Post on 29-Dec-2015
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ENM 503 Block 1 Algebraic SystemsLesson 4 – Algebraic Methods
The Building Blocks - Numbers, Equations, Functions, and other interesting things.
Did you know? Algebra is based on the concept of unknown values called variables, unlike arithmetic which is based entirely on known number values.
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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} Irrational Numbers
{non-terminating, non-repeating decimals} e.g. transcendental numbers – irrational numbers that cannot
be a solution to a polynomial equation having integer coefficients (transcends the algebraic operations of +, -, x, / ).
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More Real Numbers
Real Numbers
Rational (-4/5) = -0.8 Irrational
Transcendental (e.g. e = 2.718281… = 3.1415927…)
Integers (-4)
Natural Numbers (5)
2 1.41421...
Did you know? The totality of real numbers can be placed in a one-to-one correspondence with the totality of the points on a straight line.
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Numbers in sets
transcendental numbers
Did you know? That irrational numbers are far more numerous thanrational numbers? Consider where a and b are integers / , 1, 2,3,...n a b n
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Algebraic Operations Basic Operations
addition (+) and the inverse operation (-) multiplication (x) and the inverse operation ( )
Commutative Law a + b = b + a a x b = b x a
Associative Law a + (b + c) = (a + b) + c a(bc) = (ab)c
Distributive Law a(b + c) = ab + ac
Law and order will prevail!
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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 f(x).
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
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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)
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1 2
31 2 2
1
( )
( ) 3 4 / ln(.2 )
( , ,..., )
( , )
n
y f x
y f x x x x
z f x x x
af x x bx
x
dependentvariable
independentvariable
examples: function ofn variables
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On DomainsSuppose that the function f is specified
algebraically by the formula
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).
( )1
xf x
x
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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).
x
y
| | | | | |
_______
a linear function
the zero of the functionor roots of the equation f(x) = 0
y intercept
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Graph of a nonlinear function
3 2( )f x ax bx cx d
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Polynomials in one variable
Polynomials are functions having the following form:
2 30 1 2 3
0 1
20 1 2
( ) ...
( )
( )
nnf x a a x a x a x a x
f x a a x
f x a a x a x
nth degree polynomial
linear function
quadratic function
Did you know: an nth degree polynomial has exactly n roots; i.e. solutions to the equation f(x) = 0
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Facts on Polynomial Equations
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.
2 30 1 2 3 ... 0n
na a x a x a x a x
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The Quadratic Function
Graphs as a parabola vertex: x = -b/2a if a > 0, then convex (opens upward) if a < 0, then concave (opens downward)
Solving quadratic equations: Factoring Completing the square Quadratic formula
2 0ax bx c
2( ) , 0f x ax bx c a
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The Quadratic Formula
2
2
2
( )
0
4
2
f x ax bx c
ax bx c
b b acx
a
Then it has two solutions.This is a 2nd
degree polynomial.
Quick student exercise: Derive the quadratic formulaby completing the square
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A Diversion – convexity versus concavity
Concave:
Convex:
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More on quadratics
If a, b, and c are real numbers, 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
2 4
2
b b acx
a
discriminant
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Equations Quadratic in form
4 2
2 2
2
2
12 0
( 3)( 4) 0
3 0 and 3
4 0 and 4 2
x x
x x
x x
x x i
quadratic in x2
factoring
of no interest
A 4th degree polynomial will have 4 roots
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The General Cubic Equation
3 2
3 2
( )
0
f x ax bx cx d
ax bx cx d
…and the cubic equation has three roots, at least
one of which will always be real.
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The easy cubics to solve:
3
3
0
0
b c
ax d
dx
a
3 2 0ax bx cx d
3 2
2
2
0
0
( ) 0
0; 0
d
ax bx cx
x ax bx c
x ax bx c
3 2
2
0
0
( ) 0
0; 0
c d
ax bx
x ax b
x ax b
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The Power Function(learning curves, production functions)
( ) ; 0, 0by f x ax x a 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)
( ) ; 0, 0bb
ay f x ax x b
x
The Graph
21( ) ; 0, 0by f x ax x a
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Exponential Functions(growth curves, probability functions)
1
1
0
0
( ) ; 0
( )
c x
c x
f x c a a
f x c e
often the base is e = 2.7181818…
For c0 > 0, f(x) > 0For c0 > 0, c1 > 0, f(x) is increasingFor c0 > 0, c1 < 0, f(x) is decreasingy intercept = c0
The Graph
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10( ) c xg x c e
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Law of Exponents
( )
m n m n
mm n m n
n
m n mn
a a a
aa a a
a
a a
You must obey these
laws.
1mma a More on radicals
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Properties of radicals
1
1
( )
n n n n
n nn
n
n n n
ab a b ab
a a a
b bb
c a d a c d a
Who are you calling a radical?
but note:
n n na b a b
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Logarithmic Functions(nonlinear regression, probability likelihood functions)
0
0 0
( ) log , 1
( ) log ln
a
e
f x c x a
f x c x c x
natural logarithms, base ebase
note that logarithms are exponents: If x = ay then y = loga x
For c0 > 0, f(x) is a monotonically increasingFor 0 < x < 1, f(x) < 0For x = 1, f(x) = 0 since a0 = 1For x 0, f(x) is undefined
Graph of a log function
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Properties of Logarithmsln( ) ln ln
ln ln ln
ln lna
xy x y
xx y
y
x a x
The all important change of bases: loglog log log
logb
a b ab
xx x b
a
1/ 1since letting log ; then ; and or logy y
b ay a a b a b by
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The absolute value function
for( )
( ) for
x a x af x x a
x a x a
xa
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Properties of the absolute value
|ab| = |a| |b||a + b| |a| + |b||a + b| |a| - |b||a - b| |a| + |b||a - b| |a| - |b|
Quick “bright” student exercise: demonstrate the inequality
really nice example problem: solve |x – 3| = 5then x - 3 = 5and – (x - 3) = 5 or –x + 3 = 5therefore x = -2 and 8
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Non-important Functions
Trigonometric, hyperbolic and inverse hyperbolic functions
Gudermannian function and inverse gudermannian
1( ) 2 tan2
xgd x e I bet you
didn’t know this one!
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Composite and multivariate functions(multiple regression, optimal system design)
2 3 2( ) lnc
f x ax bx d x e xx
A common everyday composite function:
A multivariate function that may be found lying around the house:
2 2 20 1 2 3 4 5 6( , , )f x y z a a x a x a y a y a z a z
Why this is just a quadratic in 3
variables. Is this some kind of a trick
or what?
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A multi-variable polynomial
1 2
20 1,1 1 1,2 1 .
,1 0
( , ,..., )
...
m
nm n m
m nj
i j ii j
f x x x
a a x a x a x
a x
Gosh, an m
variable polynomial of
degree n. Is that something or what!
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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 where c < 0
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More on inequalities
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.
An Inequality Example
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2 2
2
3 8 7 2 3 1
( ) 5 6 0
roots : 2,3
x x x x
f x x x
x
For x < 2; f(x) > 0For 2 < x < 3, f(x) < 0For x > 3, f(x) > 0Therefore X<2 and X>3
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An absolute inequality
example problem: solve |x – 3| < 5
for x > 3, (x-3) < 5 or x < 8
for x 3, -(x-3) < 5 or –x < 5 - 3 or x > -2
therefore -2 < x < 8
I would rather solve algebra problems than do just about anything
else.
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Implicit and Inverse Functions
2 2
Explicit function:
( )
Implicit function:
( , )
by f x a
x
f x y ax by cxy
2
1
Inverse Function:
( )
by a
xb
xy a
bx f x
y a
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Finding your roots…2 3 5
0 1 2 3 4
2 3
( ) ln 0
10( ) 100 2 .4 .008 1.5ln 0
af x a a x a x a x a x
x
f x x x x xx
Find an x such that Min f(x)2
Professor, I just don't
think it can be done.
See the Solver tutorialOn finding your roots
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We End with the Devil’s Curve
y4 - x4 + a y2 + b x2 = 0
An implicit relationship thatis not single-valued
This is my curve.
Did you know: There are not very many applications of this curve in the ENM or MSC program.
Quick student exercise: confirm the graph!