Transcript Saint Omer 62

ENM 503
Fundamentals
Numbers, Bases, Algebra, Functions,
Equations and other Calculus Concepts
rd
1
Primitives & Axioms
Is the number 6 larger than the number
3?
Has anyone ever seen a number?
Is the word "cheese" on the blackboard "chalk" ?
Has anyone ever seen a point?
Axioms are assumptions made about primitives.
Bird Testing of Number
An Example of doing Mathematics
Sum of first 100 integers : 1 + 2 + 3 + 4 + 5 + 6
1 + 6 = 2 + 5 = 3 + 4 = 7 => n(n+1)/2
2
rd
Numbers
Cardinal: zero, one, two, … used for counting
Ordinal: first, second, … denote position in sequence
Integers: negative, zero and positive whole numbers
… -3 -2 -1 0 1 2 3 …
Fractions: parts of whole, ½ ¼ ¾ etc.
Numerals: symbols describing numbers
Digits: specific symbols to denote numbers
Arabic Numerals: 0 1 2 3 4 5 6 7 8 9
Roman Numerals: I II III IV V VI VII VIII IX X …
10
99
100
googol
1 googol = 10 ; 1 googolplex = 10
= 1010 ; 99
3
rd
Types of Numbers
Rational
Prime
Perfect
Algebraic – roots of equations with integer coefficients
Irrational 2½ is algebraic since x2 – 2 = 0
Imaginary and Complex, i =
Transcendental – Liouville; e, ; most
1 frequent,
not algebraic, not roots of integer
polynomials
Transfinite Numbers - Cantor
Figurate Numbers
Omega ; aleph null 0; aleph-one 1
4
rd
CASTING OUT NINES
+
28
1
39
3
42
6
109
10
1
1 Checks
Same procedure for subtraction and multiplication
25 * 25 = 625 ~ 4 after casting out 9's
7 * 7 = 49 ~ 4 after casting out 9's
5
rd
The Real Number System
natural numbers
N = {1, 2, 3, …}
Integers
I = {… -3, -2, -1, 0, 1, 2, … }
Rational Numbers
R = {a/b | a, b  I and b  0}
Algebraic Numbers
Irrational Numbers
{non-terminating, non-repeating decimals} e.g. 2
Transcendental numbers ~ irrational numbers that cannot be
a solution to a polynomial equation having integer
coefficients
transcends the algebraic operations of +, -, x, /
6
rd
Binary Arithmetic
Sum
1011  11
+ 101  5
10000 16
Difference
1011 11
- 101 - 5
110
6
Product
1011
11
X 101
5
1011
55
0000
1011
110111
Quotient
10.00110…
101 1011,0000000
-1010
01 000
101
110
101
#b11011 = 27; #o27 = 23; #xAB = 171; #7r54 = 39
7
rd
1=2
Let x = y
xy = y2
xy – x2 = y2 – x2
x(y – x) = (y – x)(y + x)
x=y+x
1=2
qed. Quad erat demonstrandum meaning
which was to be demonstrated.
8
rd
Three Classical Insolvable Problems
Using only straight edge and compass
1. Construct a square whose area equals a circle.
2. Double the volume of a given cube.
3. Trisect an angle
9
rd
Multinomials
Find the coefficient of x3yz2 in the expansion of
(x + y + z)6.
 6 

  60
 3!1!2!
(poly^n #(x #(y #(z 0 1) 1) 1) 6) 
#(X #(Y #(Z 0 0 0 0 0 0 1) #(Z 0 0 0 0 0 6) #(Z 0 0 0 0
15) #(Z 0 0 0 20) #(Z 0 0 15) #(Z 0 6) 1) #(Y #(Z 0 0 0
0 0 6) #(Z 0 0 0 0 30) #(Z 0 0 0 60) #(Z 0 0 60) #(Z 0
30) 6) #(Y #(Z 0 0 0 0 15) #(Z 0 0 0 60) #(Z 0 0 90)
#(Z 0 60) 15) #(Y #(Z 0 0 0 20) #(Z 0 0 60) #(Z 0 60)
20) #(Y #(Z 0 0 15) #(Z 0 30) 15) #(Y #(Z 0 6) 6) 1)
10
rd
Poly^n
(x + y + z)3 = x3 + 3x2y + 3x2z + 3xy2 + 6xyz + 3xy2 + y3
+ 3y2z + 3yz2 + z3
(poly^n #(x #(y #(Z 0 1) 1) 1) 3)
#(X #(Y #(Z 0 0 0 1) #(Z 0 0 3) #(Z 0 3) 1) #(Y #(Z 0 0 3)
#(Z 0 6) 3) #(Y #(Z 0 3) 3) 1)
x3 + 3x2y + 3x2z + 3xy2 + 6xyz + 3xy2 + y3 + 3y2z + 3yz2 + z3
11
rd
Joseph Liouville
Born: 24 March 1809 in Saint-Omer, France
Died: 8 Sept 1882 in Paris, France
An important area which Liouville is remembered for today is that of
transcendental numbers. Liouville's interest in this stemmed
from reading a correspondence between Goldbach and Daniel
Bernoulli. Liouville certainly aimed to prove that e is
transcendental but he did not succeed. However his
contributions were great and led him to prove the existence of a
transcendental number in 1844 when he constructed an infinite
class of such numbers using continued fractions. In 1851 he
published results on transcendental numbers removing the
dependence on continued fractions. In particular he gave an
example of a transcendental number, the number now named
the Liouvillian number:
0.1100010000000000000000010000...
where there is a 1 in place n! (n = 1,2,3, …
and 0 elsewhere.
12
rd
More Real Numbers
Real Numbers
Rational (-4/5) = -0.8
Integers (-4)
Natural Numbers (5)
Irrational
2  1.41421...
Transcendental
(e=2.718281828459045… )
(=3.141592653589793 …)
Did you know? The totality of real numbers can be placed in a one-to-one
rd points on a straight line. Dense.
13
correspondence with the totality of the
Numbers in sets
transcendental
numbers
Did you know? That irrational numbers are far more numerous than
rational numbers? Consider n a / b , n  1, 2,3,...where a and b are integers
14
rd
Identity Property
The numbers 0 and 1 play an important role in math
since they do absolutely nothing.
Any number plus 0 equals itself.
a + 0 = 0 + a = a.
One example of this is: 3 + 0 = 0 + 3 = 3.
0 is called the identity for addition.
Any number multiplied by 1 is equal to itself.
a x 1 = 1 x a = a.
One example of this is: 3 x 1 = 1 x 3 = 3
1 is called the identity for multiplication.
15
rd
Algebraic Operations
Basic Operations
addition (+) and the inverse operation (-)
multiplication (x) and the inverse operation ( )
Commutative Law
a+b=b+a
axb=bxa
* Vectors, Matrices non-commutative
Associative Law
a + (b + c) = (a + b) + c
a(bc) = (ab)c
Distributive Law
a(b + c) = ab + ac
16
rd
Functions
Functions and Domains: A real-valued function f of
a real variable is a rule that assigns to each real
number x in a specified set of numbers, called the
domain of f, a real number y = f(x) in the range.
The variable x is called the independent variable.
If y = f(x), we call y the dependent variable.
A function can be specified:
numerically: by means of a table or ordered pairs
algebraically: by means of a formula
graphically: by means of a graph
17
rd
More on Functions
A function f(x) of a variable x is a rule that assigns to
each number x in the function's domain a value
(single-valued) or values (multi-valued)
y  f ( x)
dependent
variable
examples:
18
independent
variable
y  f ( x)  3x 2  4 / x  ln(.2 x)
z  f ( x1 , x2 ,..., xn )
a
f ( x1 , x2 )   bx23
x1
rd
function of
n variables
On Domains
Suppose that the function f is specified algebraically by
the formula
x
f ( x) 
x 1
with domain (-1, 10]
The domain restriction means that we require
-1 < x ≤ 10 in order for f(x) to be defined (the round
bracket indicates that -1 is not included in the domain,
and the square bracket after the 10 indicates that 10
is included).
19
rd
A more interesting function
Sometimes we need more than a single formula to
specify a function algebraically, as in the following
piecemeal example
The percentage p(t) of buyers of new cars who used
the Internet for research or purchase since 1997 is
given by the following function. (t = 0 represents
1997).
10t  15 if 0  t  1
p(t )  
15t  10 if 1  t  4
20
rd
Functions and Graphs
The graph of a function f(x) consists of the totality of
points (x,y) whose coordinates satisfy the relationship
y = f(x).
y
_
_
_
_
_
_
_
y intercept where x = 0
a linear function
|
21
|
|
|
|
the zero of the function
or roots of the equation y = f(x) = 0
rd
|
x
Graph of a nonlinear function
Sources: Bureau of Justice Statistics, New York State Dept. of Correctional
Services/The New York Times, January 9, 2000, p. WK3.
22
rd
Polynomials in one variable
Polynomials are functions having the following form:
nth degree polynomial
f ( x)  a0  a1 x  a2 x 2  a3 x 3  ...   an x n
f ( x)  a0  a1 x
linear function
f ( x)  a0  a1 x  a2 x
2
quadratic function
Did you know: an nth degree polynomial has exactly n roots;
i.e. solutions to the equation f(x) = 0 Karl Gauss
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rd
Facts on Polynomial Equations
a0  a1 x  a2 x 2  a3 x 3  ...  an x n  0
Used in optimization, statistics (variance), forecasting,
regression analysis, production & inventory, etc.
The principle problem when dealing with polynomial
equations is to find its roots.
r is a root of f(x) = 0, if and only if f(r) = 0.
Every polynomial equation has at least one root, real or
complex (Fundamental theorem of algebra)
A polynomial equation of degree n, has exactly n roots
A polynomial equation has 0 as a root if and only if the
constant term a0 = 0.
24
rd
Make-polynomial with roots
(my-make-poly '(1 2 3))  (1 -6 11 -6)
(cubic 1 -6 11 -1)  (3 2 1)
(my-make-poly '(2 -3 7 12))  (1 -18 59 198 -504)
(quartic 1 -18 59 198 -504)  (12 7 2 -3)
(my-make-poly '(1 2 -3 7 12)) 
(1 -19 77 139 -702 504) but neither quintic nor higher
degree polynomials can be solved by formula.
25
rd
The Quadratic Function
f ( x)  ax  bx  c, a  0
2
Graphs as a parabola
vertex: x = -b/2a
if a > 0, then convex (opens upward)
if a < 0, then concave (opens downward)
b
c
ax  bx  c  0; x  x  
a
a
2
2
2
c  b 
b
 b 
2
x  x
   

a
2
a
a
2
a
 




b 2
c  b 
(x 
)   

2a
a  2a 
26
rd
2
2
The Quadratic Formula
f ( x)  ax  bx  c
2
x  4 x  3  ( x  1)( x  3)
ax  bx  c  0
2
2
b  b  4ac
x
2a
2
27
(quadratic 1 4 3)  (-1 -3)
rd
A Diversion ~ convexity versus concavity
Concave:
Convex:
28
rd
29
rd
More on quadratics
b  b  4ac
x
2a
2
If a, b, and c are real, then:
if b2 – 4ac > 0, then the roots are real and unequal
if b2 – 4ac = 0, then the roots are real and equal
if b2 – 4ac < 0, then the roots are imaginary and unequal
30
discriminant
rd
Interesting Facts about Quadratics
If x1 and x2 are the roots of a quadratic equation, then
Derived from the
quadratic formula
b
c
x1  x2 
; x1 x2 
a
a
and
x  ( x1  x2 ) x  x1 x2  0
2
31
rd
Equations Quadratic in form
x  x  12  0
4
quadratic in x2
2
( x  3)( x  4)  0
factoring
x  3  0 and x   3
Imaginary roots
2
2
2
x  4  0 and x   4  2i
2
A 4th degree polynomial has 4 roots
32
rd
The General Cubic Equation
f ( x)  ax  bx  cx  d
3
2
ax  bx  cx  d  0
3
2
Polynomials of odd degree must
have at least one real root because
complex roots occur in pairs.
33
rd
The easy cubics to solve:
ax  bx  cx  d  0
3
d 0
ax  bx  cx  0
3
2
x(ax 2  bx  c)  0
2
cd 0
bc0
ax  bx  0
ax  d  0
x 2 (ax  b)  0
d
x
a
3
2
x  0; ax 2  bx  c  0 x  0; ax  b  0
34
rd
3
3
The Power Function
(learning curves, production functions)
y  f ( x)  ax ; x  0, a  0
b
For b > 1, f(x) is convex (increasing slopes)
0 < b < 1, f(x) is concave (decreasing slopes)
For b = 0; f(x) = “a”, a constant
For b < 0, a decreasing convex function (if b = -1 then
f(x) is a hyperbola)
y  f ( x)  ax
35
b
a
 b ; x  0, b  0
x
rd
Learning Curves Cost & Time
(sim-LC 1000 10 90) 
Tn = T0 nb
Unit
Hours Cumulative
1
1000.00 1000.00
2
900.00 1900.00
3
846.21 2746.21
4
810.00 3556.21
5
782.99 4339.19
6
761.59 5100.78
7
743.95 5844.73
8
729.00 6573.73
9
716.06 7289.79
10
704.69 7994.48
The slope of 90% learning curve is -0.1520; consider any unit, say
5. 783 = 1000*5b => b = -152.
36
rd
Exponential Functions
(growth curves, probability functions)
f ( x)  c0 a ; a  0
c1 x
often the base is
e=2.718281828459045235360287471352662497757...
f ( x)  c0e
c1 x
For c0 > 0,
f(x) > 0
For c0 > 0, c1 > 0, f(x) is increasing
For c0 > 0, c1 < 0, f(x) is decreasing
y intercept = c0 ex > 0
37
rd
y = ex
y = ex y – eb = eb(x – b)
y' = ex (b, eb); intercept is x - 1
38
rd
y = ln x
39
rd
Law of Exponents
a a a
m
n
m n
2324 = 8 * 16 = 128 = 27
m
a
mn
m n

a

a
a
n
a
m n
mn
(a )  a
1
m
a  a
40
m
25/23 =32/8 = 4 = 22
(23)4 = 84 = 4096 = 212
21/2 = 1.41421356237 …
rd
Calculation Rules for Roots
Radical is
n
N  r1, r2 ,..., rn
Radicand is N, n is the root index.
41
rd
Properties of radicals
ab 
n
a

b
n
n
n
n
a
n
b   ab 
a a
 
b b
1
n
1
n
c n a  d n a  (c  d ) n a
but note:
n
42
ab  n a  n b
rd
Not a linear operator
Radar Beam
334.8F = vf where v is vehicle speed and f = 2500
megacycles.sec aimed at you. F is the difference
between the initial beam sent out and the reflected
beam. Were you speeding if the difference was 495
cycles/sec?
v = 334.8 * 495/2500 = 66.29 mph, perhaps not
speeding but driving a bit over the speed limit of 65
mph.
43
rd
Law of Exponents
a a a
m
n
m n
m
a
mn
m n
a a a
n
a
m n
mn
(a )  a
1
m
a  a
4
4
m
rd
Multiplying a Multinomial by a Multinomial
Using the distributive law, we multiply one of the
multinomials by each term in the other multinomial. We
then use the distributive law again to remove the
remaining parentheses, and simplify.
(x + 4)(x - 3) = x(x - 3) + 4(x - 3)
= x2 - 3x + 4x -12 = x2 + x -12
(x – a)(x – b)(x – c)(x – d) … (x – y) (x – z) = _______
(Poly* #(X 4 1) #(X -3 1))  #(X -12 1 1)
4
5
rd
Logarithmic Functions
(nonlinear regression, probability likelihood functions)
f ( x)  c0 log a x, a  1
base
natural logarithms, base e
f ( x)  c0 log e x  c0 ln x
note that logarithms are exponents: If x = ay then y = loga x
For c0 > 0, f(x) is a monotonically increasing
For 0 < x < 1, f(x) < 0
For x = 1, f(x) = 0 since a0 = 1
For x  0, f(x) is undefined
46
rd
Least Common Multiple (LCM)
the smallest positive integer that is divisible by the
numbers.
8
=2 2 2
8 16 24 32
40
12 = 2 2
3
12
24
36
9 =
3
3
9 18 27 36
15 =
3
5
LCM = 2 * 2 * 2 * 3 * 3 * 5 = 360
(lcm 8 12 9 15)  360
(div 360) 
(1 2 3 4 5 6 8 9 10 12 15 18 20 24 30 36 40 45 60 72 90 120 180
360)
47
rd
Greatest Common Divisor (GCD)
The largest positive integer that divides the numbers
with zero remainder
102 and 30
102 = 3 * 30 + 12
30 = 2 * 12 + 6
12 = 2 * 6 + 0
6 is gcd
(div 102)  (1 2 3 6 17 34 51 102)
(div 30)  (1 2 3 5 6 10 15 30)
48
rd
Friend Ben => Be = n
Log Base Number = Exponent
BaseExponent = Number
LogB N = E  BE = N
Log2 16 = 4  24 = 16
Log2 16 = ln 16 / ln 2 = 2.7725887/0.6931472 = 4
49
rd
Properties of Logarithms
ln( xy )  ln x  ln y
x
ln    ln x  ln y
 y
a
ln x  a ln x
log x
b
  log b x  log a b 
The all important change of bases: log a x 
log b a
log216 = ln16/ln 2 = 4
since letting y  logb a; then a  b ; and a
y
50
rd
1/ y
1
 b or log a b 
y
Properties
1. Log i(xi) = log(xi)
2. for all x and y, x > y implies log(x) > log(y)
3. for all x between 0 and 1, log(x) is negative
51
rd
Examples of logarithms
1. If Ln x = 2 Ln 3 - 3 Ln 2, then x =
52
___
2.
Log2 16 = 4 = Log2 42 = 2Log2 4 = 2*2 = 4
3.
Log2 2 = 1
4.
X=
a
log a x
Logarithms are exponents
X is called the anti-logarithm
1 can never be a base
rd
Logarithms
Common logarithm ~ lg to the base 10, log x
Ln ~ Natural logarithm with base e
Lb x Binary logarithm with base 2
Loga x logarithm of x to the base a
Loga 2 x = (loga x)2
Loga loga x = loga (loga x)
Let a = 2 and x = 16
Log2 log2 16 = log2 (log2 16) = log2 4 = 2.
(log (log 16 2) 2)  2
53
rd
Logarithm to any base using e
Find the log of 12 to the base 6. Repeat for 8 base 2
Take the log of 12 to the base e and divide by the
natural log of 6.
log612 = ln 12 /ln 6 = 1.3868…
log28 = ln 8 / ln 2 = 2.07944 /0.693147 = 3
54
rd
Logarithms
logab * logbc = logac for any a, b and c
Let a = 12, b = 37, and c = 59
Then log1237 = 1.45314026…
log 3759 =1.12922463…
log 1259 = 1.64092178…
(* 1.4531402 1.12922463)  1.64092178 …
55
rd
The absolute value function
(rectilinear distance problems, forecasting (MAD*), multicriteria decision-making)
 x  a for x  a
f ( x)  x  a  
( x  a) for x  a
*mean absolute deviation
a
56
rd
x
Properties of the absolute value
|ab| = |a| |b|
|a + b|  |a| + |b|
|a + b|  |a| - |b|
|a - b|  |a| + |b|
|a - b|  |a| - |b|
solve |x – 3| < 5
-5 < |x – 3| < 5
therefore -2 < x > 8
57
rd
Trigonometric Functions
(forecasting, bin packing problems)
f ( x)  a sin x  b cos( x)  c tan( x)
or cot, csc,sec
sin 2 x  cos 2 x  1
Identities:
csc2 x  cot 2 x  1
sec2 x  tan 2 x  1
1
1
cot x 
; csc 
tan x
sin x
1
sin x
sec x 
; tan x 
cos x
cos x
58
rd
squared relationships
c
a
b
reciprocal relations
Forecasting with Trig Functions
t = time
E(Yt )  b0  b1t  b2 sin  2 t /12  b3 cos  2 t /12
2
Quadratic trend with seasonal (monthly) effects
59
rd
sin  = DB = OE,
tan  = BC,
sec  = OC,
cos  = OD = EB,
cot  = AB,
csc  = OA
60
rd
Non-important Functions
Hyperbolic and inverse hyperbolic functions
Gudermannian function and inverse gudermannian
1
gd ( x)  2 tan e 
61
x
rd

2
Composite and multivariate functions
(multiple regression, optimal system design)
A common everyday composite function:
c
3
2
f ( x)  ax  bx   d ln x  e x
x
2
A multivariate function that may be found lying around the house:
f ( x, y, z )  a0  a1 x 2  a2 x  a3 y 2  a4 y  a5 z 2  a6 z
62
rd
The multi-variable polynomial
m
n
f ( x1 , x2 ,..., xm )   ai , j xi
j
i 1 j  0
where ai ,n  0 for at least one i
63
rd
Inequalities
An inequality is statement that one expression or
number is greater than or less than another.
The sense of the inequality is the direction, greater than
(>) or less than (<)
The sense of an inequality is not changed:
if the same number is added or subtracted from both
sides: if a > b, then a + c > b + c
if both sides are multiplied or divided by the same positive
number: if a > b, then ca > cb where c > 0
The sense of the inequality is reversed if both side sides
are multiplied or divided by the same negative number.
if a > b, then ca < cb when c < 0
64
rd
More on inequality
An absolute inequality is one which is true for all real values: x2 + 1 > 0
A conditional inequality is one which is true for certain values only:
x+2>5
Solution of conditional inequalities consists of all values for which the inequality is true.
(x – 2)(x – 3) > 0; x > 2 and x > 3
x < 2 and x < 3
2
3x 2  8 x  7  2 x  3x  1
f ( x)  x 2  5 x  6  0
roots : x  2,3
For x < 2; f(x) > 0
For 2 < x < 3, f(x) < 0
For x > 3, f(x) > 0
Therefore X < 2 and X > 3 are the solutions
65
rd
An absolute inequality
example problem: solve |x – 3| < 5
Write:
-5 < (x – 3) < 5
Conclude -2 < x < 8
for x > 3, (x - 3) < 5 => x < 8
for x  3, -(x - 3) < 5 => –x < 5 - 3 or x > -2
Therefore,
-2 < x < 8
66
rd
An important multi-valued function
(Euclidean distance problems, constrained optimization)
f ( x)  y   r  x ;  r  x  r
2
x y r
2
2
2
2
y
Pythagorean theorem
r
y
x
x y r
2
67
2
2
rd
x
Implicit and Inverse Functions
f ( x, y )  ax 2  by 2  cxy
implicit function
b
y  f ( y)  a 
x
b
ya 
x
b
x
ya
 b 
x  f ( y)  

y

a


explicit function
2
1
68
inverse function
rd
Inverse Functions
Inverse functions are symmetric around the line y = x.
Example: Let y = 2x + 3 implying the inverse function is
y = (x - 3)/2.
y = 2x + 3
y=x
y = (x – 3)/2
69
rd
The Devil’s Curve
y4 - x4 + ay2 + bx2 = 0
An implicit
relationship that
is not single-valued
70
rd
Symmetry
f(x, y) = 4x2 + 9y2 = 36
71
(-x, y)
(x, y)
(-x, -y)
(x, -y)
rd
Multiplying Polynomials
Example 6
Find the product (2t -3)(5t3 + 3t -1)
(poly*poly #(t -3 2) #(t -1 3 0 5)) 
#(T 3 -11 6 -15 10) ~ 3 – 11t + 6t2 – 15t3 +10t4
(poly^n #(x -3 2) 5) 
#(X -243 810 -1080 720 -240 32)
= -243 + 810x -1080x2 + 720x3 -240x4 +32x5
72
rd
Adding/Subtracting Polynomials
(poly+ #(x 1 2 3) #(x -3 -2 -1))
 #(X -2 0 2)
(poly- #(x 1 2 3) #(x -3 -2 -1))
 #(X 4 4 4)
73
rd
Create Poly with Roots
(my-make-poly '(1 2 3))  (1 -6 11 -6)
p(x) = x3 - 6x2 +11x - 6
(cubic 1 -6 11 -6)  (3 2 1)
Short Division
1 -6 11 -6 |3
3 -9 6
1 -3 2 0 |2
(quadratic 1 -3 2)  (2 1)
2 -2
1 -1 0 |1
1
1 0
74
rd
Binary and Decimal Base Numbers
23 22 21 20
8 4 2 1
Using only the digits 0 and 1,
1 1 0 1 = 8 + 4 + 1 = 13
1 0 1 1 = 8 + 2 + 1 = 11
1 1 0 0 0 = 16 + 8 = 24
103 102 101 100
1000 100 10 1
1
2 3 9 = 1(1000) + 2(100) + 3(10) + 9(1)
0
9 8 5 = 0(1000)+ 9(100) + 8(10) + 5(1) = 985
75
rd
Base Numbers
Write 246 in base 7
72 71 70
49 7 1
5 0 1
7 into 246 = 35 Remainder 1
7 into 35 = 5 R 0
7 into 5 = 0 R 5
Try 221 in base 3  73 R 2, 24 R 1, 8 R 0, 2 R 2, 0 R 2
22012
76
rd
Convert 324d to Base 2
32410 = 1010001002
162 0
81 0
40 1
20 0
10 0
5
0
2
1
1
0
0
1
77
rd
Base Arithmetic
For what bases does the number 121b represent a
square number? Check 1331b.
b2 + 2b + 1 = (b + 1)2
Touch all the bases to score a home run.
For example, in base 7 121 is 49 + 14 + 1 = 64
in base 9 121 is 81 + 18 + 1 = 100
in base 33 121 is 1089 + 66 + 1 = 1156 = 342
78
rd
Quadratic Equation in Base 5
Solve x2 + 3x + 2 = 0 in Base 5 where you have the
integers {0 1 2 3 4}
b  b  4ac
x
2a
2
3  9  8 3  1
x

 (1, 2)
2
2
3, 4 are the roots. Check.
79
rd
Base ?
2 3 5 11 15 21 25 ?
80
rd
Subtraction, Base 16, with
Hex Digits 0 1 2 3 4 5 6 7 8 9 A B C D E F
Base 6
54
-35
15



34
23
11
Base 16
D9  217
-AC  172
2D  45
(- #xD9 #xAC)  2D
AE16 + 768 = _________4
3230
81
rd
236
Irrational Number
Can you have an irrational number raised to an
irrational power and have the result be rational?
2
Yes, Proof: 2 is irrational and 2
is an irrational number raised to an irrational power and
is either rational or irrational. If rational, then done.
If not rational then the number below is.
 2


82
2



rd
2
Square Roots
1
1


9 16
7
5
a)
b)
12
12
83
3
c)
7
rd
2
d)
5
1
e)
5
Logarithm/Exponential Equation
Solve x + 3e2y - 8 = 0 for y in terms of x:
e2y = (8 – x)/3
2y Ln e = Ln((8 – x)/3)
y = (1/2) Ln((8 – x)/3)
8 x 
y  ln 

3


1/ 2
84
rd
Six steps to solving word problems
Picture the Problem
1.
Try to visualize the problem. Draw a diagram showing as much of the given
information as possible, including the unknown.
Understand the Words
2.

Look up the meanings of unfamiliar words in a dictionary, handbook, or
textbook.
Identify the Unknown(s) and the constants
3.

Be sure that you know exactly what is to be found in a particular problem
Summarize and write in mathematical form what is given
5. Estimate the Answer
4.

Write and Solve the Equation(s)
6.

8
5
It is a good idea to estimate or guess the answer before solving the
problem, so that you will have some idea whether the answer you finally get
is reasonable.
The unknown quantity must now be related to the given quantities by means
of equation(s).
rd
First word problem
In a group of 102 employees, there are three times as
many employees on the day shift as on the night shift,
and two more on the swing shift than on the night shift.
How many are on each shift?
Let x = number of employees on the night shift
Then 3x = number of employees on the day shift
And (x + 2) number of employees on the swing shift
x + 3x + (x + 2) = 102
5x = 100
x = 20 on the night shift
3x = 60 on the day shift
(x + 2) = 22 on the swing shift
8
6
rd
Financial Problem
A consultant had to pay income taxes of $4867 plus 28% of
the amount by which her taxable income exceeded
$32,450. Her tax bill was $7285. What was her taxable
income? Work to the nearest dollar.
Solution: Let x = taxable income (dollars).
The amount by which her income exceeded $32,450
is then x - 32,450
Her tax is 28% of that amount, plus $4867, so
tax = 4867 + 0.28(x - 32,450) = 7285
Solving for x, we get x = $41,086
8
7
rd
Mixture Problems
From 100.0 kg of solder, half lead and half zinc, 20.0 kg are
removed. Then 30.0 kg of lead are added. How much
lead is contained in the final mixture?
Solution:
initial amount of lead = 0.5(100.0) = 50.0 kg
40L
amount of lead removed = 0.5(20.0) =10.0 kg
amount of lead added = 30.0 kg
final amount of lead = 50.0 - 10.0 + 30.0 = 70.0 kg
8
8
rd
More Mixture Problems
How much steel containing 5.25% nickel must be combined
with another steel containing 2.84% nickel to make 3.25
tons of steel containing 4.15% nickel?
Let x = tons of 5.25% steel needed.
5.25 a + 2.84b = 4.15 * 3.25; a + b = 3.25
The amount of 2.84% steel is (3.25 – x)
The amount of nickel that it contains is 0.0284(3.25 – x)
The amount of nickel in x tons of 5.25% steel is 0.0525x
The sum of these must give the amount of nickel in the final
mixture. 0.0525x + 0.0284(3.25 – x) = 0.0415(3.25)
x = 1.77 t of 5.25% steel
3.25 * x = 1.48 t of 2.84% steel
8
9
rd
Sequences
90
rd
Sequences & Series
A sequence is a progression of ordered numbers: 3, 10, 19, 37, …
such that the preceding and following numbers are completely
specified.
In an arithmetic sequence the terms have a common difference:
1, 4, 7, 10, ….
In an harmonic sequence the terms are reciprocals of the terms in
an arithmetic sequence: 1, 1/4, 1/7, 1/10, ….
In a geometric sequence the terms have a common ratio:
1, 3, 9, 27, ….
A series is the sum of the terms of a sequence 1 + 3 + 9 + 27
Series are either finite or infinite; convergent or divergent
i
1 1 1
1
1
1

1






 
2 4 8 16 32
i 0  2 
5
rd
91
Arithmetic Sequences
Complete the sequences at the *
a) 2 7 12 17 * *
b) 5 13 21 * *
c) 11 15 * 23 *
d) * * 20 29 38
e) 4 * 18 * 32
f) * 33 * 65 *
g) 10 * 70
h) 10 * * 70
i) 10 * * * * 70
j) If each term of an arithmetic sequence is multiplied by a
constant, is the resulting sequence arithmetic?
a, a + d, a + 2d, a + 3d versus
ka, k(a + d) k(a + 2d) k(a + 3d)
92
rd
Arithmetic Sequences
a) The 100th term of 2 5 8 11 14 * * * is ____.
ans. 299
b) b) The 20th term of 11 15 19 23 * * * is ____.
ans. 87
c) Find the sum of the sequence: 3 7 11 15 19 23 27
Add 3 7 11 15 19 23 27
+ 27 23 19 15 11 7 3
30 30 30 30 30 30 30
=> sum = 7(30)/2 = 105
d) Find the sum of the first 100 integers. n(n+1)/2
93
rd
Arithmetic Series
Sum Sn of a finite arithmetic series is given by
Sn = n(a1 + an)/2
Example: 2 + 4 + 6 + 8 + . . . + 100 = 50(2 + 100)/2 = 2550;
where n = 100/2 = 50; a1 = 2; an = 100
1 + 5/3 + 7/3 + . . . + 201 =
1 + (n – 1)(2/3) = 201 => n = 301 terms
=> Sn = 301(1 + 201)/2 = 30,401
94
rd
.
Harmonic Series
Arithmetic sequence 1 4 7 10 13 16 19 22
Reciprocals: 1 1/4 1/7 1/10 1/13 1/16 1/19 1/22
is an harmonic sequence
(+ 1 1/2 1/4 1/8 1/16 1/32 1/64 1/128 1/256 1/512 1/1024
1/2048 1/4096 1/8192 1/16384 1/32768 1/65536 1/131072
1/262144 1/524288 1/1048576)  2097151 / 1048576
= 1.9999999999…
(let ((x 0))
(dotimes (i 100 x) (incf x (recip (expt 2 i)))))  2
95
rd
Harmonic Series
Find the 36th term of the series
1 + 1/4 + 1/7 + 1/10 + 1/13 + …
The arithmetic series is 1 4 7 10 13 …
and the 36th term is 1 + 35*3 = 106 => 1/106.
96
rd
Harmonic Series
A cyclist travels from A to B at 40 mph and returns at 60
mph. The average speed for the round trip is .
a) 48
b) 49
c) 50
d) 51 e) none of these
(1 / [(1/40 + 1/60)/ 2] = 48
Apply sensitivity analysis to explain why.
97
rd
Geometric Series
Sum Sn of a geometric series of n terms is given by
1  r n a1  ran
Sn = a1

; r  1.
1 r
1 r
Find the sum of the geometric series 1 4 16 64 256 1024.
Sum = (1 – 4 * 1024)/(1 – 4) = 1365
Find the sum of the geometric series 3 18 108 . . . 839,808.
(3 – 6 * 839,808)/(1 – 6) = 1,007,769
98
rd
Geometric Series
Find the sum of the following geometric series:
(1 + i)0 + (1 + i)1 + (1 + i)2 + (1 + i) 3
Sum = (1 + i)0 - (1 + i)(1 + i)3
1 - (1 + i)
= 1 - (1 + i)4
-i
A A A A
(F/A, 6%, 4)
= (1 + i)4 – 1
i
= F/A = [(1 + i)n – 1]/i = 4.37462 at i = 6%
99
rd
Sequences
1. Write the first 5 terms of {1 – 1/(2n)}
½ ¾ 4/5 5/6 7/8 9/10
2. Repeat #1 for {½[(-1)n + 1]}
01010
3. Write the general term of 1, 1/3, 1/5, 1/7, 1/9
Look at reciprocals 1 3 5 7 9
General term is 1/(2n – 1)
100
rd
Sequences (continuing)
1. 102 103 105 107 111 113 ?
2. 3 15 14 7 18 1 20 21 12 1 20 9 15 14 ?
3. 2 12 36 80 150 252 392 ?
4. 3 5 6 2 9 5 1 4 1 ?
5. 1 1 2 3 5 8 13 ?
101
rd
Triangle Inequality
|a + b|  |a| + |b|
If both non-negative, |a + b| = a + b = |a| + |b|
If both negative, |a| = -a; |b| = -b and a + b is negative
then |a + b| = -(a + b) = -a + (-b) + |a| + |b|
If a > 0 and b < 0, then |a| = a, |b| = -b
If |a| > |b|, then |a + b| = a + b < a – b + |a| + |b|
If |a| = |b|, then |a + b| = 0 < |a| + |b|
etc.
102
rd
US Currency
Bills: 1 2 5 10 20 50 100 500 1,000 5,000 10,000 100,000
Find a geometric sequence of bills with common ratio 10.
1 100 10,000 100,000
103
rd
Pyramid Scheme
104
rd
Fibonacci Sequence
Recursive – up a staircase one or two steps at a time
with n steps
n # of ways
1
1
2
2: 1, 2
3
3: 111,12, 21
4
5: 1111, 112, 121, 211, 22
fn = fn-1 + fn-2
For n = 5 steps, take 1 step 45 ways
11111,1112,1121,1211,122
thereafter or 2 steps 33 ways 2111, 212, 221
thereafter yielding 5 + 3 = 8
105
rd
Fibonacci Sequence
(1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584
4181 6765 10946)
Which terms are evenly divisible by 3, 5, 8, 13 and 55?
Which term is the largest cube?
106
rd
Intelligence Testing
1. John is twice as old as his sister Mary, who is now 5 years
of age. How old will John be when Mary is 30 years of
age?
2. Mary is 24 years old. She is twice as old as Ann was when
Mary was as old as Ann is now. How old is Ann?
Let x = Ann's age: 24 = 2[x – (24 – x)]
107
rd
Numerology
How much wheat can be put on a chessboard with 1
grain on the first square, 2 on the next, 4 on the third
etc.?
S = 1 + 2 + 4 + 8 + 16 + 32 + … + 263
= (1 – 264) / (1 – 2)
= 18,446,744,073,709,551,615 grains of wheat
Roughly a train reaching a thousand times around the
Earth filled with wheat.
108
rd
Differential Equation for e
y’ – y = 0
Assume y = ex = a0 + a1x + a2x2 + a3x3 + a4x4 + a5x5+ . .
Then
and
y’ = a1 + 2 a2 x + 3 a3 x2 + 4 a4 x3 + 5 a5 x4 + . . .
y(0) = 1 => a0 = 1
y’(0) = 1 => a1 = 1
y’’(0) = 1 => 2a2 = 1 => a2 ½
y’’’(0) = 1 => 6a3 = 1 => a3 = 1/6
ex = 1 + x + x2/2! + x3/3! + …
109
rd
Infinite Series
Two motorcyclists A and B, 100 miles apart, head for
each other. A travels at 40 mph and B at 60 mph. A
fly flies from A’s nose to B’s nose and back again
and again at 70 mph. How far will the fly have flown
when the two cyclists meet?
Infinite series whose terms increase in magnitude
have no attainable sum. If a sum exists, series is
said to be convergent; if not, divergent.
110
rd
Pythagorean Triples
m
n
m2 – n2
2mn m2 + n2
3
2
5
12
13
6
1
35
12
37
6
5
11
60
61
7
6
13
84
85
8
7
15
112
113
60
11
3479
1320 3721 = 612
84
13
6887
2184
7225 = 852
10
6
64 = 43 120
136
6
3
27 = 33
36
45
Try one yourself by picking an m > than n.
111
rd
Asking for a Raise
Would you rather receive a raise in salary of $300 every
6 months or $1000 every year?
112
rd
1000 Lockers
There are 1000 lockers and all are opened. Then I go
by each and reverse the state. If open, I close it; if
closed, I open it. Then I repeat for every 2 lockers,
then every 3 lockers , etc. When done, what are the
states of the lockers?
What are you modeling mathematically?
113
rd
Chord Length
Express the length L of a chord of circle with radius r as
a function of x being the distance from the center.
L = 2(r2 - x2)1/2
r
L/2
x
Find length of chord in circle of radius 13 that is 5 units
from the center. L = 2(169 – 25)1/2 = 24.
114
rd
Random
What does random mean? What is the probability that
a randomly drawn chord is shorter than the leg of the
equilateral triangle in the circle? 2/3 Repeat for the
perpendicular at the midpoint of the radius. 1/4
A
(comb 2 2)
B
C
Which answer is correct? Both are mathematically
correct. What does "at random" mean?
115
rd
Radioactive Decay
N = N0e-t for t in days
Given an element N = 100e-0.062t , find the initial amount, the
half life, and verify that the amount at half life is half of the
initial amount. How much is present after 9 days?
N0 = 100e-0.062 * 0 = 100
N/N0 = ½ = e-0.062t Solve for t to get 11.1788 days as the
half life.
N = 100e-0.062 * 11.1788 = 50
N9 = 100e-0.062 * 9 = 57.235 mg
Sanity check: As 9 is less than half life, expect more than
50 mg.
116
rd
Rationalizing Denominator
Rationalize 1/21/2
Multiply numerator and denominator by 21/2
to get 21/2 /2
117
rd
Spurious Roots
x – 4 + (x – 2)1/2 = 0
(4 – x)2 = x – 2
16 – 8x + x2 = x -2
(quadratic 1 -9 18)  (3, 6)
Check 6: 6 – 4 = (6 – 2)½
2=2
3: 3 – 4 + (3 – 2½
-1  1 reject 3; accept 6.
118
rd
Spurious Roots
(5x2 + 10x – 6)½ = 2x + 3
5x2 + 10x – 6 = 4x2 +12x + 9 :Squaring both sides
Solve to get (quadratic 1 -2 -15)  (5, -3)
Check to see that -3 is spurious and rejected, but that
root 5 checks OK.
119
rd
Logarithmic Equations
(ln x)2 - 2 ln x - 3 = 0 Solve for x
Let y = ln x
Then y2 – 2y – 3= 0 or (y - 3)(y + 1) = 0
ln x = 3; ln x = -1
x = e3; e-1
120
rd