Math Primer Course - Numbers

Math Primer Course - Numbers - Transcript

GLOBAL ECONOMIC PROSPECTIVE MATH PRIMER COURSE

Instructor

Numbers
SETS


Fundamental in mathematics is the concept of a set A set is defined as a class or collection of distinct objects a set of all university professors the set of all letters A B C D Z the English alphabet A set of numbers 1 2 3 4 A 2 4 6 8 10 is a set of even numbers less or equal to 10

For example




SETS


Individual objects of the set are called members or elements Any part of a set is called a subset of the given set The set consisting of no elements is called the empty set or null set





REAL NUMBERS


Natural numbers numbers we normally use in counting are referred to as natural numbers These numbers constitute the set of natural numbers N defined as N 1 2 3 Any subset of the set of natural numbers can be drawn on a number line E g Plot the following set A 2 3 7







Real Numbers
Whole numbers when zero is added to the set of natural numbers the set N is transformed into the set of whole numbers W defined as W 0 1 2 3 Integers the set of integers is an extension of whole numbers by incorporation of negative numbers The set of positive and negative integers and zero is called the set of integers I 3 2 1 0 1 2 3 Like natural numbers a set of integers can be plotted on a coordinate line Plot P 3 0 2

Fractions Reciprocals and Absolute values
Fractions


Real Numbers

Fractions constitute a means of representing parts of a whole They are expressed as a ration of two integers E g 1 3 1 8 7 8 2 5 7 8 etc Two numbers are reciprocal of each other if their product is equal to one The reciprocals of a 5 8 and 1 5 are 1 a 8 5 5 respectively The absolute value of an integer k denoted by k is the difference between the integer k and zero the smaller number being subtracted from the larger one





Reciprocals




Absolute values


Real Numbers
Rational and Irrational Numbers Rational numbers


The set of Rational Numbers denoted by Q is the set of all numbers that can be expressed as a ratio of two integers For example 1 10 1 9 1 10 Between any two rational numbers they is an infinite number of other rational numbers





Rational and Irrational Numbers
Irrational Numbers
Irrational numbers are the opposite of rational numbers They are numbers that cannot be expressed as a ration of two integers E g
2 3



OPERATIONS WITH NUMBERS
Arithmetic operations


Here will deal with operations of the addition subtraction multiplication and division of a set of positive and negative This means there are four possible addition and subtraction operations e g 6 2 6 2 8 6 2 6 2 4



Adding positive numbers


Subtracting positive numbers


Arithmetic operations


Adding a negative number gives the same result as subtracting a positive one e g 6 2 6 2 4



Subtracting a negative number gives the same result as adding a positive one eg 6 2 6 2 8



Arithmetic operations
The rule is


If the numbers are separated by two plus signs or by two minus signs the numbers are added together If the numbers are separated by either a plus sign followed by a minus sign or a minus sign followed by a plus sign the second number is subtracted from the first



Arithmetic operations
Multiplication and division These follow a similar pattern to addition and subtraction


The rule for both multiplication and division is

Rule 1 If the two numbers are both positive or both negative the result is positive Rule 2 If one of the numbers is positive and the other is negative the result is negative

Multiplication and Division


Examples
4 3 4 3 12 Rule 2



5 2 10 the multiplication sign is generally omitted between brackets 15 3 5



Special case


Special cases with zero

Use of variables


A variable is something whose magnitude changes In some cases the value of some quantity say Income may be unknown A variables say X or Y or any other symbol or letter can be used to stand for a quantity whose value we do not know The use of such a symbol or variable is illustrated in the following example







Examples


How much must I save for the next fifteen weeks if I want 500 to spend on a holiday

Solution what is not know here is weekly savings so if we let Y weekly savings then 15 Y 500 15Y 500 Dividing both sides of the equation by 15 Y 33 333

EXAMPLE


So you need to save 33 33 a week A variable which is constrained to take a particular value or values is sometimes called an unknown In the above example Y is the unknown Particular symbols are generally used to denote commonly used variables such as P for price and Q for quantity Y for income C for consumption e t c



Use of variables


Arithmetic operation may be carried out on terms containing variables Note that when you are adding or subtracting variable terms the terms must have exactly the same variables in them eg 5x 2x 3x the variable x in both terms is the same 3x 2y the variables are different so the terms cannot be added together 5x 2xy the variable terms are different the second has a y in addition to the x









Use of variables


2x 8x 6 all the terms are different the first 2 term has x x is squared but the second has only a single x and the third has no variable 3x 9x 4 2x y 3x 4xy 5 which may be 2 simplified to 5x 6x y 4xy 9 allows for addition subtraction of terms with same variables
2 2

2



POWERS


By definition a power function is of the form Y Xa where Y is the dependent variable X independent variable and a is the index or exponent X is sometimes called the base e g 2 3 which is called the index power or exponent tells us the number of times the base 2 must be multiplied by itself i e 2 2 2 2 8 power 3
3 3





2 cubed or 2 to the



Powers


X4 x x x x
2

x to the power 4 x 2 squared or
3



x 2 x 2 x 2 x 2 to the power 2
3



2x 2x 2x 2x 8x 2x cubed or 2x to the power 3

Powers


When negative numbers are raised to higher powers e g 2 3 4 a pattern emerges 2 2 2 4 positive
3 2



negative negative



2 2 2 2 4 2 8 negneg positive then posneg negative 2 2 2 2 2 4 2 2 8 2 16 2 2 2 2 2 2 4 2 2 2 8 2 2 16 2 32
5 4





Powers
RULE


When a negative number is raised to an even positive power the result is positive When a negative number is raised to an odd positive power the result is negative When a positive number is raised to either a positive even or odd power the result is positive When any integer is raised to a power of 1 the result is the reciprocal of the integer Remember 1 1 1 1 0 0
2 2 2









Examples


2 1 2 1 1 2 2 2 1 2 2 2 2 1 2 2 1 4







Square roots


Square rooting a number is the reverse of squaring e g 22 4
4 2 x
2 2

and

2 2 4

4x

2 and 2 x x and x 2 x 2x and 2x

Square roots


For positive numbers there are always two square roots a positive one and a negative one In practice you may only require the positive root but the negative one still exists Calculators only give the positive root so don t forget to include the negative one if needed



Square roots


In general it is not possible to find the square root where there is more than one term under a single square root sign as you cannot simply square root each term e g



x2 9


The square root of is NOT x 3 We cannot find the square root of

Square roots
Square roots of negative numbers


Since any number positive or negative gives a positive result when squared negative numbers do not have real square roots They have complex roots we shall not go into this



Cubic roots


the cube of a positive number is a positive the cube of a negative number is a negative number Since finding a cube root is the reverse of cubing a number it follows that both positive and negative numbers have one cube root e g 3 8 2 3 8 2 Higher roots follow the same pattern







BODMAS


BODMAS brackets of Division Multiplication Addition and Subtraction e g simplify7 2 4 x2 10x2 4 13 7 6x2 10x2 4 13 7 12 5 13 27




BODMAS


Calculate 3 x 1 3 1 2 4
2 3 2

2

3

3 x 1 4 2 4 brackets where possible 3 x 2x 1 64 2 4 3x 6x 3 32 4 R 3x2 6x 31
2

orders divide and multiply L to add and subtract L to R

Dealing with brackets in multiplication
i A bracket multiplied by a single term


eg 5 x 4 2 3x 2y p p 8q 3r 4x x y 5 x 2 2m m 7n Multiply each term inside the bracket by the term outside the bracket making sure you get the signs correct So 5 x 4 5x 20 2 3x 2y 6x 4y
2 2







p p 8q 3r p 8pq 3pr 4x x y 4x 4xy 5 x 2 5x 10



Dealing with brackets in multiplication
ii Pairs of brackets


eg

a b c d



Each term in the first bracket is used in turn to multiply each term in the second bracket a b c d a c d b c d ac ad bc bd 2x 3 3x y 2x 3x y 3 3x y 3x2 2xy 9x 3y This method can be used for brackets containing many terms x 2 x2 3x 5 x x2 3x 5 2 x2 3x 5 x3 3x2 5x 2x2 6x 10 x3 x2 11x 10