fhmm1014 chapter 1 number and set

FHMM1014 Mathematics I*FHMM1014 Mathematics ICentre For Foundation StudiesDepartment of Sciences and EngineeringChapter 1Number and Set

FHMM1014 Mathematics I*Content 1.1 Real Numbers System.

1.2 Indices and Logarithm

1.3 Complex Numbers

1.4 Set

FHMM1014 Mathematics I*1.1 Real Numbers

*Real NumbersWhat number system have you been using most of your life?The real number system.

A real number is any number that has a decimal representation.FHMM1014 Mathematics I

*Set of Real Numbers(i)Natural NumbersCounting numbers (also called positive integers)

(ii)IntegersNatural numbers, their negatives, and 0.

N= { 1, 2, 3, }Z= {, 2, 1, 0, 1, 2, }Whole Numbers:FHMM1014 Mathematics I

*Set of Real Numbers(iii)Rational Numbers,Numbers that can be represented aswhere a and b are integers andAll rational number can be represented by:(a)terminating decimal numberssuch as(b)nonterminating repeating decimal numberssuch asQFHMM1014 Mathematics I

*Set of Real Numbers(iv)Irrational NumbersNumbers which cannot be expressed as a ratio of two integers. They are non-terminating & non-repeating decimal numbers.

(v)Real Numbers,All rational and irrational numbers.RNote: The square roots of all natural numbers which are not perfect squares are irrational.FHMM1014 Mathematics I

*Set of Real NumbersNZIRQFHMM1014 Mathematics I

*Real Number Line04884OriginFHMM1014 Mathematics I

FHMM1014 Mathematics I*Example 1 (a)Identify each number below as an integer, or natural number, or rational number or irrational number.

FHMM1014 Mathematics I**(i)Commutative Law* Addition :* Multiplication :Operations on Real Numbers

FHMM1014 Mathematics I**(ii)Associative Law* Addition :* Multiplication :Operations on Real Numbers

FHMM1014 Mathematics I**(iii)Distributive LawOperations on Real Numbers

FHMM1014 Mathematics I**(iv)Identity Law* Addition :* Multiplication :Operations on Real Numbers

FHMM1014 Mathematics I**(v)Inverse Law*Addition :*Multiplication :Operations on Real Numbers

Example 1 (b)FHMM1014 Mathematics I*Example 1 (b)Identify the law that justifies each of the following statements:

FHMM1014 Mathematics I**For any two different real numbers, a and b, with a < b:The open interval is defined as the set

The closed interval is defined as the set

The half-closed (or half-open) interval is defined as

Interval Notations for Real Numbersabxabxabxabx

FHMM1014 Mathematics I*Example 2(i) Express each interval in terms of inequalities, and then graph the interval. a) [1, 8) b) [2.5, 8] c) (3, )

(ii) Graph each set.(a) (1, 3) [2, 8](b) (1, 3) [2, 8]

FHMM1014 Mathematics I**The absolute value (or modulus) of a real number, x is denoted by .Absolute Values

FHMM1014 Mathematics I**Absolute Values

FHMM1014 Mathematics I**Find the values of x if(i)(ii)Example 3

FHMM1014 Mathematics I*1.2 Indices and Logarithms

FHMM1014 Mathematics I*1.2 ExponentsIf a is any real number and n is a positiveinteger, then the nth power of a is: (multiply a n times).

The number a is called the base and n is called the exponent.

FHMM1014 Mathematics I**Properties of Exponents

FHMM1014 Mathematics I**An equation with a variable in the exponent is called an exponential equation.

Property :

Note : Both bases must be the same!!****Exponential Equation

FHMM1014 Mathematics I**Solve(a)(b)Example 4

FHMM1014 Mathematics I**Solve the equationExample 5

FHMM1014 Mathematics I*Exponential FunctionsThe exponential function with base a is defined for all real numbers x by: where a > 0 and a 1.

FHMM1014 Mathematics I*Example 6If

FHMM1014 Mathematics I**Definition of e :As m becomes larger and larger, becomes closer and closer to the number e, whose approximate value is 2.71828...Natural Exponential Base

FHMM1014 Mathematics I**Natural Exponential Base

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0.11111111110.250.1353352832

0.19245008970.35355339060.2231301601

0.33333333330.50.3678794412

0.57735026920.70710678120.6065306597

111

1.73205080761.41421356241.6487212707

322.7182818285

5.19615242272.82842712474.4816890703

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15.58845726815.656854249512.1824939607

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-0.50.57735026920.70710678120.6065306597

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1322.7182818285

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FHMM1014 Mathematics I**Graphs of Exponential Function

FHMM1014 Mathematics I*Natural Exponential Functions Find the values of

FHMM1014 Mathematics I**Definition of logarithm :For

**

**Logarithm

FHMM1014 Mathematics I**Graphs of Logarithmic Functions

FHMM1014 Mathematics I**Properties of Logarithms

FHMM1014 Mathematics I**Solve the equationsExample 7

FHMM1014 Mathematics I*Natural Base Logarithms Common Logarithm

Natural Logarithm is when the base,

Note :

FHMM1014 Mathematics I*Example 8

FHMM1014 Mathematics I*1.3 Complex Numbers

FHMM1014 Mathematics I*1.3 Complex NumbersFor example, the equation has no realsolution. If we try to solve this equation, we will get

But this is impossible, since square of any realnumber is positive. Hence Mathematiciansinvented the complex number system to solve allquadratic equations.

FHMM1014 Mathematics I**A complex number :

wherea, b are real numbers and(real part)(imaginary part)Complex Numbers

FHMM1014 Mathematics I*ConjugatesFor the complex number

we define its complex conjugate to be:

FHMM1014 Mathematics I*Operations of Complex NumbersAddition:

Subtraction:

FHMM1014 Mathematics I*Operations of Complex NumbersMultiplication:

Division:

FHMM1014 Mathematics I*Square Root of Negative NumbersWe have Therefore, for

i.e square root of a negative number will have 2 roots, same as square root of a positive number.

FHMM1014 Mathematics I*Argand DiagramTo graph the complex number a + bi (or x + yi) we plot the ordered pair of numbers (a, b) or (x, y) in this Cartesian plane/form.Imaginary axisReal axisbiaa + bir=length

FHMM1014 Mathematics I*Complex NumberGraph the complex numbers:

FHMM1014 Mathematics I*ModulusThe modulus (or absolute value) of the complex number is:

*Argument The argument of z, denoted by arg(z), is the angle between OX and OP.

The principal arguments is

The angle is positive if counterclockwise and negative if clockwise.

*Example (a): Argument 1. What is arg(z) if z = 1 + i?From the diagram,

*Example (b): Argument 2. What is arg(z) if z = -1 + i?From the diagram,

*Example (c): Argument 3. What is arg(z) if z = 1 i?From the diagram,

*Example (d): Argument 4. What is arg(z) if z = 1 i?From the diagram,

FHMM1014 Mathematics I*Square Roots of a Complex NumberHow to find the square root ?

First, let .

Equating the real & imaginary parts will produce 2 new equations.

Therefore x and y can be obtained by solving these 2 equations.

FHMM1014 Mathematics I*Example 13Find

FHMM1014 Mathematics I*If is an angle in standard position whoseterminal side coincides with this line segment,by the definitions of sine and cosine x = r cos and y = r sin So, z = r cos + ir sin z = r(cos + i sin )This is the polar form.

Polar or Trigonometric Form

FHMM1014 Mathematics I*Example 14Write these complex numbers in Cartesian formsinto Polar (trigonometric) form.

FHMM1014 Mathematics I*1.4 Sets

FHMM1014 Mathematics I**SET=Any collection of objects specified in such a way that we can tell whether any given object is or is not in the collection.Each object in a set is called a member, or element, of the set.Capital letters are often used to designate particular sets.Set

FHMM1014 Mathematics I**means a is an element of set A means a is not an element of set A Set

FHMM1014 Mathematics I**Let set A : { x x is an even positive integer which is less than 13 }.Set A = { 2, 4, 6, 8, 10, 12}

Set

FHMM1014 Mathematics I**If each element of a set A is also an element of set B, then A is a subset of B.If set A and set B have exactly the same elements, then the two sets are said to be equal.Notation :means A is a subset of B means A is not a subset of B Subset

FHMM1014 Mathematics I**A set that contains all the elements of the set in a specific discussion is called the universal set. It is represented by:Universal Set

FHMM1014 Mathematics I**A set without any elements is called the empty, or null, set. It is represented by:Note :is a subset of every set.Empty Set

FHMM1014 Mathematics I**If A = { 3, 2, 2, 3 } , B = { 3, 3, 2, 2 } , and C = { 3, 2, 1, 0, 1, 2, 3 }.Indicate whether the following relationships are TRUE (T) or FALSE (F):-

Example 17

FHMM1014 Mathematics I**(a)Which of the following is False?

(b)List all the subsets of the set { 1, 2, 3, 4 }.Example 18

FHMM1014 Mathematics I**UnionIntersectionDifferenceComplementOperations of Sets

FHMM1014 Mathematics I**The union of sets A and B , denoted by is the set of all elements formed by combining all the elements of A and all the elements of B into one set.

x may be an element of set A or set B or both.Union

FHMM1014 Mathematics I**The intersection of sets A and B , denoted by is the set of elements in set A that are also in set B .

x is an element of both set A and set B .

If , the sets A and B are said to be disjoint / mutually exclusive.Intersection

FHMM1014 Mathematics I**The difference between set A and set B,is the set of elements in set A but not in set B.

Difference between 2 Sets

FHMM1014 Mathematics I**The complement of A , denoted by is the set of elements in that are not in A .

Complement

FHMM1014 Mathematics I**Union :BAVenn Diagram

FHMM1014 Mathematics I**Intersection :BAVenn Diagram

FHMM1014 Mathematics I**Intersection : (A and B are mutually exclusive)ABVenn Diagram

FHMM1014 Mathematics I**Complement :AVenn Diagram

FHMM1014 Mathematics I**If A = { 2, 4, 6 } , B = { 1, 2, 3, 4, 5 } , C = { 3, 8, 9 } , and = { 1, 2, 3, 4, 5, 6, 7, 8, 9}.Find :-(i)(ii)(iii)(iv)(v)(vi)Example 19

FHMM1014 Mathematics I**Given that

Find:

Example 20

*Example 21Find, in interval notation, each of the following sets: FHMM1014 Mathematics I

FHMM1014 Mathematics I**Commutative lawAssociative lawDistributive lawDe Morgans lawAlgebraic Laws on Sets

FHMM1014 Mathematics I**For any two sets A and B,Commutative Law

FHMM1014 Mathematics I**For any three sets A, B and C,Associative Law

FHMM1014 Mathematics I**For any three sets A, B and C,Distributive Law

*For any two sets A and B,First law :Complement of the union is the intersection of the complements.Second law : Complement of the intersection is the union of the complements.FHMM1114 General Mathematics IDe Morgans Law

FHMM1114 General Mathematics I

FHMM1014 Mathematics I**

By using set algebra, prove that, for any sets A and B

Example 22

FHMM1014 Mathematics I**

By using set algebra, prove that, for any sets A and B

Example 23

FHMM1014 Mathematics I*The End OfChapter 1

FHMM1014 AlgebraFHMM1014 Algebra*FHMM1014 AlgebraFHMM1014 Algebra*FHMM1014 AlgebraFHMM1014 Algebra*

fhmm1014 chapter 1 number and set

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