Number System (Matriculation)

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Matriculation QS015 2014

S.Y.Chuah

June 18, 2014

Chapter

11 :

: Num

Number

ber Sys

System

tem

1.1 Real Numbers

1.1 Real Numbers (a)

(a) DeﬁDeﬁne ne NatuNatural Numberal Numbersrs NN, Whole Number, Whole Number WW, Integers, Integers ZZ, Prime Numbers,, Prime Numbers, Rational Numbers

Rational Numbers QQ, Irrational Numbers., Irrational Numbers. (b)

(b) RepresenRepresent rational numbers and irrationat rational numbers and irrational numbers in decimal forms.l numbers in decimal forms. (c)

(c) ReprRepresenesent t the relatithe relationsonship of hip of nunumber sets in mber sets in a a real numreal number ber syssystem diagratem diagram- m-matically.

matically. (d)

(d) ReprRepresenesent t open, open, closclosed ed and and halfhalf-open inter-open intervvals als and and theitheir r reprrepresenesentatitations ons onon the number line.

the number line. (e)

(e) SimpliSimplify fy unionunion

_∪∪

, intersection, intersection

_∩∩

of two or more intervals with the aid of numberof two or more intervals with the aid of number line.

line.

1.2 Complex Numbers 1.2 Complex Numbers

(a)

(a) RepresenRepresent a t a complex ncomplex number in Cartesian form.umber in Cartesian form. (b)

(b) Deﬁne the equalitDeﬁne the equality of two complex numy of two complex numbers.bers. (c)

(c) DetermiDetermine the conjugate of a complex nune the conjugate of a complex number,mber, ¯ ¯Z Z .. (d)

(d) PerfoPerform algebraic operations on complerm algebraic operations on complex numbers.x numbers. (e)

(e) ReprRepresenesent t a a comcomplex numplex number ber in polar formin polar form Z Z == rr((cosθcosθ + + i i sinθsinθ) where) where r r << 00 and

and

₋

π < θ < π..π < θ < π

1.3 Indices, Surds and Logarithms 1.3 Indices, Surds and Logarithms

(a)

(a) StatState the e the rulerules of s of indindiceices.s.

(b) Explain the meaning of surd and its conjugate and to carry out algebraic (b) Explain the meaning of surd and its conjugate and to carry out algebraic

operations on surds. operations on surds. (c)

(c) StatState e the laws of logarithe laws of logarithmsthms.. (d)

(d) ChaChange the nge the basbase e of logariof logarithmsthms..

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1.1 Real Numbers

R

1.1.1 Sets of Real Numbers

Deﬁnition 1 (Real Numbers) The set of real numbers,

Deﬁnition 1 (Real Numbers) The set of real numbers,RR, comprises rational numbers and, comprises rational numbers and irrational numbers. irrational numbers.     00 Deﬁ

Deﬁnitinition on 1.1 1.1 NatNatural ural nunumbermbers,s,NN, , are are posposititivive e nnumumbers bers ththat at are are usused ed fofor r coucountntining:g:

N

N ==

_{{

1,2,3,1,2,3,

_{· · · }}

..

Deﬁnition 1.2 Whole numbers,

Deﬁnition 1.2 Whole numbers,WW, are natural numbers including the number zero:, are natural numbers including the number zero:

W

W ==

_{{

0,1,2,3,0,1,2,3,

_{· · · }}

.. Deﬁnition 1.3 Integers,

Deﬁnition 1.3 Integers,ZZ, are whole numbers including their negatives:, are whole numbers including their negatives:

Z

Z ==

_{···

,,-2,-1,0,1,2-2,-1,0,1,2,,

_{· · · }}

..

Deﬁnition 1.4 Prime numbers are natural numbers greater than 1 that can be divided by Deﬁnition 1.4 Prime numbers are natural numbers greater than 1 that can be divided by itself and 1 only.

itself and 1 only. Primenumbers

Primenumbers = =

_{{

2,3,5,7,112,3,5,7,11,,

_{· · · }}

.. Deﬁniti

Deﬁnition on 1.5 1.5 Rational numbersRational numbers,,QQ, are numbers that can be written in the form, are numbers that can be written in the form pp_q_q where where p and q are integers and

p and q are integers and q q

_

= = 0.0.

Q

Q ==

_{{

p p_q_q p,p, q q

_∈

ZZ,, q q

_

= = 00

_}}

In decimal form, rational numbers may be a

In decimal form, rational numbers may be a terminating decimal terminating decimal , such as, such as 33₄₄ = = 00..75 or a75 or a rep

repeating eating decimal decimal , such as, such as 33 11

11 = = 00..2727272727272727

·· ·· ··

, in which a group of one or more digits, in which a group of one or more digits

repears indeﬁnitely. Examples of rational numbers are

₋

33,,

₋

33₄₄,, 22₅₅,, 00..66,, 11..212121212121

_{·· ·· ··}

,, 66,, 20.20. Deﬁnition 1.6 Irrational numbers,

Deﬁnition 1.6 Irrational numbers, ¯¯QQ, are numbers that cannot be written in the form, are numbers that cannot be written in the form

p p q

q where p and q are integers and where p and q are integers and q q



= 0. For example,= 0. For example, ππ, e and, e and

√

3.3.

Figure 1: Real Number System Figure 1: Real Number System

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Figure 2: Venn Diagram represents diﬀerent types of real numbers

Exercise 1: Determine whether each statement is true or false.

(a) N

_⊂

W (b) Z

_⊂

N (c)

√

3

_∈

Q (d) 8.2525

_{· · · ∈}

Q¯ (e) 0.21212212 . . . /

_∈

Q (f) 0.23

_∈

Q

1.1.2 Intervals of Real Numbers

Intervals of real numbers can be illustrated using

1. Set notation denoted by

_{}

. The solution to the inequality x

_≥

2 can be expressed in set notation as follows:

{

x : x

_≥

2

_}

It is read as : The set of all x such that x is greater than or equal to 2. 2. Real number line denoted by

3. Interval notation denoted by [ ], ( ), [ ) or ( ]. (a,b) - open interval

[a, b] - closed interval

(a,b], [a,b) - half-open interval (a,

_∞

) - inﬁnite interval

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Exercise 2: Summary of Set notation and Interval notation Problem 1

Note : The symbol

_∞

is not numerical. When we write [a,

_∞

), we are simply referring to the interval starting from a and continuing indeﬁnitely to the right.

Problem 2

Graph all real numbers x such that (i) (

₋

20,

₋

5) (ii) (

₋

2,

_∞

) (iii) (

_−∞

,

₋

7) (iv) [0, 6] (v) [

₋

6, 1) (vi) [10,

_∞

) Problem 3

Graph each of the following on a number line. (i) All integers x such that

₋

3 < x < 3 (ii) All whole numbers x such that x

_≤

4

(iii) All natural number x such that

₋

2

_≤

x

_≤

3 (iv) All real numbers x such that

₋

1

_≤

x < 5

(v) x :

₋

3 < x

_≤

9, x

_∈

primenumber (vi) x : 2 < x < 10, x

_∈

R

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1.1.3 Combining Intervals

Using the symbol of union (

_∪

) and intersection (

_∩

).

The intersection of two intervals is the set of real numbers that belong to both intervals. EXAMPLE 1:

(1, 5)

_∩

(3, 9)

Hence, (1, 5)

_∩

(3, 9) = (3, 5).

The union of two intervals is the set of real numbers that belong to one, or the other, or both of the intervals.

EXAMPLE 2: (1, 5)

_∪

(3, 9) Hence, (1, 5)

_∪

(3, 9) = (1, 9). EXAMPLE 3: Given A=

_{

x :

₋

2 < x

_≤

5, x

_∈

R

_}

B=

_{

x : 0

_≤

x < 6, x

_∈

R

_}

C=

_{

x :

₋

3

_≤

x

_≤

4, x

_∈

Z

_}

Find (i) (A

_∪

B)

_∩

C (ii) (A

_∩

C )

_∪

B

Exercise 3: Write each union or intersection as a single interval.

(a) (

₋

3,

₋

5)

_∪

[0, 10) (b) (

_−∞

, 10]

_∩

(

₋

5, 7)

(c) [0, 15]

_∪

(

₋

5, 1] (d) [2,

_∞

)

_∩

[

₋

2, 10)

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1.2 Complex Numbers

C

x2 ₌

−

1 has no solution because square of real numbers cannot be negative. Therefore i is introduced to replace

√

₋

1, i.e. i =

√

₋

1. Hence, i2 ₌

−

1. Numbers which contain i is a complex number.

Let the complex number, z = a + bi, a, b

_∈

R, a is known as the real part and b is known as the imaginary part.

Re(a + bi) = a, Im(a + bi) = b

Names for Particular Kinds of Complex Numbers

Let a + bi be a complex number, a and b are real numbers. If b

_

= 0, then a + bi is a complex number.

If a = 0, then 0 + bi = bi is a pure complex number. If b = 0, then a + 0i = a is a real number.

If a = 0, b = 0 then 0 + 0i = 0 is called a complex zero number.

Square Roots of Negative Numbers

For any positive real number b,

√

−

b = i

√

b Example 1: Write in standard form, a + ib (A)

√

₋

4 =

√

4

_×

√

₋

1 (B) 4 +

√

₋

5 = 4 + (

√

5

_×

√

₋

1) = (C)

√

₋

7 +

√

₋

27 = (D)

−

2

−

√

−

48 2 = CAUTION!!

√

a

_×

√

b =

√

ab but

√

₋

a

_×

√

₋

b

_

=

√

₋

a

_{× −}

b Thus

√

9

_×

√

4 =

√

36 = 6 or

√

9

√

4 = 3

_×

2 = 6 But

√

₋

9

_×

√

₋

4

_

=

√

₋

9

_{× −}

4

_

=

√

36

_

= 6

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1.2.1 The Equality(Uniqueness) of Complex Numbers

If two complex numbers are equal, their real parts are equal and their imaginary parts are equal.

(a + bi) = (c + di)

_⇔

(a + bi)

₋

(c + di) = 0

_⇔

(a

₋

c) + (b

₋

d)i = 0

⇔

a

₋

c = 0, b

₋

d = 0

⇔

a = c, b = d

EXAMPLE 3 Solve the following equations. (i) 2 + 3yi = (x

₋

1) + 3i

By comparing the real and imaginary part,

2 = x

₋

1 and 3y = 3

∴ x = 3 and y = 1.

(ii)

₋

x + 2yi = (2

₋

i)2

Expand the right hand side of the equation,

−

x + 2yi = (2

₋

i)2

= 4

₋

4i + i2

= 3

₋

4i By comparing the real and imaginary part,

−

x = 3 and 2y =

₋

4

∴ x =

−

3 and y =

−

2

1.2.2 Operations with complex numbers

When you add, substract, multiply or divide two complex numbers a + bi and c + di, the result is another complex number.

Addition and substraction

By usual rules of algebra,

(a + bi)

_±

(c + di) = a + bi

_±

c

_±

di = a

_±

c + bi

_±

di = (a

_±

c) + (b

_±

d)i

Since a,b,c,d are real numbers, so are a

_±

c and b

_±

d. The expression at the end of the lines therefore has the form p + qi where p and q are real.

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Multiplication

By the usual rules for multiplying out brackets,

(a + bi)

_×

(c + di) = ac + a(di) + (bi)c + (bi)(di) = ac + adi + bci + bdi2

= (ac

₋

bd) + (ad + bc)i

Since, a,b,c,d are real numbers, so are ac

₋

bd and ad + bc. The product is therefore of the form p + qi where p and q are real.

An important special case is

(a + bi)

_×

(a

₋

bi) = (aa

₋

b(

₋

b)) + (a(

₋

b) + ba)i = (a2 + b2) + 0i

= a2 + b2

So with complex numbers, the sum of two squares, a2 _{+ b}2 _{can be factorised as}

(a + bi)(a

₋

bi).

Division

First, we take a + bi

c + di and consider two special cases. If d = 0, then a + bi c + 0i = a + bi c = a c + b ci

And if c = 0, you can simplify the expression by multiplying numerator and denominator by i: a + bi 0 + di = a + bi di = (a + bi)i (di)i = ai + bi2 di2 = . . .

In the general case a + bi

c + di the trick is to multiply numerator and denominator by c

−

di.

Natural powers of i

Natural powers of i take on particularly simple forms: i i2 ₌

−

1 i3 _= i2

·

i =

₋

i i4 _= i2

·

i2 _{= 1} i5 _= i4

·

i = i6 _= i4

·

i2 ₌ i7 ₌ i8 ₌

In general, what are the possible values for in_{, n a natural number? Then evaluate each}

of the following.

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Exercise 4: Uniqueness and Operations of Complex Numbers Problem 1

If p = 3+4i, q = 1

₋

i, r =

₋

2 + 3i, solve the following equations for the complex number z .

(A) p + z = q (B) qz = r

Problem 2

Solve these pairs f simultaneous equations for the complex numbers z and w. (1 + i)z + (2

₋

i)w = 3 + 4i

iz + (3 + i)w =

₋

1 + 5i

Problem 3

Simplify the following (A) 2i10

−

4i49 _{(B) (3i)}3 _{+ (i}5₎10

Problem 4

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1.2.3 Complex Conjugates

If z = x + yi, then its complex conjugate, denoted by z ∗ _{or ¯}_{z has the same real part as z}

but an imaginary part of the opposite sign, written as z ∗ _= x

−

yi. Theorem 1 Product of a Complex Number and Its Conjugate

(a + bi)(a

₋

bi) = a2 _+ b2 _[_{A real number}_]

Theorem 2 Sum of a Complex Number and Its Conjugate (a + bi) + (a

₋

bi) = 2a [A real number ]

Theorem 3 Diﬀerence of a Complex Number and Its Conjugate (a + bi)

₋

(a

₋

bi) = 2bi [An imaginary number ] Proof. (Try to complete the proof of Theorem 1,2 and 3.)

EXAMPLE 1

(A) z = 3

₋

5i (B) z =

₋

1

₋

3i

Conjugate complex numbers have important properties. Suppose for example, that s = a + bi and t = c + di are two complex numbers, so that s∗ _= a

−

bi and t∗ _= c

−

di, then (a) (s

_±

t)∗ ₌ (b) (st)∗ ₌ (c)



s t



∗ =

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1.2.4 Complex Numbers in Polar Form

Geometrical representation of complex numbers

There are two ways of representing a complex number by using a plane. The complex number z = a + bi can either be represented by a translation of the plane, a units in the x-direction and b units in the y-direction (see diagram on the left) or as the point z with coordinates (a, b)(see diagram on the right).

The second of these representations is called an Argand diagram, named after John-Robert Argand (1768

₋

1822).

The axes are called the real axis (x-axis) and the imaginary axis (y-axis). These con-tain all the points representing real numbers and imaginary numbers respectively.

Points representing the conjugates pairs a

_±

ib are .

The modulus of z , written as

_|

z

_|

, is the length of the line from the origin to the point representing the complex number on an Argand Diagram.

|

z

_|

=

√

a2 _+ b2

The argument of z is the angle θ between the positive x-axis and the line from the origin to the point representing the complex number on an Argand diagram such that

−

π < θ < π. It is denoted as arg(z ).

arg(z ) = θ = tan−1b

a,

−

π < θ < π For example, given complex number z = 9 + 6i,

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Exercise 5: Plot the complex numbers in the Argand diagram and calculate the modulus and argument of each

(A) z = 2 + 2

√

3 (B) z =

₋

4 + i

(C) z = 2

₋

3i (D) z =

₋

1

₋

i

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Polar Form

If z is a complex number with modulus r and θ then z can be written as z = r(cosθ + isinθ);

₋

π < θ < π, r

_≥

0

or z = reiθ_;

−

π < θ < π, r

_≥

0 Example

Find the modulus and argument of the following complex numbers. Hence, ﬁnd its polar form.

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1.3 Indices, Surds and Logarithms

1.3.1 Indices

Large or small numbers are better expressed in terms of indices. A given number can be written as a base raised to the index, (base)index_{. For example,}

Deﬁnition 1 an_{, n an integer and a a real number}

1. For n a positive interger: an _= a

×

a

_×

a

_{× · · · ×}

a 59049 = 95 _{= 9}

×

9

_×

9

_×

9

_×

9 2. For n = 0: a0 _{= 1 for a}



= 0 1320 _{= 1} 00 _{is not deﬁned}

3. For n a negative integer: an ₌ 1

a−n for a



= 0 7

−3 ₌ 1

73

Theorem 1 Properties of Integer Indices 1. am

×

an _= am+n 2. (am₎n _= amn 3. (ab)m _= am_bm 4.



a b



m = a m bm 5. am an =



am−n, 1 an−m, a = 0

EXAMPLE 1 Using Index Properties

Simplify using index properties, and express answers using positive indices only. (A) (3a5_)(2a−3) =

(B) 6x

−2

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Theorem 2 Further Index Properties For any a and b any real numbers and m, n and p any intergers (Excluding division by 0):

1. (am_bn₎ p _= a pm_b pn 2.



a m bn



p ₌ a pm b pn 3. a −n b−m = bm an 4.



a b



−n =



b a



n

Proof. (Try to complete the proof of Theorem 2)

EXAMPLE 2 Using Index Properties

Simplify using index properties, and express answers using positive indices only. (A) (2a−3_b2₎−2 (B)



a 3 b5



−2 (C) 4x −3_y−5 6x−4y3 (D)



m −3_m3 n−2



(E) (x + y)−3 (F) x −2

−

y−2 x−1 + y−1

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Deﬁnition 2 bmn and b− m

n, Fractional Indices

For m and n natural numbers and b any real number (except b cannot be negative when n is even): bmn = (b 1 n)m b− m n = 1 bmn

EXAMPLE 3 Using Fractional Indices

Simplify, and express answers using positive indices only. All letters represent positive real numbers. (A) 823 (B) (

₋

8)53 (C)



4x 1 3 x12



12 (D)



u12

−

2v12

 

3u 1 2 + v 1 2



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2.1.1 Index Equations

[This topic appears in in Chapter 2]

If both sides of an index equations can be expressed in the same base, then equate the powers and solve the resulting equation.

If they cannot be expressed in the same base, then take log of both sides of the equa-tion(see example on log equations later).

EXAMPLE 5 Solve the following equations. (A) 91−x _{= 9}3 (B) 4x−3 = 8 (C) 27x+1 _{= 9} (D) 3x2x−3x _{= 81} (E) 42x2 +2x _{= 8} (F) 2x ₌ 16 −2x 8

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1.3.2 Surds

A surd is a radical that is not evaluated, or cannot be precisely evaluated. The radicand is often a constant, such as the square root of two:

We know that the square root of 2 is 1.4142.. But why do we leave it as a radical and not convert it to the number?

Deﬁnition 3

√

n

b For n a natural number greater than 1 and b a real number, we deﬁne

n

√

_{b to be the principle nth root of b, that is}

n

√

_b = b1

n

Theorem 3 Properties of Surds For n a natural number greater than 1, and x and y positive real numbers:

1.

√

n xn _= x 2.

√

n xy =

√

n x

×

√

n y 3.



n x y = n

√

_x n

√

_y **Additional 1.

√

x

_{× √}

x = x 2.

√

x +

√

x = 2

√

x 3.

√

a

_÷

√

b =



a b 4. (

√

a +

√

b)2 _{= a + b + 2}

√

_ab 5. (

√

a +

√

b)(

√

a

₋

√

b) = a

₋

b EXAMPLE 6 Simplifying Surds Simplify

(A) 8



(3x2_y)8 ₌ _(B)

√

₁₀

√

_{5 =}

(C)



3 x

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Deﬁnition 4 Simpliﬁed Surd Form

1. Non radicand contains a factor to a power greater than or equal to the index of the surd.

2. No power of the radicand and the index of the surd have a common factor other than 1

3. No surd appears in a denominator. 4. No fraction appears within a surd. Deﬁnition 5 Conjugate Surds

The conjugate of a +

√

b is

√

a

₋

√

b as the product (

√

a +

√

b)(

√

a

₋

√

b) = a

₋

b is a rational number.

Rationalising Operations

For example, we have this algebraic fractions 3

√

₅

√

_x3

−

1 3



2a2 3b2 6 2

₋

√

3

√

₂

−

1 2 +

√

2

Here, to solve, we need to eliminate a surd from a denominator hence we refer this as rationalizing denominator.

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Example 7 Simplifying the expression below which involves rationalizing surds. (A)

_√

3 5 (B)



3 2a 2 3b2 (C)

_√

3 x

₋

1 (D) 6 2

₋

√

3 (E) 1 2

√

5

₋

3

√

2 +

√

_{6 + 2}

√

₃ 2

√

6

₋

√

3 (F)

√

₂

−

1 2 +

√

2 +

√

₂

−

1 2

₋

√

2

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2.1.2 Surds Equations

There are equations involving surds. To solve surd equations, we have to look if the equations have one, two or three surds in the equation.

Now, we will consider the three cases which have 4 equal steps to solve the equation. 4 STEPS TO SOLVE SURDS EQUATION:

1. Square both sides of the equation and isolate any remaining surds. 2. Square the equation again to remove any remaining surds.

3. Solve the resulting equation. 4. Check your answers

CASE 1: ONE SURD in the equation

If there is only one surd in the equation, put it on one side before starting the 4 STEPS solution.

√

_{5x + 1 + 1 = x}

CASE 2: TWO SURDS in the equation

If there is only two surds in the equation, move one to the other sidebefore starting the 4 STEPS solution.

√

_5x

₋

₁

₋

√

_{x + 2 = 1}

CASE 3: THREE SURDS in the equation

If there is only three surds in the equation, make sure one of them is on one sidebefore starting the 4 STEPS solution.

√

_{8x + 17}

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1.3.3 Logarithm

Deﬁnition 6 Deﬁnition of Logarithm For b > 0 and b

_

= 1,

Logarithm form Index form

y = log_b x is equivalent to x = by

y = log₁₀ x is equivalent to x = 10y

y = log_e x is equivalent to x = ey

** log_e x = ln x , this is called a Natural Logarithm .

A logarithm is an index or in other words, we can say that logarithm form is equivalent to index form so in order to solve any problem related to logarithm and index, logarithm form and index form are interchangeable.

EXAMPLE 7 Solve these equations by interchanging logarithm form and index form. (A) log₁₀ x =

₋

2 (B) ln(2 + x) = 1 (C) log₂(x2

−

3x

₋

2) = 3

Theorem 4 Properties of Logarithm 1. log_b 1 = 0

2. log_b b = 1

3. log_b bx _{= x log bb = x}

4. blogb x = x

5. log_b M N = log_b M + log_b N 6. logb

M

N = logb M

−

logb N 7. log_b M P _= P log

b M

(23)

EXAMPLE 8 Solve the following by using the properties of logarithm. (A) 2log₁₀ 5 + log₁₀ 70 + log₁₀ 45

35

−

log10 45

2

(B) Given log₂ 3 = 1.59 and log₂ 5 = 2.32, without using calculator, evaluate: (a) log₂ 0.6 (b) log₂ 30 _{(c) log}

2 3

3

5 (d) log2

3

√

_1.5

Change-of-Base Formula Let y = logb N where N and b are positive and N



= 1.

y = log_b N by = N logy_b = log_N log_a by = log_a N y loga b = loga N y = loga N log_a b

EXAMPLE 9 Solve these by changing the base of logarithm. (A) log₈16 (B) log₂₇81 (C) log₃ 7

(24)

EXAMPLE 10 Solve. (A) log5 81

log₅ 27

(B) (log₂ 2)3 _{(C) log}

4(13 + 3)

CAUTION 3 common errors in logarithm

(a) logb M logb N



= log_b M

₋

log_b N

(b) log_b(M + N )

_

= log_b M + log_b N (c) (logb M ) p



= p logb M

2.1.3 Logarithm Equations

Logarithm equations can be solved by considering all the properties of logarithm carefuly. We will look at more examples involving logarithm equations.

EXAMPLE 11 Solve logarithm equations. (A) log₇ 4x

₋

log₇(x + 1) = 1

2 log7 4 (B) 2x

·

8 = 3x

·

5x (C) log₉ x 3 = log₉ x log₉ 3