Symbol
Factor
1 tera
T
1 000 000 000 000
10 12
1 giga
G
1 000 000 000
10 9
1 mega
M
1 000 000
10 6
1 kilo
k
1 000
10 3
1 hecto
h
100
10 2
1 deca
da
10
10 1
base unit
base unit
1
10 0
1 deci
d
0.1
10 -1
1 centi
c
0.01
10 -2
1 milli
m
0.001
10 -3
1 micro

0.000 001
10 -6
1 nano
n
0.000 000 001
10 -9
1 pico
p
0.000 000 000 001
10 -12
1 femto
f
0.000 000 000 000 001
10 -15
1 atto
a
0.000 000 000 000 000 001
10 -18
Re-arranging formulas
Given the following formula,
q
Ek 2
r
Solve for q:
Er
q
k
2
Given the following formula,
4 2 r
ac  2
T
Solve for T:
4 r
T
ac
Given the following formula,
d1
v

2
f
 vi  t
Solve for vf:
2d
vf 
 vi
t
Trigonometry
Trigonometry is the study and solution of Triangles. Solving
a triangle means finding the value of each of its sides and
angles.
B
c
a
ө
C
b
Side Opposite
Hypothenuse
Side Adjacent
Cos θ= Hypothenuse
Side Opposite
Tan θ= Side Adjacent
Sinθ=
A
=
a
c
b
c
=
a
b
=
Find the angles of the following triangle.
B
α
6
10
β
C
θ
A
8
Opp 6
sin  
  0.6
Hyp 10
sin 
1
sin    sin   0.6
1
  39.6  40
β is a right angle
therefore it is 90° and
because all the angles
of a triangle add up to
180° ά must be 50°
The measurements have changed. Find side BA and side AC
B
Sin34=2/BA
2
α
0.559=2/BA
0.559BA=2
β
34º
C
A
BA=2/0.559
BA~3.578
The Pythagorean theorem when used in this triangle states that…
(BC)2+(AC)2=(AB)2
(AC)2=(AB)2-(BC)2
(AC)2=12.802-4=8.802
(AC)~3