For Quarantined students developed by AKHSS, HUnza ( Four Lessons )

Lesson 1

Rational and Irrational Numbers Definition

What are Rational numbers?

The rational numbers are numbers which can be expressed as a fraction and also as positive numbers, negative numbers and zero. It can be written as p/q, where q is not equal to zero.

Rational word is derived from the word ‘ratio’, which actually means a comparison of two or more values or integer numbers and is known as a fraction. In simple words, it is the ratio of two integers.

Examples: 3/2 is a rational number. It means integer 3 is divided by another integer 2

           All terminating decimals are rational : 3.235 , 0.343434 and 1.4321879 are rational.

             Non terminating and recurring decimals are rational: 2.3333333…..,  0.3434343……..

            and   1.234234234………. are rational

What are Irrational Numbers?

The numbers which are not a rational number are called irrational numbers. Now, let us elaborate, irrational numbers could be written in decimals but not in fractions which means it cannot be written as the ratio of two integers.

Irrational numbers have endless non-repeating digits after the decimal point. Below is the example of the irrational number:

Examples: √8=2.8284271……                              = 1.7320508……..         e=2.71828182……..
  Non-terminating and non-recurring decimals are rational.

How to Classify Rational and Irrational Numbers?

Let us see how to identify rational and irrational numbers based on below given set of examples.

As per the definition, The rational numbers include all integers, fractions and repeating decimals. For every rational number, we can write them in the form of p/q, where p and q are integers value.

Difference Between Rational and Irrational Numbers

It is expressed in the ratio, where both numerator and denominator is the whole number	It is impossible to express irrational numbers in fractions or in a ratio of two integers.
It includes perfect squares	It includes surds.
The decimal expansion for rational number executes finite or recurring decimals	Here, non-finite and non-recurring decimals are executed

 

Examples

The list of examples of rational and irrational numbers are given here.

List of Rational Numbers

• number 9 can be written as 9/1 where 9 and 1 both are integers.
• 0.5 can be written as ½, 5/10 or 10/20 and in the form of all termination decimals.
• √81 is a rational number, as it can be simplified to 9 and can be expressed as 9/1.
• 0.7777777 is recurring decimals and is a rational number

List of Irrational Numbers

Similarly, as we have already defined that irrational numbers cannot be expressed in fraction or ratio form, let us understand the concepts with few examples.

• 5/0 is an irrational number, with the denominator as zero.
• π is an irrational number which has value 3.142…and is a never-ending and non-repeating number.
• √2 is an irrational number, as it cannot be simplified.
• 0.212112111…is a rational number as it is non-recurring and non terminating.

There are a lot more examples apart from above-given examples, which differentiate rational numbers and irrational numbers.

Rational and Irrational Numbers Rules

Here are some rules based on arithmetic operations such as addition and multiplication performed on rational number and irrational number.

#Rule 1: The sum of two rational numbers is also rational.

Example: 1/2 + 1/3 = (3+2)/6 = 5/6

#Rule 2: The product of two rational number is rational.

Example: 1/2 x 1/3 = 1/6

#Rule 3: The sum of two irrational numbers is not always irrational.

Example: √2+√2 = 2√2 is irrational

2+2√5+(-2√5) = 2  is rational

#Rule 4: The product of two irrational numbers is not always irrational.

Example: √2 x √3 = √6 (Irrational)

√2 x √2 = √4 = 2 (Rational)

  0.343434……..……… can be written in the form of 

      Let  x = 0.343434…………….

      Multiplying  100 on both sides

             100x = 34.343434…………

             100x=  34+ 0.343434…………..

              100x = 34 + x

               100x-x = 34

                   99x= 34

                     x = 

       0.343434 ……………..  = 

Students will solve following questions:

1. Which of the following numbers is irrational?

A.                                B.                                C.                      D.

2.Which of the following numbers is rational?

                     A.  2.16789543256……     B. 0.123123123…….     C.                   D.  

 

3. Show that 0.333333……………….=

 

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Lesson 2

REAL NUMBERS:

Union of set of rational and irrational numbers is called set of real numbers.

     R = Q Q^/
A number whose square is non-negative is called real number.

-1 is real number because its square is positive.    ,   is not real

because  (negative)  is an imaginary number.

 

Addition Properties of Real Numbers

Suppose a, b, and c represent real numbers.

 

1) Closure Property of Addition

• Property: a + b is a real number
• Verbal Description: If you add two real numbers, the sum is also a real number.
• Example: 3 + 9 = 12  where 12 (the sum of 3 and 9) is a real number.

2) Commutative Property of Addition

• Property: a + b = b + a
• Verbal Description: If you add two real numbers in any order, the sum will always be the same or equal.
• Example: 5 + 2 = 2 + 5 =10

3) Associative Property of Addition

• Property: (a + b) + c = a + (b + c)
• Verbal Description: If you are adding three real numbers, the sum is always the same regardless of their grouping.
• Example: (1 + 2) + 3 = 1 + (2 + 3) = 6

4) Additive Identity Property of Addition

• Property: a + 0 = a
• Verbal Description: If you add a real number to zero, the sum will be the original number itself.
• Example: 3 + 0 = 3  or  0 + 3 = 3

5) Additive Inverse Property

• Property: a + (– a) = 0
• Verbal Description: If you add a real number and its opposite, you will always get zero.
• Example: 13 + (– 13) = 0

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Multiplication Properties of Real Numbers

Suppose a, b, and c represent real numbers.

6) Closure Property of Multiplication

• Property: a × b is a real number.
• Verbal Description: If you multiply two real numbers, the product is also a real number.
• Example: 6 × 7 = 42  where 42 (the product of 6 and 7) is a real number.

7) Commutative Property of Multiplication

• Property: a × b = b × a
• Verbal Description: If you multiply two real numbers in any order, the product will always be the same or equal.
• Example:  9 × 4 = 4 × 9 = 36

8) Associative Property of Multiplication

• Property: (a × b) × c = a × (b × c)
• Verbal Description: If you are multiplying three real numbers, the product is always the same regardless of their grouping.
• Example: (5 × 3) × 2 = 5 × (3 × 2) = 30

9) Multiplicative Identity Property of Multiplication

• Property: a × 1 = a
• Verbal Description: If you multiply a real number to one (1), you will get the original number itself.
• Example: 25 × 1 = 25  or  1 × 25 = 25

10) Multiplicative Inverse Property

• Property: a × (1/a) = 1  but  a ≠ 0
• Verbal Description: If you multiply a nonzero real number by its inverse or reciprocal, the product will always be one (1).
• Example: 2 × (1/2) = 1

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The Property of Multiplication together with Addition

11) Distributive Property of Multiplication over Addition

Suppose a, b, and c represent real numbers.

• Property: a(b + c) = ab + ac or (a+b)c = ac + bc 
• Verbal Description: The operation of multiplication distributes over addition operation.
• Example: 4 (5 + 8) = 4 × 5 + 4 × 8 or (5 + 8) 4 = 5 × 4 + 8 × 4 

 

 

 

Students are supposed to solve the following questions.

1.         Show that  is a real number.

2.         Show that  is not real.  

3.         If  y²-9 = 0then show that y is real.

4.         If x² + 4=0 then show that x is an imaginary number.

5.        Identify the properties of real numbers in the following.

        i. 3+7=7+3     ii.   1 4=4 1=4       iii.    2 =          iv.  3+0=0+3+3

       v.  x+(-x)= -x+x=0  vi. 3+(4+5) = (3+4)+5   vii. 2(6+2)=2 6 + 2

6.         Which of the following sets is closed under addition?

         A.  {1,0}             B.  {1,2,3}      C. {-1,1}   D. Set of natural numbers.

7.   Show that the set {0,1} is closed under multiplication.

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Less0n 3

Laws of Exponents

 

The laws of exponents are explained here along with their examples.

1. Multiplying powers with same base

For example: x² × x³, 2³ × 2⁵, (-3)² × (-3)⁴

In multiplication of exponents if the bases are same then we need to add the exponents.

 

Consider the following: 

1. 2³ × 2²= (2 × 2 × 2) × (2 × 2) = 2³⁺² = 2⁵

2. 3⁴ × 3² = (3 × 3 × 3 × 3) × (3 × 3) =3⁴⁺² = 3⁶

3. (-3)³ × (-3)⁴ = [(-3) × (-3) × (-3)] × [(-3) × (-3) × (-3) × (-3)] =  (-3)⁷

4. m⁵ × m³ = (m × m × m × m × m) × (m × m × m) = m⁵⁺³ = m⁸

 

From the above examples, we can generalize that during multiplication when the bases are same then the exponents are added.

aᵐ × aⁿ =

In other words, if ‘a’ is a non-zero integer or a non-zero rational number and m and n are positive integers, then

aᵐ × aⁿ = a^m+n

Similarly, (a/b)ᵐ × (a/b)ⁿ = (a/b)^m+n

Note:

(i) Exponents can be added only when the bases are same.

(ii) Exponents cannot be added if the bases are not same like

m⁵ × n⁷, 2³ × 3⁴

For example:

1. 5³ ×5⁶

= (5 × 5 × 5) × (5 × 5 × 5 × 5 × 5 × 5)

= 5³⁺⁶ , [here the exponents are added] 

= 5⁹

2. (-7)¹⁰ × (-7)¹²

= [(-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7)] × [( -7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7)].

= (-7) ¹⁰⁺¹²  , [exponents are added] 

= (-7)²²

3. (1/2)⁴ × ( 1/2)³

=[(1/2) × ( 1/2) × ( 1/2) × ( 1/2)] × [ ( 1/2) × ( 1/2) × ( 1/2)]

=(1/2)⁴⁺³

=(1/2)⁷

4. 3² × 3⁵

= 3²⁺⁵  

= 3⁷

5. (-2)⁷ × (-2)³

= (-2)⁷⁺³  

= (-2)¹⁰

6. (4/9)³ × (4/9)²

= (4/9)³⁺²      

= (4/9)⁵

We observe that the two numbers with the same base are

multiplied; the product is obtained by adding the exponent.

2. Dividing powers with the same base

For example:

3⁵ ÷ 3¹, 2² ÷ 2¹, 5(²) ÷ 5³

In division if the bases are same then we need to subtract the exponents.

Consider the following:

2⁷ ÷ 2⁴ = 2⁷/2⁴ = (2 × 2 × 2 × 2 × 2 × 2 × 2)/(2 × 2 × 2 × 2) = 2^7-4 = 2³

5⁶ ÷ 5² = 5⁶/5² = (5 × 5 × 5 × 5 × 5 × 5)/(5 × 5) = 5^6-2 = 5⁴

10⁵ ÷ 10³ = 10⁵/10³ = (10 × 10 × 10 × 10 × 10)/(10 × 10 × 10) = 10^5-3= 10²

7⁴ ÷ 7⁵ = 7⁴/7⁵ = (7 × 7 × 7 × 7)/(7 × 7 × 7 × 7 × 7) = 7^4-5 = 7^-1

Let a be a non zero number, then

a⁵ ÷ a³ = a⁵/a³ = (a × a × a × a × a)/(a × a × a) = a^5-3 = a²

again, a³ ÷ a⁵ = a³/a⁵ = (a × a × a)/(a × a × a × a × a) = a^-(5-3)

 

 = a^-2

Thus, in general, for any non-zero integer a,

aᵐ ÷ aⁿ = aᵐ/aⁿ = a^m-n        

Note 1:

Where m and n are whole numbers and m > n;

aᵐ ÷ aⁿ = aᵐ/aⁿ = a^-(n-m)

Note 2:

Where m and n are whole numbers and m < n;

We can generalize that if ‘a’ is a non-zero integer or a non-zero rational number and m and n are positive integers, such that m > n, then

aᵐ ÷ aⁿ =  if m < n, then aᵐ ÷ aⁿ = 

Similarly, (a/b)ᵐ ÷ (a/b)ⁿ =

For example:

1. 7¹⁰ ÷ 7⁸

= (7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7)/(7 × 7 × 7 × 7 × 7 × 7 × 7 × 7)

= 7^10-8, [here exponents are subtracted] 

= 7²


2. p⁶ ÷ p¹=p⁶/p¹

= (p × p × p × p × p × p)/p

= p6−1, [here exponents are subtracted] 

= p⁵


3. 4⁴ ÷ 4² = 4⁴/4²

= (4 × 4 × 4 × 4)/(4 × 4)

= 4^4-2 , [here exponents are subtracted] 

= 4²


4. 10² ÷ 10⁴ = 10²/10⁴

= (10 × 10)/(10 × 10 × 10 × 10)

= 10^-(4-2) , [See note (2)] 

= 10^-2

5. 5³ ÷ 5¹

= 5^3-1

= 5²

6. 3⁵/3²

= 3^5-2

= 3³


7. (-5)⁹/(-5)⁶

= (-5)^9-6

= (-5)³


8. (7/2)⁸ ÷ (7/2)⁵

= (7/2)^8-5

= (7/2)³

3. Power of a power

For example: (2³)², (5²)⁶, (3² )−3−3

In power of a power you need multiply the powers.

Consider the following

(i) (2³)⁴

Now, (2³)⁴ means 2³ is multiplied four times

i.e. (2³)⁴ = 2³ × 2³ × 2³ × 2³

=

=2¹²

Note: by law (l), since aᵐ × aⁿ = a^m+n

(ii) (2³)²

Similarly, now (2³)² means 2³ is multiplied two times

i.e. (2³)² = 2³ × 2³

= ,   [since aᵐ × aⁿ = a^m+n] 

= 2⁶

Note: Here, we see that 6 is the product of 3 and 2 i.e,(2³)² = 23×23×2= 2⁶

(iii) ( )³

Similarly, now ( )³ means

 is multiplied three times

i.e. ( )³ =  ×  ×

=

= 4^-6

Note: Here, we see that -6 is the product of -2 and 3 i.e, ( )³ =  =

For example:

1.(3²)⁴ =  = 3⁸

2. (5³)⁶ =  = 5¹⁸

3. (4³)⁸ = = 4²⁴

4. ( )⁴ =  = a⁴ᵐ

5. (2³)⁶ =  = 2¹⁸

6. (xᵐ)^-n = x^m(-n)= x^-mn

In general, for any non-integer a, (aᵐ)ⁿ= aᵐⁿ. Thus where m and n are whole numbers. 

If ‘a’ is a non-zero rational number and m and n are positive integers, then {(a/b)ᵐ}ⁿ = (a/b) ᵐⁿ

For example:

[(-2/5)³]²
= (-2/5)⁶

4. Multiplying powers with the same exponents

For example: 3² × 2², 5³ × 7³

We consider the product of 4² and 3², which have different bases, but the same exponents.

(i) 4² × 3² [here the powers are same and the bases are different]

= (4 × 4)×(3 × 3)

= (4 × 3)×(4 × 3)

= 12 × 12

= 12²

Here, we observe that in 12², the base is the product of bases 4 and 3.

 

We consider,

(ii) 4³ × 2³

=(4 × 4 × 4)×(2 × 2 × 2)

=(4 × 2)× ( 4 × 2)× (4 × 2)

=8 × 8 × 8

=8³

(iii) We also have, 2³ × a³

= (2× 2 × 2)×(a × a × a)

= (2 × a)×(2 × a)×(2 × a)

= (2 × a)³

= (2a)³ [here 2 × a = 2a]

(iv) Similarly, we have, a³ × b³

= (a × a × a)×(b × b × b)

= (a × b)× (a × b)× (a × b)

= (a × b)³

= (ab)³ [here a × b = ab]

Note: In general, for any non-zero integer a, b.

aᵐ × bᵐ

= (a × b)ᵐ

= (ab)ᵐ [here a × b = ab]

Note: Where m is any whole number.

(-a)³ × (-b)³

= [(-a) × (-a) × (-a)]×[(-b) × (-b) × (-b)]

= [(-a) × (-b)]× [(-a) × (-b)]× [(-a) × (-b)]

= [(-a)×(-b)]³

= (ab)³ [here a × b = ab and two negative become positive,

(-) × (-) = +]

5. Negative Exponents

If the exponent is negative we need to change it into positive exponent by writing the same in the denominator and 1 in the numerator.

If ‘a’ is a non-zero integer or a non-zero rational number and m is a positive integers, then a−m is the reciprocal of aᵐ, i.e., 

a−m = a^m, if we take ‘a’ as p/q then (p/q)−m = 1/(p/q)ᵐ = (q/p)ᵐ

again, a^-m = aᵐ

Similarly, (a/b)^-n = (b/a)ⁿ, where n is a positive integer

Consider the following

2^-1 = 1/2

2^-2  = 1/2² = 1/2 × 1/2 = 1/4

 

[So in negative exponent we need to write 1 in the numerator and in the denominator 2 multiplied to itself five times as 2−5−5. In other words negative exponent is the reciprocal of positive exponent] 

For example:

1. 10^-3

= 1/10^-3, [here we can see that 1 is in the numerator and in the denominator 10³ as we know that negative exponent is the reciprocal] 

= 1/10 × 1/10 × 1/10 [here 10 is multiplied to itself 3 times]

= 1/1000

2. (-2)^-4

= 1/(-2)⁴ [here we can see that 1 is in the numerator and in the denominator

(-2)⁴]

=(- 1/2) × (- 1/2) × (- 1/2) × (- 1/2)

= 1/16

6. Power with exponent zero

If the exponent is 0 then you get the result 1 whatever the base is.

For example: 8⁰, ( a/b)⁰, m⁰…....

If ‘a’ is a non-zero integer or a non-zero rational number then,

a⁰ = 1

Similarly, (a/b)⁰ = 1

Consider the following

a⁰ = 1 [anything to the power 0 is 1] 

(a/b)⁰ = 1

(-2/3)⁰ = 1

(-3)⁰ = 1

For example:

1. (2/3)³ × (2/3)^-3

= (2/3)3+(−3), [here we know that aᵐ × aⁿ = am+nm+n] 

= (2/3)3−3

= (2/3)⁰

= 1

2. 2⁵ ÷ 2⁵

= 2⁵/2⁵

= (2 × 2 × 2 × 2 × 2)/(2 × 2 × 2 × 2 × 2)

= 2^5-5, [here by the law aᵐ ÷ aⁿ =a^m-n] 

= 2⁰

= 1

3. 4⁰ × 3⁰

= 1 × 1, [here as we know anything to the power 0 is 1]

= 1

4. aᵐ × a^-m

= a^m-m

= a⁰

= 1

5. 5⁰ = 1

6. (-4/9)⁰ = 1

7. (-41)⁰= 1

8. (3/7)⁰ = 1

7. Fractional Exponent

In fractional exponent we observe that the exponent is in fraction form.

Consider the following:

2¹ = 2 (it will remain 2). 

2^1/2 =√2 (square root of 2). 

2^1/3  =∛2 (cube root of 2). 

 For example:

1. a^1/n, [Here a is called the base and 1/n is called the exponent or power] 

= ⁿ√a [nth root of a]

2. 3^1/2 = √3 [square root of 3] 

3. 5^1/3 = ∛5 [cube root of 5]

 

Students will solve the following questions.

1.      Simplify the following by using laws of exponents.

a.    z^{-9 3} 

b.     

c.        

d.       

e.      

f.       

 

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           Lesson 4                                   

  Complex Numbers

 

A complex number is a number that can be written in the form a+bi, where a and b are real numbers and i is the imaginary unit defined by  = and  ,

a is real part and b is imaginary part of complex number z = a+b.

A complex number is denoted by ‘z’ and set of complex numbers is denoted by C.

The set of complex numbers, denoted by C, includes the set of real numbers R and the set of pure imaginary numbers.

 

 RC

 

 

 

In complex number z = 5 , 5 is real part of z and -7 is imaginary part of z.

Find real and imaginary parts of following complex numbers.

i.          z  = 3+ +a

 

Real part of z =3+a

Imaginary part of z= 5

ii.                   

Real part of = 2

Imaginary part of =

 

Students will solve the following questions.

1.   Find real and imaginary part of z =8+ -2

2.      If real part of z is -5 and imaginary part of z is 3 the find z,z+3 and .

3.   If 3z= 6 +  then find real and imaginary part of z.

 

Equality of two complex numbers

Two complex numbers  = a + ib and  = x + iy are equal if and only if a = x and b = y i.e., Real part of  = Real part of and imaginary part of  = imaginary part of

Thus,  =  ⇔ Re ( ) = Re  ) and Im ( )= Im ( ).

For example, if the complex numbers   = x + iy and   = -5 + 7i are equal, then x = -5 and y = 7.

Solved examples on equality of two complex numbers:

1. If  = 5 + 2yi and  = -x + 6i are equal, find the value of x and y.

Solution:

The given two complex numbers are  = 5 + 2yi and   = -x + 6i.

We know that, two complex numbers   = a + ib and   = x + iy are equal if a = x and b = y.

  =  

⇒ 5 + 2yi = -x + 6i

⇒ 5 = -x and 2y = 6

⇒ x = -5 and y = 3

Therefore, the value of x = -5 and the value of y = 3.

 2. If a, b are real numbers and 7a + i(3a - b) = 14 - 6i, then find the values of a and b.

Solution:

Given, 7a + i(3a - b) = 14 - 6i

⇒ 7a + i(3a - b) = 14 + i(-6)

Now equating real and imaginary parts on both sides, we have

7a = 14 and 3a - b = -6

⇒ a = 2 and 3 ∙ 2 – b = -6

⇒ a = 2 and 6 – b = -6

⇒ a = 2 and – b = -12

⇒ a = 2 and b = 12

Therefore, the value of a = 2 and the value of b = 12.

Students will solve the following questions

1.      If = 5+(3 and =+ 6i are equal then find the value of k.

2.   If  then find the value of m and n.

3.      If = 3+(p,  and  then find the value of p and q.

4.   If  a+3 4+(b) then find the value of  a and b.

5.   If z = c+2d and  then find c and d.

 

 

 

Conjugate of a complex number:

 

Conjugate of complex number z = a+is given  = a

Every complex number has associated with another complex number known as its complex conjugate. You find the complex conjugate simply by changing the sign of the imaginary part of the complex number.

 Example: To find the complex conjugate of 4+7i we change the sign of the imaginary part. Thus the complex conjugate of 4+7i is 4 − 7i.

 

Example: To find the complex conjugate of 1−3i we change the sign of the imaginary part. Thus the complex conjugate of 1 − 3i is 1+3i.

 

Example: To find the complex conjugate of −4 − 3i we change the sign of the imaginary part. Thus the complex conjugate of −4 − 3i is −4 + 3i.

The complex conjugate has a very special property. Consider what happens when we multiply a complex number by its complex conjugate.

 For example, multiplying (4+7i) by (4 − 7i): (4 + 7i)(4 − 7i) = 16 − 28i + 28i − 49i 2 = 16 + 49 = 65 We find that the answer is a purely real number - it has no imaginary part. This always happens when a complex number is multiplied by its conjugate - the result is real number. Example (1 − 3i)(1 + 3i) = 1 + 3i − 3i − 9i 2 = 1 + 9 = 10 Once again, we have multiplied a complex number by its conjugate and the answer is a real number. This is a very important property which applies to every complex conjugate pair of numbers. We will use this property in the next unit when we consider division of complex numbers.

 

Students are supposed to solve the following questions.

1.

 

 

 

 

 

 

;

 

 

 

 

;