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Number and Algebra lecture 11

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.

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Number and Algebra lecture 11

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  1. Number and Algebra lecture 11 Polynomial rings, Functions

  2. 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.

  3. 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.

  4. Euler (1707 – 1783) regarded a function as any equation or formula.

  5. 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.

  6. 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

  7. Function • A relation satisfying the univalence property. • Univalence Property:  x  domain(f),  a unique y  range(f) such that f(x) = y.

  8. Interpretation Verbal Numeric Graphic Algebraic Process Object Function Concept Table Representation

  9. To From Verbal Numeric Graphic Algebraic Verbal Measuring Sketching Modeling Numeric Reading Plotting Fitting data Graphic Interpret Graph Reading Values Curve Fitting Algebraic Recognize Formula Computing Values Curve Sketching Function Translation

  10. X 1 2 3 4 5 6 7 8 9 Y 3 5 7 8 2 1 4 6 7 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.

  11. 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

  12. Functions must be smooth, they cannot have corners. y = | x | • Functions must be continuous.

  13. Function Tests • Geometric: Vertical Line Test

  14. Function Tests • Algebraic: f is a function iff x1 = x2 implies that f(x1) = f(x2). • Function Diagram Domain Range

  15. Domain Input x Function Output f(x) Range Process Interpretation of Function • A function is a dynamic process assigning each domain value a unique range value.

  16. 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

  17. Constant Function Ex: f(x) = 5 Domain: Range:

  18. Identity Function f(x) = x Domain: Range:

  19. Square Function f(x) = x2 Domain: Range:

  20. Cube Function f(x) = x3 Domain: Range:

  21. Square Root Function Domain: Range:

  22. Reciprocal Function Domain: Range:

  23. Absolute Value Function Domain: Range:

  24. Object Interpretation of Function A function is a static object or thing Allows for: • Trend Analysis • Classification • Operation

  25. School distance time Home Function as Object: Trend Analysis The graph below represents a trip from home to school. Interpret the trends.

  26. x 0 2 -2 7 -7 y 5 3 3 -9 -9 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.

  27. Classify as even or odd: 2. 3. y = x2 + 5 4. y = x5 + 3x3 - x

  28. Function as Object: Operation Given two functions f(x) and g(x), we can combine them to get a new function:

  29. Inverse • Inverse: to turn inside out, to undo • Additive Inverse: a + (-a) = 0 • Multiplicative Inverse: a • (1/a) = 1 • Pattern: (element) * (inverse) = identity

  30. 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)?

  31. Function Inverse Given identity is i(x)=x, f -1(x) is a function such that

  32. 1. x 1 2 3 4 y 2 8 7 5 2. x 1 -1 3 7 y 2 2 5 8 What is the inverse for the function in table/numeric form?

  33. 1. What is the inverse for the function in graphic form? 2.

  34. What is the inverse for the function f(x)=3x+5 in algebraic form?

  35. 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

  36. 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.

  37. ((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).

  38. 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

  39. 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

  40. Coefficients ai. • Polynomial in x over S. • Term of Polynomial aixi .

  41. Francis Sowerby Macaulay Born: 11 Feb 1862 in Witney, EnglandDied: 9 Feb 1937 in Cambridge, Cambridgeshire, England

  42. 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 polynomialrings.

  43. Wolfgang Krull Born: 26 Aug 1899 in Baden-Baden, GermanyDied: 12 April 1971 in Bonn, Germany

  44. 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 polynomialrings.

  45. 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

  46. x -2 + 2x – 5 • x1/2 + ½ x2 + 3 • ni=0aixi • 2 + x3 – 2x5

  47. Polynomial Ring • Is (S [x],+,• ) a polynomial ring? • Is (S [x],+,• ) a commutative ring? • Is (S [x],+,• ) a ring with unity?

  48. Closure +

  49. Closure •

  50. Commutative & Associative for + and •

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