Econ 103

Mathematics for Economics Preliminary stuff

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What we assume you already know well Basic Arithmetic Basic arithmetic operators

+ addition- subtractionx or * multiplication or / division

Comparison operators

= equal to< strictly less than inequality less than or equal to inequality> strictly greater than inequality greater than or equal to inequality not equal to

n

n

ii xxxx +++= ...21

sum of n numbers

product of n numbers

n

n

ii xxxx ...21 =

Rules of Basic Arithmetic Some basic rules of arithmetic

Subtracting a negative number is the same as adding. For example: 5 - ( -10) = 5 + 10 = 15

Multiplying or dividing a positive number by a negative number gives a negative number For example, 2 *( -5 )= -10 and 5/ (-10 )= -0.5

Multiplying or dividing a negative number by a positive number gives a negative number For example: ( -2 )* 5 = -10 and (-5)/10 = -0.5

Multiplying or dividing a negative number by a negative number gives a positive number For example: ( -2)*( -5) = 10 and (-5)/(-10) = 0.5

Division by zero gives infinity For example: 1/0.00000000000000002 = 500000000000000 is very large (infinity)

Rounding

In many examples, you will need to round off your answer to a suitable number of decimal places (d.p.). Decimal places Decide on the number n of decimal places (eg 2) Look at the number in the next decimal place (3rd in our eg) If it is less than 5 then the n (eg 2) decimal places remain the same If it is 5 or more, then the nth decimal place needs to rise by 1 Examples 9.5467 to 2 decimal places (9.55) 12.3891 to 2 decimal places (12.39) 26.8549 to 2 decimal places (26.85 why?)

Basic Algebra

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Expressions

In algebra, letters are used to denote variables, with and being the most used letters.

The choice of letters in algebraic expressions has to be meaningful, such as for price or for quantity.

Example:

= 0(1 + 100) Where = balance (amount) in the account at time ; 0 = the principal (the original sum invested); = the interest rate; = the investment horizon (number of years, for example) In algebraic expressions, the multiplication sign is usually suppressed.

Examples: = ; 10 = 10 To evaluate algebraic expressions it is necessary to assign numerical values to the letters used: for ex, = 10.

Order of Operations When we evaluate an algebraic expression or when we calculate the

final value of a numerical expression, the operations involved have to be performed in the following order:

Brackets (B) Indices (I) Division and Multiplication (DM) Addition and Subtraction (AS) Acronym: BIDMAS

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Compare the results of evaluating 103 and (10)3 when = 10 103 =

More on algebraic expressions

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In algebraic expressions, like terms are multiples of the same letter (letters). For example, 3,5 and 0.25 This is important in solving equations: if an algebraic expression contains like terms, which are added and (or) subtracted together then it can be simplified to produce an equivalent shorter (simpler) expression

Simplify:

a) 2 + 10 0.1 b) 3 + 32 + 9 + 4 8

Brackets

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Removing the brackets: expanding brackets or multiplying out brackets

The distributive law applied to addition:

( + ) = + Verify this for = 2, = 10, = 3 Expand:

a) + 5 (2 ) b) ( + )( )

Be careful when you remove the brackets: ( + ) ( + ) = ( + ) ( )

Factorisation

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The reverse procedure: restoring the brackets is called factorization: 6 3 + 9 5 32 The difference of two squares formula:

2 2 = ( + )( ) Key Terms: Associative Law, Distributive Law, Expanding Brackets, Factorization

Fractions

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numeratordenominator Algebraic fractions: both the numerator and the denominator are letters: 2+1

1 Two fractions are said to be equivalent if they represent the same numerical value: for example, 34 and 68. A fraction is in its simplest form (also called reduced to its lowest terms) when there are no factors common to both the numerator and the denominator

Example: reduce the following fractions to their lowest term:

a) 1260 b) 363 c) 2(2)(2+1) d) 3

+3

Operations with Fractions

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Multiplication of fractions: multiply their corresponding numerators and denominators

=

Division of fractions: to divide by a fraction, turn it upside down and multiply

=

Without using a calculator, evaluate:

a) 23 89 b) 89 16

Rule for addition and subtraction of fractions: to add (or subtract) two fractions, first write them as equivalent fractions with a common denominator and then add (subtract) their numerators

Examples:

a) 712 58 b)

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Equations An equation is any mathematical expression that contains an equals sign Examples: 3x + 10 = -17; y - 3x + 2 = 0 These are examples of as none of the variables that appear in them are raised to any power other than power 1. An equation may contain one or more unknowns or variables and also some given or known values, called constants, coefficients or parameters. y 3x = -2 (two variables and two coefficients) y ax + b = 0 (two variables and two parameters; general way of writing an equation as parameters a and b can be any number)

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Equations

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2 + 3 + 32 10 = 42 7: represents an identity By comparison,

7 1 = 20 is an equation, as this is true only for some value of . This is a linear equation, with one solution

2 2 = 1 is a quadratic equation How to solve an equation: RULE: you can apply whatever mathematical operation you like to an equation provided that you do the same thing to both sides

Examples: solve the following equations

a) 6 + 1 = 7 + 9 b) 9

+2 = 72+1

The Concept of Set Def: A Set is any well defined combination of objects; the

elements (members) of a set are the objects in the set. Usually we denote sets with upper-case letters and elements with lower-case letters. Notation to show membership: x S x is a member of set S x S x is not a member of set S

Ways of describing sets: enumeration: S = {1,2,3} description: A is the set of all positive integers from 1 to 5 inclusive: A = { integer x | 1 x 5} Sets: Finite or Infinite (denumerable or non-denumerable)

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Example: The Real-Number System

A-Integers: 1, 2 (positive), -1,-2 (negative), 0

B-Fractions: ratios of two integers: 2/3, 4/5, -1/2

(A) and (B): Rational numbers Irrational numbers: Rational and Irrational numbers = the set of all real numbers: R

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Relationships between Sets Equality : A = B if they contain identical elements (the order of

elements does not matter) Subsets: Ex: S = {1,3,5,7} and T = {3} then each element of

T is also an element of S: T is a subset of S: T S (T is contained in S) or S T (S includes T)

Note that for two sets A and B: A B and B A iff (if and only if) A = B so subset: and proper subset:

The null (empty) set: ; is a subset of any set S The universal set: = the set of all elements currently under

consideration (see in macroeconomics: the information set) Disjoint (mutually exclusive) sets: two sets that have no

elements in common

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Operations on Sets Union: A B A B = {x | x A and x B} Intersection: A B A B = {x | x A or x B} Complement of a set: A complement or not A: set of all elements not in A An illustration using Venn diagrams

A

{ }A and = xxxA

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Relations and Functions Relation: ordered pairs with the property

that for (x, y) any x value determines more than one value of y

Function: a set or ordered pairs with the property that for (x, y) any x value uniquely determines a single y value.

A function is also called a mapping or transformation: y= f(x) or f: x y

where f is a rule of mapping

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Levels of Generality of Functions

Specific function 1: specific form and specific parameters

Y = 10 - .5X

Specific function 2: specific form and general parameters

Y = a bX

General function: general form and no parameters

y = f(x) f maps x into a unique value of y

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General Functions

y = f (x) y is value of the function or dependent

variable f is the function or a rule for mapping X into a

unique Y x is argument or the independent variable

( the set of all permissible values that x can take is called the domain of the function)

In economics, behavioral relations are usually represented as functions

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Specific Functions Types of functions

y = a (constant: fixed costs) y = a+ bx (linear: consumption) y = a + bx + cx2 (quadratic: total cost) y = a + bx + cx2 + dx3 (cubic: total cost) y = a/x (hyperbolic) y = axb (power: production) y = ax (exponential: interest) lny = ln(a) + bln(x) (logarithmic)

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Econ 103 Mathematics for EconomicsPreliminary stuffWhat we assume you already know wellBasic Arithmetic Rules of Basic Arithmetic RoundingBasic AlgebraOrder of OperationsMore on algebraic expressionsBracketsFactorisationFractionsOperations with FractionsEquationsEquationsThe Concept of SetExample: The Real-Number SystemRelationships between SetsOperations on SetsRelations and FunctionsLevels of Generality of FunctionsGeneral FunctionsSpecific Functions