FHMM1114_Chapter1_Number_Set_Part_A Gneral Mathematic

FHMM 1114General Mathematics I

Lecture Chapter 1(Number and Set)

Part A

2

Topics

Real Numbers Set of Real Numbers Operations on Real Numbers Intervals Absolute Values (Modulus)

Exponents and Logarithms Exponents Logarithms

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Real Numbers

What 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.

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Set of Real Numbers

(i) Natural Numbers

Counting numbers (also called positive integers)

(ii) Integers

Natural numbers, their negatives, and 0.

N = { 1, 2, 3, …… }

Z = {……, –2, –1, 0, 1, 2, ……}

Whole Numbers:

{0} {0,1,2,3, }W N

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Set of Real Numbers

(iii) Rational Numbers,

Numbers that can be represented aswhere a and b are integers and

All rational number can be represented by:(a) terminating decimal numbers

such as

(b) nonterminating repeating decimal numbers

such as

,ba.0b

Q

75.043,5.021,5.225

...1333.0152...,666.032

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Set of Real Numbers

(iv) Irrational Numbers

Numbers which cannot be expressed as a ratio of two integers. They are nonterminating & nonrepeating decimal numbers.

(v) Real Numbers,

Rational and irrational numbers.

I ,,,2 e

R

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Set of Real Numbers

N

Z

I

RQ

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Real Number Line

0 4 8–8 –4

Origin

534

21

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Example 1.1

Change the rational number 0.141414… as a ratio of two integers.

Answer

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Example 1.2

Change the rational number 0.168168168 … as a ratio of two integers.

Answer

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Operations on Real Numbers

(i) Commutative Law

abba * Addition :

abba * Multiplication :

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(ii) Associative Law

cbacba )()(* Addition :

cabbca )()( * Multiplication :

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(iii) Distributive Law

acabcba )()1(

acabcba )()2(

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(iv) Identity Law

00a a a * Addition :

11a a a * Multiplication :

a + identity = a

a identity = a

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(v) Inverse Law

( ) ( ) 0a aa a *Addition :

11 1aaa a *Multiplication :

a + inverse = identity

a inverse = identity

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Real Number Intervals

For any two different real numbers, a and b,

with

The open interval is defined as the set

The closed interval is defined as the set

:ba

}:{),( bxaxba

}:{],[ bxaxba

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Absolute Values

The absolute value (or modulus) of a real number, x

is denoted by .x

0

0

xifx

xifxx

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Absolute Values

axaax

axaxax ,

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Example 1.3

Find the values of x if

(i)

(ii)

(iii)

(iv)

513 x

532 x

32

x

x

325 x

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Exponents

For n a natural number (positive integer)and a any real number,

where n is called the exponent anda is called the base.

aaaaan

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Properties of Exponents

nm,For Qnmnm aaa i.

,and a R+

nmnm aaa .iimnnm aa )(iii.

1iv. 0 an

aan

1v.

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Exponential Equation

An equation with a variable in the exponent, called an exponential equation.

Property :

Note : Both bases must be the same!!

.then , and,1,0 If yxaaaa yx

.then , and,1,0 If yx aayxaa **

**

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Example 1.4

Solve (a)

(b)

(c)

279 x

312 12832 xx

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9)6.0( x

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Example 1.5

Solve the equation

01)5(65)(

.032)2(52)(12

32

xx

xx

b

a

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Example 1.6

If show that,4832 rqp .)4( pqrpq

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Natural Exponential Base

Definition of e :

As m becomes larger and larger, becomes

closer and closer to the number e, whose approximate value is 2.71828...

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n

n

0

1lim 1

Alternatively,

1 1 1 1 1 1

0! 1! 2! 3! 4! !

n

n

k

en

ek

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0

2

4

6

8

10

-2 -1 0 1 2

x

** is between

Natural Exponential Base

xe ,3and2 xx because e is between 2 and 3.

** For , the graphs show that0x .23 xxx e ** For , the graphs show that0x .23 xxx e

** All three graphs have y-intercept (0, 1).

xy 2xy 3xey

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Logarithms

Definition of logarithm :

For , , an 0,1 d0a a x

nxax an logmeans

**

**

01log1 0 aa

1log1 aaa a

When a=10 => common logarithm

When a=e => natural logarithm

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Properties of Logarithms

yxxy aaa logloglogi.

yx aayx

a logloglogii.

xpx ap

a loglogiii.

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Properties of Logarithms

a

cc

b

ba log

loglogiv.

ab

ba log

1logv.

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Example 1.7

(a) Solve .1)3(loglog 22 xx

(b) Find x in term of b, given that

1)2(log2log bxx bb

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Example 1.8

Solve the equation

.02log2)419(log 2 xx

Note: xxx 10loglglog

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Example 1.9

Solve the equation

3log4log)(

74log12log)(

48loglog)(

3

4

2

x

x

x

xc

xb

xa

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Example 1.10

Find the smallest integer of n such that

.999.2)31(3 n

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Example 1.11

Given that

yxyx

yx

lglg3lg2

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Find the values of x and y.

FHMM1114_Chapter1_Number_Set_Part_A Gneral Mathematic

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