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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!