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ENM 503 Fundamentals

ENM 503 Fundamentals. Numbers, Bases, Algebra, Functions, Equations and other Calculus Concepts. Primitives & Axioms. Is the number 6 larger than the number 3 ? Has anyone ever seen a number? Is the word " cheese " on the blackboard " chalk " ? Has anyone ever seen a point?

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ENM 503 Fundamentals

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  1. ENM 503 Fundamentals Numbers, Bases, Algebra, Functions, Equations and other Calculus Concepts rd

  2. Primitives & Axioms Is the number 6 larger than the number 3? Has anyone ever seen a number? • Is the word "cheese" on the blackboard "chalk" ? Has anyone ever seen a point? Axioms are assumptions made about primitives. Bird Testing of Number An Example of doing Mathematics Sum of first 100 integers : 1 + 2 + 3 + 4 + 5 + 6 1 + 6 = 2 + 5 = 3 + 4 = 7 => n(n+1)/2 rd

  3. Numbers • Cardinal: zero, one, two, … used for counting • Ordinal: first, second, … denote position in sequence • Integers: negative, zero and positive whole numbers • … -3 -2 -1 0 1 2 3 … • Fractions: parts of whole, ½ ¼ ¾ etc. • Numerals: symbols describing numbers • Digits: specific symbols to denote numbers • Arabic Numerals: 0 1 2 3 4 5 6 7 8 9 • Roman Numerals: I II III IV V VI VII VIII IX X … • 1 googol = 10100 ; 1 googolplex = 10googol = rd

  4. Types of Numbers • Rational • Prime • Perfect • Algebraic – roots of equations with integer coefficients • Irrational 2½ is algebraic since x2 – 2 = 0 • Imaginary and Complex, i = • Transcendental – Liouville; e, ; most frequent, • not algebraic, not roots of integer polynomials • Transfinite Numbers - Cantor • Figurate Numbers • Omega ; aleph null 0; aleph-one 1 rd

  5. CASTING OUT NINES • + • 28 1 • 39 3 • 426 • 109 10 • 1 1 Checks • Same procedure for subtraction and multiplication • 25 * 25 = 625 ~ 4 after casting out 9's • 7 * 7 = 49 ~ 4 after casting out 9's rd

  6. 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} Algebraic Numbers • 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, / rd

  7. Binary Arithmetic • Sum Difference1011  11 1011 11+ 101 5- 101 - 5 10000 16 110 6 • Product Quotient 1011 11 10.00110…X 101 5 101 1011,0000000 1011 55 -1010 0000 01 0001011 - 101 110111 110 - 101 • #b11011 = 27; #o27 = 23; #xAB= 171; #7r54 = 39 rd

  8. 1=2 • Let x = y • xy = y2 • xy – x2 = y2 – x2 • x(y – x) = (y – x)(y + x) • x = y + x • 1 = 2 • qed. Quad erat demonstrandum meaning • which was to be demonstrated. rd

  9. Three Classical Insolvable Problems • Using only straight edge and compass • 1. Construct a square whose area equals a circle. • 2. Double the volume of a given cube. • 3. Trisect an angle rd

  10. Multinomials • Find the coefficient of x3yz2 in the expansion of • (x + y + z)6. • (poly^n #(x #(y #(z 0 1) 1) 1) 6)  • #(X #(Y #(Z 0 0 0 0 0 0 1) #(Z 0 0 0 0 0 6) #(Z 0 0 0 0 15) #(Z 0 0 0 20) #(Z 0 0 15) #(Z 0 6) 1) #(Y #(Z 0 0 0 0 0 6) #(Z 0 0 0 0 30) #(Z 0 0 0 60) #(Z 0 0 60) #(Z 0 30) 6) #(Y #(Z 0 0 0 0 15) #(Z 0 0 0 60) #(Z 0 0 90) #(Z 0 60) 15) #(Y #(Z 0 0 0 20) #(Z 0 0 60) #(Z 0 60) 20) #(Y #(Z 0 0 15) #(Z 0 30) 15) #(Y #(Z 0 6) 6) 1) rd

  11. Poly^n • (x + y + z)3 = x3 + 3x2y + 3x2z + 3xy2 + 6xyz + 3xy2 + y3 + 3y2z + 3yz2 + z3 • (poly^n #(x #(y #(Z 0 1) 1) 1) 3) • #(X #(Y #(Z 0 0 0 1) #(Z 0 0 3) #(Z 0 3) 1) #(Y #(Z 0 0 3) #(Z 0 6) 3) #(Y #(Z 0 3) 3) 1) • x3 + 3x2y + 3x2z + 3xy2 + 6xyz + 3xy2 + y3 + 3y2z + 3yz2 + z3 rd

  12. Joseph Liouville • Born: 24 March 1809 in Saint-Omer, FranceDied: 8 Sept 1882 in Paris, France • An important area which Liouville is remembered for today is that of transcendental numbers. Liouville's interest in this stemmed from reading a correspondence between Goldbach and Daniel Bernoulli. Liouville certainly aimed to prove that e is transcendental but he did not succeed. However his contributions were great and led him to prove the existence of a transcendental number in 1844 when he constructed an infinite class of such numbers using continued fractions. In 1851 he published results on transcendental numbers removing the dependence on continued fractions. In particular he gave an example of a transcendental number, the number now named the Liouvillian number: 0.1100010000000000000000010000... where there is a 1 in place n! (n = 1,2,3, … and 0 elsewhere. rd

  13. More Real Numbers Real Numbers Rational (-4/5) = -0.8 Irrational Integers (-4) Transcendental (e=2.718281828459045… ) (=3.141592653589793 …) Natural Numbers (5) 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. Dense. rd

  14. Numbers in sets transcendental numbers Did you know? That irrational numbers are far more numerous than rational numbers? Consider where a and b are integers rd

  15. Identity Property • The numbers 0 and 1 play an important role in math since they do absolutely nothing. • Any number plus 0 equals itself. a + 0 = 0 + a = a. One example of this is: 3 + 0 = 0 + 3 = 3. 0 is called the identity for addition. • Any number multiplied by 1 is equal to itself. a x 1 = 1 x a = a. One example of this is: 3 x 1 = 1 x 3 = 3 1 is called the identity for multiplication. rd

  16. 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 * Vectors, Matrices non-commutative • Associative Law • a + (b + c) = (a + b) + c • a(bc) = (ab)c • Distributive Law • a(b + c) = ab + ac rd

  17. 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 y = f(x) in the range. • 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 rd

  18. 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) independent variable dependent variable examples: function of n variables rd

  19. On Domains Suppose 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). rd

  20. A more interesting function • Sometimes we need more than a single formula to specify a function algebraically, as in the following piecemeal example • The percentage p(t) of buyers of new cars who used the Internet for research or purchase since 1997 is given by the following function. (t = 0 represents 1997). rd

  21. 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). y _______ y intercept where x = 0 a linear function | | | | | | x the zero of the function or roots of the equation y = f(x) = 0 rd

  22. Graph of a nonlinear function Sources: Bureau of Justice Statistics, New York State Dept. of Correctional Services/The New York Times, January 9, 2000, p. WK3. rd

  23. Polynomials in one variable Polynomials are functions having the following form: 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 Karl Gauss rd

  24. Facts on Polynomial Equations • Used in optimization, statistics (variance), forecasting, regression analysis, production & inventory, etc. • 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. rd

  25. Make-polynomial with roots • (my-make-poly '(1 2 3)) (1 -6 11 -6) • (cubic 1 -6 11 -1) (3 2 1) • (my-make-poly '(2 -3 7 12))  (1 -18 59 198 -504) • (quartic 1 -18 59 198 -504) (12 7 2 -3) • (my-make-poly '(1 2 -3 7 12))  • (1 -19 77 139 -702 504) but neither quintic nor higher degree polynomials can be solved by formula. rd

  26. 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) rd

  27. The Quadratic Formula (quadratic 1 4 3) (-1 -3) rd

  28. A Diversion ~ convexity versus concavity Concave: Convex: rd

  29. Concave vs. convex rd

  30. More on quadratics If a, b, and c are real, 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 discriminant rd

  31. Interesting Facts about Quadratics If x1 and x2 are the roots of a quadratic equation, then Derived from the quadratic formula rd

  32. Equations Quadratic in form quadratic in x2 factoring Imaginary roots A 4th degree polynomial has 4 roots rd

  33. The General Cubic Equation Polynomials of odd degree must have at least one real root because complex roots occur in pairs. rd

  34. The easy cubics to solve: rd

  35. The Power Function(learning curves, production functions) 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) rd

  36. Learning Curves Cost & Time (sim-LC 1000 10 90)  Tn = T0nb Unit Hours Cumulative 1 1000.00 1000.00 2 900.00 1900.00 3 846.21 2746.21 4 810.00 3556.21 5 782.99 4339.19 6 761.59 5100.78 7 743.95 5844.73 8 729.00 6573.73 9 716.06 7289.79 10 704.69 7994.48 The slope of 90% learning curve is -0.1520; consider any unit, say 5. 783 = 1000*5b => b = -152. rd

  37. Exponential Functions(growth curves, probability functions) often the base is e=2.718281828459045235360287471352662497757... For c0 > 0, f(x) > 0 For c0 > 0, c1 > 0, f(x) is increasing For c0 > 0, c1 < 0, f(x) is decreasing y intercept = c0ex > 0 rd

  38. y = ex • y = ex y – eb = eb(x – b) • y' = ex (b, eb); intercept is x - 1 rd

  39. y = ln x rd

  40. Law of Exponents 2324 = 8 * 16 = 128 = 27 25/23 =32/8 = 4 = 22 (23)4 = 84 = 4096 = 212 21/2 = 1.41421356237 … rd

  41. Calculation Rules for Roots Radical is Radicand is N, n is the root index. rd

  42. Properties of radicals but note: Not a linear operator rd

  43. Radar Beam • 334.8F = vf where v is vehicle speed and f = 2500 megacycles.sec aimed at you. F is the difference between the initial beam sent out and the reflected beam. Were you speeding if the difference was 495 cycles/sec? • v = 334.8 * 495/2500 = 66.29 mph, perhaps not speeding but driving a bit over the speed limit of 65 mph. rd

  44. Law of Exponents rd

  45. Multiplying a Multinomial by a Multinomial Using the distributive law, we multiply one of the multinomials by each term in the other multinomial. We then use the distributive law again to remove the remaining parentheses, and simplify. (x + 4)(x - 3) = x(x - 3) + 4(x - 3) = x2 - 3x + 4x -12 =x2 + x -12 (x – a)(x – b)(x – c)(x – d) … (x – y) (x – z) = _______ • (Poly* #(X 4 1) #(X -3 1))  #(X -12 1 1) rd

  46. Logarithmic Functions(nonlinear regression, probability likelihood functions) base natural logarithms, base e note that logarithms are exponents: If x = ay then y = logax For c0 > 0, f(x) is a monotonically increasing For 0 < x < 1, f(x) < 0 For x = 1, f(x) = 0 since a0 = 1 For x  0, f(x) is undefined rd

  47. Least Common Multiple (LCM) • the smallest positive integer that is divisible by the numbers. • 8 = 2 2 2 8 16 24 32 40 • 12 = 2 2 3 12 24 36 • 9 = 3 3 9 18 27 36 • 15 = 3 5 • LCM = 2 * 2 * 2 * 3 * 3 * 5 = 360 • (lcm 8 12 9 15) 360(div 360) • (1 2 3 4 5 6 89 10 1215 18 20 24 30 36 40 45 60 72 90 120 180 360) rd

  48. Greatest Common Divisor (GCD) • The largest positive integer that divides the numbers with zero remainder • 102 and 30 • 102 = 3 * 30 + 12 • 30 = 2 * 12 + 6 • 12 = 2 * 6 + 0 • 6 is gcd • (div 102)  (1 2 3 617 34 51 102) • (div 30)  (1 2 3 5 6 10 15 30) rd

  49. Friend Ben => Be = n Log Base Number = Exponent BaseExponent = Number LogB N = E  BE = N Log2 16 = 4  24 = 16 Log2 16 = ln 16 / ln 2 = 2.7725887/0.6931472 = 4 rd

  50. Properties of Logarithms The all important change of bases: log216 = ln16/ln 2 = 4 rd

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