1 ENM 503 Block 1 Algebraic Systems Lesson 4 – Algebraic Methods The Building Blocks - Numbers, Equations, Functions, and other interesting things. Did you know? Algebra is based on the concept of unknown values called variables, unlike arithmetic which is based entirely on known number values.

2 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} 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, / ).

3 More Real Numbers Real Numbers Rational (-4/5) = -0.8Irrational Transcendental (e.g. e = …  = …) Integers (-4) 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.

4 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

5 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 Associative Law a + (b + c) = (a + b) + c a(bc) = (ab)c Distributive Law a(b + c) = ab + ac Law and order will prevail!

6 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 f(x). 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

7 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) dependent variable independent variable examples: function of n variables

8 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).

9 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). x y |||||||||||| ______________ a linear function the zero of the function or roots of the equation f(x) = 0 y intercept

10 Graph of a nonlinear function

11 Polynomials in one variable Polynomials are functions having the following form: n th 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

12 Facts on Polynomial Equations 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 a 0 = 0.

13 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) Solving quadratic equations: Factoring Completing the square Quadratic formula

14 The Quadratic Formula Then it has two solutions. This is a 2 nd degree polynomial. Quick student exercise: Derive the quadratic formula by completing the square

15 A Diversion – convexity versus concavity Concave: Convex:

16 More on quadratics If a, b, and c are real numbers, then: if b 2 – 4ac > 0, then the roots are real and unequal if b 2 – 4ac = 0, then the roots are real and equal if b 2 – 4ac < 0, then the roots are imaginary and unequal discriminant

17 Equations Quadratic in form quadratic in x 2 factoring of no interest A 4 th degree polynomial will have 4 roots

18 The General Cubic Equation …and the cubic equation has three roots, at least one of which will always be real.

19 The easy cubics to solve:

20 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)

The Graph 21

22 Exponential Functions (growth curves, probability functions) often the base is e = … For c 0 > 0, f(x) > 0 For c 0 > 0, c 1 > 0, f(x) is increasing For c 0 > 0, c 1 < 0, f(x) is decreasing y intercept = c 0

The Graph 23

24 Law of Exponents You must obey these laws. More on radicals

25 Properties of radicals Who are you calling a radical? but note:

26 Logarithmic Functions (nonlinear regression, probability likelihood functions) natural logarithms, base e base note that logarithms are exponents: If x = a y then y = log a x For c 0 > 0, f(x) is a monotonically increasing For 0 < x < 1, f(x) < 0 For x = 1, f(x) = 0 since a 0 = 1 For x  0, f(x) is undefined

Graph of a log function 27

28 Properties of Logarithms The all important change of bases:

29 The absolute value function x a

30 Properties of the absolute value |ab| = |a| |b| |a + b|  |a| + |b| |a + b|  |a| - |b| |a - b|  |a| + |b| |a - b|  |a| - |b| Quick “bright” student exercise: demonstrate the inequality really nice example problem: solve |x – 3| = 5 then x - 3 = 5 and – (x - 3) = 5 or –x + 3 = 5 therefore x = -2 and 8

31 Non-important Functions Trigonometric, hyperbolic and inverse hyperbolic functions Gudermannian function and inverse gudermannian I bet you didn’t know this one!

32 Composite and multivariate functions (multiple regression, optimal system design) A common everyday composite function: A multivariate function that may be found lying around the house: Why this is just a quadratic in 3 variables. Is this some kind of a trick or what?

33 A multi-variable polynomial Gosh, an m variable polynomial of degree n. Is that something or what!

34 Inequalities An inequality is statement that one expression or number is greater than or less than another. The sense of the inequality is the direction, greater than (>) or less than (<) The sense of an inequality is not changed: if the same number is added or subtracted from both sides: if a > b, then a + c > b + c if both sides are multiplied or divided by the same positive number: if a > b, then ca > cb where c > 0 The sense of the inequality is reversed if both side sides are multiplied or divided by the same negative number. if a > b, then ca < cb where c < 0

35 More on inequalities An absolute inequality is one which is true for all real values: x > 0 A conditional inequality is one which is true for certain values only: x + 2 > 5 Solution of conditional inequalities consists of all values for which the inequality is true.

An Inequality Example 36 For x 0 For 2 < x < 3, f(x) < 0 For x > 3, f(x) > 0 Therefore X 3

37 An absolute inequality example problem: solve |x – 3| < 5 for x > 3, (x-3) < 5 or x < 8 for x  3, -(x-3) -2 therefore -2 < x < 8 I would rather solve algebra problems than do just about anything else.

38 Implicit and Inverse Functions

39 Finding your roots… Find an x such that Min f(x) 2 Professor, I just don't think it can be done. See the Solver tutorial On finding your roots

40 We End with the Devil’s Curve y 4 - x 4 + a y 2 + b x 2 = 0 An implicit relationship that is not single-valued This is my curve. Did you know: There are not very many applications of this curve in the ENM or MSC program. Quick student exercise: confirm the graph!