Number and Algebra lecture 11
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Transcript of Number and Algebra lecture 11
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Number and Algebra lecture 11
Polynomial rings,
Functions
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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.
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• 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.
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• Euler (1707 – 1783) regarded a function as any equation or formula.
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• 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.
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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
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Function
• A relation satisfying the univalence property.
• Univalence Property: x domain(f),
a unique y range(f) such that
f(x) = y.
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Function Concept Table
Representation
Object
Process
AlgebraicGraphic Numeric VerbalInterpretation
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Function Translation
Curve Sketching
Computing Values
Recognize Formula
Algebraic
Curve Fitting
Reading Values
Interpret Graph
Graphic
Fitting dataPlottingReadingNumeric
ModelingSketchingMeasuringVerbal
AlgebraicGraphicNumericVerbalTo
From
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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.
7 6 4 1 2 8 7 5 3 Y
9 8 7 6 5 4 3 2 1 X
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• 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
More Function Misconceptions
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• Functions must be smooth, they cannot have corners.
y = | x |
• Functions must be continuous.
0,1
0,
11
)1)(1(
2
xx
xxy
xy
x
xxy
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Function Tests
• Geometric: Vertical Line Test
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Function Tests
• Algebraic: f is a function iff
x1 = x2 implies that f(x1) = f(x2).
• Function Diagram
Domain Range
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Process Interpretation of Function
• A function is a dynamic process assigning each domain value a unique range value.
FunctionDomain
Range
Input x
Output f(x)
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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
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Constant Function
Ex: f(x) = 5
Domain:
Range:
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Identity Function
f(x) = x
Domain:
Range:
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Square Function
f(x) = x2
Domain:
Range:
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Cube Function
f(x) = x3
Domain:
Range:
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Square Root Function
Domain:
Range:
xxf )(
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Reciprocal Function
Domain:
Range:
xxf
1)(
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Absolute Value Function
Domain:
Range:
xxf )(
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Object Interpretation of Function
A function is a static object or thing
Allows for:
• Trend Analysis
• Classification
• Operation
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Function as Object: Trend Analysis
The graph below represents a trip from home to school. Interpret the trends.
School
Hometime
distance
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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 0 2 -2 7 -7y 5 3 3 -9 -9
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Classify as even or odd:
2. 3. y = x2 + 5
4. y = x5 + 3x3 - x
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Function as Object: Operation
Given two functions f(x) and g(x), we can combine them to get a new function:
))(())((
)(/)())(/(
)()())((
)()())((
)()())((
xgfxgf
xgxfxgf
xgxfxgf
xgxfxgf
xgxfxgf
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Inverse
• Inverse: to turn inside out, to undo
• Additive Inverse: a + (-a) = 0
• Multiplicative Inverse: a • (1/a) = 1
• Pattern: (element) * (inverse) = identity
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Function Identity
Let i(x) represent the identity, then for any function f(x) we have
Ex: f(x) = 5x + 2, then
What is i(x)?
)()()( xfxixf
2)]([5))(())(( xixifxif
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Function Inverse
Given identity is i(x)=x, f -1(x) is a function such that
xxff ))(( 1
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What is the inverse for the function in table/numeric form?
1. x 1 2 3 4y 2 8 7 5
2. x 1 -1 3 7y 2 2 5 8
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What is the inverse for the function in graphic form?
1. 2.
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What is the inverse for the function f(x)=3x+5 in algebraic form?
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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
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• 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 non-commutative algebras. These are algebras in which the operation of multiplication is not required to be commutative.
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• ((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 non-commutative algebra).
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• 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
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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
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• Coefficients ai.
• Polynomial in x over S.
• Term of Polynomial aixi .
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Francis Sowerby MacaulayBorn: 11 Feb 1862 in Witney,
EnglandDied: 9 Feb 1937 in Cambridge,
Cambridgeshire, England
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• 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.
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Wolfgang KrullBorn: 26 Aug 1899 in Baden-
Baden, GermanyDied: 12 April 1971 in Bonn,
Germany
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• 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.
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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
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• x -2 + 2x – 5
• x1/2 + ½ x2 + 3
• ni=0 aixi
• 2 + x3 – 2x5
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Polynomial Ring
• Is (S [x],+,• ) a polynomial ring?
• Is (S [x],+,• ) a commutative ring?
• Is (S [x],+,• ) a ring with unity?
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Closure +
r
i
iii xbaxgf
0
)())((
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Closure •
nm
i
ii
kkik xbaxgf
0 0
)())((
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Commutative & Associative for + and •
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Identity +
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Inverse +
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Identity •