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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 (f (f (f (f g )( x) f ( x) g ( x) g )( x) f ( x) g ( x) g )( x) f ( x) g ( x) / g )( x) f ( x) / g ( x) 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 •