Chapter 1 by islamdansainsa

CHAPTER 1 : NUMBER SYSTEM

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25 JUNE – 3 JULY 2018

Upon completion of this chapter, readers should

be able to:

1. Define and state the rules of indices, surds

and logarithms.

2. Perform the algebraic operations of indices,

3. Solve equations involving indices, surds,

logarithms and complex numbers. snash’s

surds, logarithms and complex numbers.

TYPES OF NUMBER INTERVAL OF REAL NUMBER COMPLEX NUMBER

INDEX RULES / OPERATIONS

OPERATIONS

SURD SOLVING EQUATION

SOLVING EQUATION

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LOG

real NUMBER SYSTEM

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TYPES OF NUMBERS REAL;

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Number that can’t be

A number that has a decimal representation. Examples: 1, 0.5, 2.11111 , 2/3

Number that can be

represented as fraction represented of two as a fraction RATIONAL; IRRATIONAL; ’ integers. of , and a,b . -ve integers, 0 & +ve integers Examples: 2.3154… , e,Examples: -1, , 0.2,

INTEGER;

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NATURAL;

Zero and Natural numbers Examples: 0, 1, 2, … Counting numbers Example: 1,2,..

WHOLE;W

Examples: …, -1, 0, 1, 2, …

Example 1:

List the numbers in the set that are a) b) c) d) e) f)

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Natural numbers Whole numbers Integers Rational numbers Irrational numbers Real numbers

Example 2: Express the following decimal numbers as fractions.

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1.651d)

0.725

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REAL NUMBER LINE & INTERVALS

INTERVALS

OPEN

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HALF-OPEN/ HALF-CLOSE

INFINITE

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Example 3:

Rewrite each of the following inequalities by using interval notation and illustrate them on the real number line. a) c)

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b) d)

Exercise:

Example 4:

Write each of the following intervals as inequalities. Use as the variable. a)

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Indices

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INDICES

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If a is any real number, a  R (a > 0) and n is a positive integer, then The integer n is called the index or exponent and a is the base.

( Read a as ‘a to the nth power’) n

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n  Z    ve integer 

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POSITIVE

ZERO

n0

INDICES

NEGATIVE  n  Z (-ve integer)

FRACTION snash’s

For n=0

a0  1 , a  0 0

0 is not defined. For n positive integer

a n  a  a   a 1st

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2nd

nth

 n factors of a

For n is negative integer ; i.e: n = –m

m

1  m a

;

m

a �0

�a � �b � �b �  �a � ; a �0, b �0 ��

x For n is in the form of fraction ;i.e: n  y a a snash’s

x y 1 y

 a  y

 a

 a y

, a  0.

, a0

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RULES FOR INDICES

1) 2)

a a a am

 amn

  m

 ab 

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EXAMPLE 5c)

a

EXAMPLE 5e)

 ambm

EXAMPLE 6c)

EXAMPLE 5b)

a� a �  m � � b� b �

m n

m n

, bEXAMPLE �0 6e)

Example 7:

Simplify: a) c)

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b) d)

Example 10:

If , prove that .

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Example 8:

Exercise:

Simplify: b) c) Example 11: Show that is divisible by 7 if is a positive integer.

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surds

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SURDS

Surd is a number that contain one or more root sign; (or radical sign)

 It cannot be simplified into a fraction of two integers, .  Example, is a surd but is not a surd since .

 Surd is an irrational number.

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SURDS n

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�

a a

nth root of a

1 n

Surd is expressed in simplest form. For examples:

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 Examples of surd: a) b) 4 d) d) e)

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PROPERTIES OF SURD 1)

m n



a b

a a

2) m a m b 

n m

a  mn a

5) a c �b c   a �b  snash’s

CAUTION!!

but and but and

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Notes: Notes:

Example 12:

Identify which of the followings are surd.

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Example 13:

Simplify: b) d)

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Example 14:

Simplify: c)

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Exercise:

Example 13: a) Example 14: b)

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CONJUGATE OF SURD a b

a b

a b

a b

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a b

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RATIONALISING DENOMINATOR

 process of eliminating surd in denominator so that the denominator is a rational number.  Simplest form  denominator free from surd  Multiplication between conjugates will result in a rational number (no surd expression exists):

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RATIONALISING DENOMINATOR

 Problem arise when surd exists in denominator Solution: 

Multiply the numerator and denominator by suitable factor

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 Multiply the numerator and denominator with the conjugate of denominator

Example 15:

Simplify the expressions: a) c)

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Example 16:

b) Given and , find the value of .

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Exercise

Example 15: Simplify the expression: b)

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logarithm

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A number which is in an index form can be



LOGARITHM

written in a logarithmic form.  x is the logarithm of b to the base a is written as and it is equivalent to .

where and .

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LOGARITHM

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LOGARITHM a b x

Index Form

loga b  x Logarithmic Form

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LOGARITHM

Logarithmic with base 10 is called common logarithm and it can be written without the base 10:

 Logarithmic with base e is called natural logarithm and

it can be written as :

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LOGARITHMIC RULES Let , and

loga xy  loga x  loga y

x 2 / loga  loga x  loga y y

3 / loga xn  nloga x

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loga a  1

loga 1  0

aloga x  x

Example 19:

Evaluate without using calculator: b)

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Example 20:

Expand using the rules of logarithm. a) c)

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Example 21:

Rewrite the following expressions in a single logarithmic expression. b)

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CHANGE OF LOG BASE

Let , then Let , then

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

Example 22:

Evaluate by converting into ln. Leave the answer in four decimal places.

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No such thing as … loga  x �y  �loga x �loga y x loga x loga � y loga y

loga x � loga x  loga x loga x  loga y � loga y loga x  loga y � loga x   loga y  n

loga  2 x  �2loga x snash’s

loga  xy  � loga x  � loga y 

Exercise

Example 19: Evaluate without using calculator: c) Example 20: Expand using the rules of logarithm. b) Example 21: Rewrite the following expressions in a single log expression. c)

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SOLVING INDEX, SURD & LOG EQUATIONS

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COMMON BASE

COMMON BASE WITH ± BETWEEN TERMS

 convert into common base  simplify  compare index

 convert into common base  substitution

EXAMPLE 9b:

EXAMPLE 9c:

REMEMBER! an > 0

DIFFERENT BASES

cannot be converted into common base take log(or ln) both sides  apply log property

EXAMPLE 25a:

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INDEX EQUATION

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SURD EQUATION 2 SURDS

1 surd at one side

1 surd each side

3 SURDS

1 SURD

Square both sides and simplify Square both sides again (if there is a remaining surd) Solve the equation

EXAMPLE 17b ?

check for validity  LHS  RHS snash’s

  

1 surd at one side, 2 surds at the other side

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LOG EQUATION

1/COMMON LOG EXPRESSIONS

EXAMPLE 23a:

 change the base  apply property  substitution/ comparison EXAMPLE 23b:

check for validity  a, b > 0 for loga b snash’s

 simplify  use antilog or  change the base

MORE THAN 1 LOG WITH ± BETWEEN TERMS

Example 26:

Solve the simultaneous equations below. b)

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Exercise

Example 18: Find the value(s) of x: Example 24: Solve the equation:

Example 25: Solve the equation and leave the answer correct to 3 s.f. b) Example 26: Solve the simultaneous equations below. a) snash’s

EXPLORE INDEX You can explore this website for more information: 1. https://revisionmaths.com/gcse-maths-revision/algebra/indices 2. http://mathematics.laerd.com/maths/indices-1.php 3. https://www.youtube.com/watch?v=sbwSKpJkR2s SURD You can learn more about simplifying surd, the simple fact and history about surd at: http://www.mathslearn.co.uk/core1surds.html LOGARITHM Browse the website to study about the proof of logarithmic rules: https://www.onlinemathlearning.com/logarithms-properties.html SOLVING EQUATIONS You can check your workings and answers for Examples 9, 17, 18, 23, 24 and 25 via the online equation calculator: https://www.symbolab.com/solver/equation-calculator

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Complex number

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IMAGINARY NUMBER 2

i  1

i 3  i 2 i  i 4

 

4 1

i  i

  1  1 2

i i

i i i

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Example 28:

Write the following imaginary numbers in the form of . a) b) c) d)

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COMPLEX NUMBER

complex number = real part + (imaginary part)(imaginary number) z = a + bi a = Re (z) b = Im (z)

real part imaginary part snash’s

a + bi

COMPLEX NUMBER

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Set of complex number is denoted by C or �

Complex Real Part Number26

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Note

-4

0 5 0 5 0

0 0 0 −6 0 −6

are real numbers are real numbers Purely Real Purely Imaginary Purely Real Purely Imaginary

5 −6i 5 −6i

Imaginary Part

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Equalityof ofaaComplex ComplexNumber: Number: Equality

Zerosof ofaaComplex ComplexNumber: Number: Zeros

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CONJUGATE OF COMPLEX NUMBER

Suppose z = a + bi, then the conjugate of z is given by:

z  a  bi • Conjugate of a complex number is a complex number where i is being replaced by –i

• i,e change the sign of imaginary part

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Complex Number

Conjugate

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CONJUGATE OF COMPLEX NUMBER

Suppose z = a + bi, then the conjugate of z is given by:

z  a  bi Properties:

z z a b snash’s

z  z  2a

Example:

If , find: a) b) c) d)

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OPERATIONS OF COMPLEX NUMBERS

SUBTRACTION -

MULTIPLICATION ×

DIVISION ÷

ADDITION +

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z2  c �id

z1 �z2

  a  ib  � c  id    a �c   i  b �d  example 29 : example 30 :  3  2i    1  7 i3  5i   4  3i

  3  1  i  2  7 3  4   i  5   3    4  9i

 1  8i

ADDITION /

and

SUBTRACT ION

Let z1  a �ib

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  a  ib   c  id   ac   i  bc  ad  i2  bd    ac  bd  i  bc  ad example 32 :

 3  2i  1  7i  3  2i  21i  14i2  3  23i  14  1  11  23i snash’s

MULTIPLICATION

z1z2

operation

rationalizing a  ib    denominator  c  id a  ib  �  c  id � Multiply  with � �  c  id �c  id � conjugate

M example 36 : 5  2i �7  4i � 35  14i  20i  8i2 = � � 7  4i �7  4i � 72  42

 

DIVISION

z1 z2

35  8  6i = L 49  16 operation

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Find

Example 31: a) b)

Exercise

Example 33: Multiply each of the following and write the answers in standard form. b) c) Example 35: Find the product of the following complex number and its conjugate. a) b) Example 37: Express the following in the form of b) d) snash’s

ARGAND DIAGRAM

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• Complex numbers can be shown on the complex

number plane known as an Argand diagram • Each number is represented by a point. • The real part is plotted on the horizontal axis and the imaginary part on the vertical axis.

from the origin to the point.

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• Sometimes the number itself is represented as a line

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Example 38:

Plot the following complex numbers on an Argand diagram. a) b) e)

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z1  z2 a  bi  c  di \ ac bd

b � b2  4ac x 2a

EQUATION OF COMPLEX NUMBER

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Solving Quadratic equation when discriminant is negative

Equality of Complex numbers Find the value(s) of unknown

Example 40:

Solve the equation in the complex number system.

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Example:

Solve for p where  4i  p  2  i   4  3i let p  a 3bi p  2  i   4  3i  3  4 i 7i� 3  4i   a  bi1 2 i2  i4�  3i p �2  i � 2  i 3  4i   2a  2bi  � ai  b �   4  3i 2  14i  i  7  3  2a  bp  i  4  2b  a   4  3i 4 1 3  2a  b  49 13  4  2b  a  3 p  i 2a  b  1 5 L  51 a  2b  7 L  2  M snash’s

Example 42:

Find the square roots of .

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Exercise

Example 38: Plot the following complex numbers on an Argand diagram. c) d) f) Example 41: Solve the equation in the complex number system. Example 42: … .Use the results to solve the quadratic equation

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