Transcript Slide 1

Number and Algebra lecture
11
Polynomial rings,
Functions
History Of Function Concept
• CA 200 BC Function concept has origins
in Greek and Babylonian mathematics.
• Babylonian Tablets for finding squares and
roots.
• Middle Ages: mathematicians expressed
generalized notions of dependence
between varying quantities using verbal
descriptions.
• Late 16th – Early 17th Century – Galileo
and Kepler study physics, notation to
support this study lead to algebraic
notation for function.
• Leibniz (1646 – 1716) introduces term
“function” as quantity connected to a
curve.
• Bernoulli(1718) interprets function as any
expression made up of a variable and
constants.
• Euler (1707 – 1783) regarded a function
as any equation or formula.
• Clairant (1734) developed notation f(x),
functions were viewed as well-behaved
(smooth & continuous).
• Dirichlet (1805-1859) introduced concept
of variables in a function being related as
well as each x having a unique image y.
Question
• What is your definition of function?
• Which of the following are functions under
Euler’s definition? Under Dirichlet’s
definition?
• x2 + y2 = 25
• f(x) = 0 if x is rational
1 if x is irrational
Function
• A relation satisfying the univalence
property.
• Univalence Property:  x  domain(f),
 a unique y  range(f) such that
f(x) = y.
Function Concept Table
Representation
Interpretation
Process
Object
Verbal
Numeric
Graphic
Algebraic
Function Translation
To Verbal
Numeric
Graphic
Algebraic
Measuring
Sketching Modeling
From
Verbal
Numeric
Reading
Graphic
Interpret
Graph
Plotting
Reading
Values
Algebraic Recognize Computing Curve
Formula
Values
Sketching
Fitting data
Curve
Fitting
Function Misconceptions
• Functions must have an algebraic rule.
For every value of x choose a
corresponding value of y by rolling a
die.
• Tables are not functions.
X
1
2
3
4
5
6
7
8
9
Y
3
5
7
8
2
1
4
6
7
More Function Misconceptions
• Functions can have only one rule for all
domain values.
x + 1 if x  0
y = 2x + 1 if x > 0
• Functions cannot be a set of disconnected
points.
x
if x is even
y = 2x if x is odd
• Any equation represents a function.
x2 + y2 = 25
• Functions must be smooth, they cannot
have corners.
y= |x|
• Functions must be continuous.
( x  1)(x  1)
y
x 1
1
y
x
 x2 , x  0
y
 x  1, x  0
Function Tests
• Geometric: Vertical Line Test
Function Tests
• Algebraic: f is a function iff
x1 = x2 implies that f(x1) = f(x2).
• Function Diagram
Domain
Range
Process Interpretation of
Function
• A function is a dynamic process assigning
each domain value a unique range value.
Domain
Input x
Function
Output f(x)
Range
Process Interpretation Tasks
• Evaluating a function at a point
– Ex: Find f(2) when f(x) = 3x - 5
• Determining Domain and Range
– Ex: Determine the domain and range of the
seven basic algebraic functions
Constant Function
Ex: f(x) = 5
Domain:
Range:
Identity Function
f(x) = x
Domain:
Range:
Square Function
f(x) = x2
Domain:
Range:
Cube Function
f(x) = x3
Domain:
Range:
Square Root Function
f ( x)  x
Domain:
Range:
Reciprocal Function
1
f ( x) 
x
Domain:
Range:
Absolute Value Function
f ( x)  x
Domain:
Range:
Object Interpretation of Function
A function is a static object or thing
Allows for:
• Trend Analysis
• Classification
• Operation
Function as Object:
Trend Analysis
The graph below represents a trip from
home to school. Interpret the trends.
distance
Home
School
time
Function as Object:
Classification
•A function that is symmetric to the y-axis is
said to be even.
•A function that is symmetric about the origin
is said to be odd.
•Classify the following as even or odd:
1.
x
y
0
5
2
3
-2 7
3 -9
-7
-9
Classify as even or odd:
2.
3. y = x2 + 5
4. y = x5 + 3x3 - x
Function as Object: Operation
Given two functions f(x) and g(x), we can
combine them to get a new function:
( f  g )(x)  f ( x)  g ( x)
( f  g )(x)  f ( x)  g ( x)
( f  g )(x)  f ( x)  g ( x)
( f / g )(x)  f ( x) / g ( x)
( f  g )(x)  f ( g ( x))
Inverse
• Inverse: to turn inside out, to undo
• Additive Inverse: a + (-a) = 0
• Multiplicative Inverse: a • (1/a) = 1
• Pattern: (element) * (inverse) = identity
Function Identity
Let i(x) represent the identity, then for any
function f(x) we have
f ( x)  i ( x)  f ( x)
Ex: f(x) = 5x + 2, then
( f  i)(x)  f (i( x))  5[i( x)]  2
What is i(x)?
Function Inverse
Given identity is i(x)=x, f -1(x) is a function
such that
1
( f  f )(x)  x
What is the inverse for the function in
table/numeric form?
1.
x
y
1
2
2
8
3
7
4
5
2.
x
y
1
2
-1 3 7
2 5 8
What is the inverse for the function in
graphic form?
1.
2.
What is the inverse for the function
f(x)=3x+5 in algebraic form?
Abstract Algebra
• In the 19th century British mathematicians
took the lead in the study of algebra.
• Attention turned to many "algebras" - that
is, various sorts of mathematical objects
(vectors, matrices, transformations, etc.)
and various operations which could be
carried out upon these objects.
MORE INFO
• http://www.math.niu.edu/~beachy/aaol/frames_index.html
• Thus the scope of algebra was expanded
to the study of algebraic form and
structure and was no longer limited to
ordinary systems of numbers.
• The most significant breakthrough is
perhaps the development of noncommutative algebras. These are algebras
in which the operation of multiplication is
not required to be commutative.
• ((a,b) + (c,d) = (a+b,c+d) ;
• (a,b) (c,d) = (ac - bd, ad + bc)).
• Gibbs (American, 1839 -1903) developed
an algebra of vectors in three-dimensional
space.
• Cayley (British, 1821-1895) developed an
algebra of matrices (this is a noncommutative algebra).
• The concept of a group (a set of
operations with a single operation which
satisfies three axioms) grew out of the
work of several mathematicians
• …and then came the concepts of rings
and fields
Polynomial in x with coefficients
in S
• Let S be a commutative ring with unity
• Indeterminate x – symbol interpretation of
variable.
• A polynomial is an algebraic expression of
the form
ao xo + a1x1+ a2x2 + …. + anxn
where n  Z+ U {0} ai  S
• Coefficients ai.
• Polynomial in x over S.
• Term of Polynomial aixi .
Francis Sowerby Macaulay
Born: 11 Feb 1862 in Witney,
England
Died: 9 Feb 1937 in Cambridge,
Cambridgeshire, England
• Macaulay wrote 14 papers on algebraic
geometry and polynomial ideals.
• Macaulay discovered the primary
decomposition of an ideal in a
polynomial ring which is the analog of
the decomposition of a number into a
product of prime powers in 1915.
• In other words, in today's terminology,
he is examining ideals in polynomial
rings.
Wolfgang Krull
Born: 26 Aug 1899 in BadenBaden, Germany
Died: 12 April 1971 in Bonn,
Germany
• Krull's first publications were on rings
and algebraic extension fields.
• He was quickly recognized as a
decisive advance in Noether's
programme of emancipating abstract
ring theory from the theory of
polynomial rings.
Question
Which of the following are polynomials?
• Let S = {ai  ai is an even integer}, then is
ao xo + a1x1+ a2x2 + …. + anxn
a polynomial?
• Let S = Z, then is
ao xo + a1x1+ a2x2 + …. + anxn
a polynomial?
• 5x3 – ½ x2 + 2i x + 5 where S = C
• x -2 + 2x – 5
• x1/2 + ½ x2 + 3
• ni=0 aixi
• 2 + x3 – 2x5
Polynomial Ring
• Is (S [x],+,• ) a polynomial ring?
• Is (S [x],+,• ) a commutative ring?
• Is (S [x],+,• ) a ring with unity?
Closure +
r
( f  g )(x)   (ai  bi ) x i
i 0
Closure •
m n
i
i 0
k 0
( f  g )(x)   ( ak  bi k ) x i
Commutative & Associative for
+ and •
Identity +
Inverse +
Identity •