Componente de Software para la Firma Digital de Documentos en formato PDF Software Component for Digital Signature of Documents in PDF format

Componente de Software para la Firma Digital de Documentos en formato PDF Software Component for Digital Signature of Documents in PDF format Alexei Z

DIGITAL TEXTBOOK

PREPARED BY

Anwar Sadiq

STANDARD IX MATHEMATICS

National Anthem Jana-gana-mana-adhinayaka jaya he Bharata-bhagya-vidhata Punjaba-Sindhu-Gujarata-Maratha Dravida-Utkala-Banga Vindhya-Himachala-Yamuna-Ganga Uchchala-jaladhi-taranga Tava shubha name jage, tava shubha asisa mage,

Gahe tava jaya-gatha. Jana-gana-mangala-dayaka jaya he Bharata-bhagya-vidhata. Jaya he, Jaya he, Jaya he,

Jaya jaya jaya, jaya he. Salute to Nation

PLEDGE India is my country. All Indians are my brothers and sisters. I love my country and I am proud of its rich and varied heritage. I shall strive to be worthy of it. I shall respect my parents, teachers and all elders and treat everyone with courtesy. To my country and all my people, I pledge my devotion. In their well-being and prosperity alone lies my happiness.

1. to abide by the constitution and respect its ideal and institutions; 2. to cherish and follow the noble ideals which inspired our national struggle for freedom; 3. to uphold and protect the sovereignty, unity and integrity of India; 4. to defend the country and render national service when called upon to do so; 5. to promote harmony and the spirit of common brotherhood amongst all the people of India transcending religious, linguistic and regional diversities, to renounce practices derogatory to the dignity of women; 6. to value and preserve the rich heritage of our composite culture; 7. to protect and improve the natural environment including forests, lakes, rivers, and wild-life and to have compassion for living creatures; 8. to develop the scientific temper, humanism and the spirit of inquiry and reform; 9. to safeguard public property and to abjure violence; 10. to strive towards excellence in all spheres of individual and collective activity, so that the nation constantly rises to higher levels of endeavour and achievement. Further, one more Fundamental duty has been added to the Indian Constitution by 86th Amendment of the constitution in 2002.

11. who is a parent or guardian , to provide opportunities for education to his child, or as the case may be, ward between the age of six and fourteen years.

DEAR STUDENTS, Man invented various types of numbers to understand the world through measurements and the relations between measures. You have already seen how natural numbers and fractions evolved like this and how their operations were defined based on the physical contexts in which they were used. In this book you can get acquainted with knowledge of Real Numbers and it's operations

CONTENT REAL NUMBERS DISTANCE BETWEEN NUMBERS MIDPOINT OF NUMBERS ABSOLUTE VALUE OF NUMBER

Definition 1: Real Numbers, Rational Numbers irrational Numbers A rational number is the one which can be represented in the form of P/Q where P and Q are integers and Q ≠ 0. But an irrational number cannot be written in the form of simple fractions. Natural numbers, fractions, and their negatives with zero as well are collectively called Rational Numbers and all other numbers are called Irrational Numbers. Rational Numbers and Irrational Numbers together are called Real Numbers.

Figure 1: Representation of rational Numbers and irrational Numbers

Example 1 Is √5 is an irrational number? Answer: Assuming √5 as a rational number, that is, can be written in the form a/b where a and b are integers with no common factors other than 1 and b is not equal to zero. It means that 5 divides a². This has arisen due to the incorrect assumption as √5 is a rational number. Therefore, √5 is irrational.

Example 2 State if each number is rational, irrational, or not a real number. a) 23

b) 9/0

Answer: a) 23 is a natural number. So it is a rational number b) 9/0 is undefined. So not a real number

Definition 2: Number line Number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a point.

Figure 2: Number Line

Definition 3: Distance between two points on number line The distance between any two points on the number line is the smaller of the numbers denoting them subtracted from the larger. Example 3 Let's do an activity a) Find the distance between -2½ and -5 ¼? b) Find the distance between 5 ¼ and -2½? Answer: a) The large number is -2 ½; Subtracting the smaller number -5 ¼. From it gives -2 ½ - (-5¼) = -2 ½+5¼= 5¼ - 2½=2 ¾ b) 5¼ is the large number and -2½ is the smaller 5 ¼ - (-2½) = 5¼+2½=7¾

Definition 4: Midpoint of two points on number line The midpoint of two points on the number line is that point denoted by half the sum of the numbers denoting these points Example 4 Let's see this examples below Find midpoint of -2½ and 4 ¾? Answer: ½ (-2½+4¾) = ½ x 2 ¼ =9/8

Activity 1 1. Find the distance between the points on the number line, denoted by each pair of numbers given Below: i) 1,-5

ii) ¾, ½

iii) ¾,-2

iv) 5 ,-4

2. The part of the number line between the points denoted by the numbers 3 and 2 is divided into four equal parts. Find the numbers denoting the ends of each such part.

Definition 5: Absolute value of a number On the number line, the distance between the point denoted by 0 and that point denoted by another number is the absolute value of that number. The distance between two points on the number line is the absolute value of the difference of the numbers denoting these points Example 5 1. The distance between the points 2 and 5 on the number line is, | 2-5 |=| -3 |=3 2. The distance between 2 and -5 is, | 2-(-5) |=| 2+5 |=| 7 |=7 Example 6 Prove that | X |2 = X2 for any number X? If X is a positive number, then | X |= X and so | X |2 = X2 If X is a negative number, then |X|= -X and so |X|2= (-X) 2= (-X)(-X)=X2

Finally, if X=0, then |X|=0 and |X|2 = 02 = 0 Again, since X = 0, we have X2- 02 = 0 Thus in this case |X2|=X2. Hence Proved. Example 7 Find X satisfying each of the equations below a) |X-1|=|X-3|

b) |X-3|=|X-4|

Answer: a) X is the midpoint of 1 and 3, because the above equations implies that distance between X and 1 and distance between X and 3 are equal. So X = (1+3)/2 = 2 b) X is the midpoint of 3 and 4 So X = (3+4)/2=3.5 Example 8 What are the numbers X for which | X-2 |+| X- 8| =6?

We solve this equation in four cases Case 1: X-2 and X-8 are positive numbers X-2+ X-8-6 2X-10=6 2X = 16 X=8 Case 2: X-2 and X-8 are negative numbers -(X-2) + - (X-8)=6 -X+2-X+8=6 -2X+10=6 2X=4 X=2

Case 3: X-2 is negative number and X-8 is positive number -(X-2)+X-8=6 -X+2+X-8=6 -X+2+X-8=6 Equation is not valid

Case 4: X-2 is positive number and X-8 is negative number (X-2)-(X-8) = 6 X-2-X +8= 6 Not valid Here, the possible values of X is 2 and 8 Activity 2

1. Find the X satisfying the each of the equations below:

a) |X+2|=|X-4|

b) |X|=|X+1|

c) |X-2|+|X-8|=10

2. Prove that if X 7, then | X-Y | > 4 3. Find two numbers X and Y such that | X+Y |=|X|+|Y│?

4. Are there numbers X and Y such that | X + Y | >|X|+|Y|? 5. Are there numbers X and Y such that | X+Y |

PREPARED BY

Anwar Sadiq

STANDARD IX MATHEMATICS

National Anthem Jana-gana-mana-adhinayaka jaya he Bharata-bhagya-vidhata Punjaba-Sindhu-Gujarata-Maratha Dravida-Utkala-Banga Vindhya-Himachala-Yamuna-Ganga Uchchala-jaladhi-taranga Tava shubha name jage, tava shubha asisa mage,

Gahe tava jaya-gatha. Jana-gana-mangala-dayaka jaya he Bharata-bhagya-vidhata. Jaya he, Jaya he, Jaya he,

Jaya jaya jaya, jaya he. Salute to Nation

PLEDGE India is my country. All Indians are my brothers and sisters. I love my country and I am proud of its rich and varied heritage. I shall strive to be worthy of it. I shall respect my parents, teachers and all elders and treat everyone with courtesy. To my country and all my people, I pledge my devotion. In their well-being and prosperity alone lies my happiness.

1. to abide by the constitution and respect its ideal and institutions; 2. to cherish and follow the noble ideals which inspired our national struggle for freedom; 3. to uphold and protect the sovereignty, unity and integrity of India; 4. to defend the country and render national service when called upon to do so; 5. to promote harmony and the spirit of common brotherhood amongst all the people of India transcending religious, linguistic and regional diversities, to renounce practices derogatory to the dignity of women; 6. to value and preserve the rich heritage of our composite culture; 7. to protect and improve the natural environment including forests, lakes, rivers, and wild-life and to have compassion for living creatures; 8. to develop the scientific temper, humanism and the spirit of inquiry and reform; 9. to safeguard public property and to abjure violence; 10. to strive towards excellence in all spheres of individual and collective activity, so that the nation constantly rises to higher levels of endeavour and achievement. Further, one more Fundamental duty has been added to the Indian Constitution by 86th Amendment of the constitution in 2002.

11. who is a parent or guardian , to provide opportunities for education to his child, or as the case may be, ward between the age of six and fourteen years.

DEAR STUDENTS, Man invented various types of numbers to understand the world through measurements and the relations between measures. You have already seen how natural numbers and fractions evolved like this and how their operations were defined based on the physical contexts in which they were used. In this book you can get acquainted with knowledge of Real Numbers and it's operations

CONTENT REAL NUMBERS DISTANCE BETWEEN NUMBERS MIDPOINT OF NUMBERS ABSOLUTE VALUE OF NUMBER

Definition 1: Real Numbers, Rational Numbers irrational Numbers A rational number is the one which can be represented in the form of P/Q where P and Q are integers and Q ≠ 0. But an irrational number cannot be written in the form of simple fractions. Natural numbers, fractions, and their negatives with zero as well are collectively called Rational Numbers and all other numbers are called Irrational Numbers. Rational Numbers and Irrational Numbers together are called Real Numbers.

Figure 1: Representation of rational Numbers and irrational Numbers

Example 1 Is √5 is an irrational number? Answer: Assuming √5 as a rational number, that is, can be written in the form a/b where a and b are integers with no common factors other than 1 and b is not equal to zero. It means that 5 divides a². This has arisen due to the incorrect assumption as √5 is a rational number. Therefore, √5 is irrational.

Example 2 State if each number is rational, irrational, or not a real number. a) 23

b) 9/0

Answer: a) 23 is a natural number. So it is a rational number b) 9/0 is undefined. So not a real number

Definition 2: Number line Number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a point.

Figure 2: Number Line

Definition 3: Distance between two points on number line The distance between any two points on the number line is the smaller of the numbers denoting them subtracted from the larger. Example 3 Let's do an activity a) Find the distance between -2½ and -5 ¼? b) Find the distance between 5 ¼ and -2½? Answer: a) The large number is -2 ½; Subtracting the smaller number -5 ¼. From it gives -2 ½ - (-5¼) = -2 ½+5¼= 5¼ - 2½=2 ¾ b) 5¼ is the large number and -2½ is the smaller 5 ¼ - (-2½) = 5¼+2½=7¾

Definition 4: Midpoint of two points on number line The midpoint of two points on the number line is that point denoted by half the sum of the numbers denoting these points Example 4 Let's see this examples below Find midpoint of -2½ and 4 ¾? Answer: ½ (-2½+4¾) = ½ x 2 ¼ =9/8

Activity 1 1. Find the distance between the points on the number line, denoted by each pair of numbers given Below: i) 1,-5

ii) ¾, ½

iii) ¾,-2

iv) 5 ,-4

2. The part of the number line between the points denoted by the numbers 3 and 2 is divided into four equal parts. Find the numbers denoting the ends of each such part.

Definition 5: Absolute value of a number On the number line, the distance between the point denoted by 0 and that point denoted by another number is the absolute value of that number. The distance between two points on the number line is the absolute value of the difference of the numbers denoting these points Example 5 1. The distance between the points 2 and 5 on the number line is, | 2-5 |=| -3 |=3 2. The distance between 2 and -5 is, | 2-(-5) |=| 2+5 |=| 7 |=7 Example 6 Prove that | X |2 = X2 for any number X? If X is a positive number, then | X |= X and so | X |2 = X2 If X is a negative number, then |X|= -X and so |X|2= (-X) 2= (-X)(-X)=X2

Finally, if X=0, then |X|=0 and |X|2 = 02 = 0 Again, since X = 0, we have X2- 02 = 0 Thus in this case |X2|=X2. Hence Proved. Example 7 Find X satisfying each of the equations below a) |X-1|=|X-3|

b) |X-3|=|X-4|

Answer: a) X is the midpoint of 1 and 3, because the above equations implies that distance between X and 1 and distance between X and 3 are equal. So X = (1+3)/2 = 2 b) X is the midpoint of 3 and 4 So X = (3+4)/2=3.5 Example 8 What are the numbers X for which | X-2 |+| X- 8| =6?

We solve this equation in four cases Case 1: X-2 and X-8 are positive numbers X-2+ X-8-6 2X-10=6 2X = 16 X=8 Case 2: X-2 and X-8 are negative numbers -(X-2) + - (X-8)=6 -X+2-X+8=6 -2X+10=6 2X=4 X=2

Case 3: X-2 is negative number and X-8 is positive number -(X-2)+X-8=6 -X+2+X-8=6 -X+2+X-8=6 Equation is not valid

Case 4: X-2 is positive number and X-8 is negative number (X-2)-(X-8) = 6 X-2-X +8= 6 Not valid Here, the possible values of X is 2 and 8 Activity 2

1. Find the X satisfying the each of the equations below:

a) |X+2|=|X-4|

b) |X|=|X+1|

c) |X-2|+|X-8|=10

2. Prove that if X 7, then | X-Y | > 4 3. Find two numbers X and Y such that | X+Y |=|X|+|Y│?

4. Are there numbers X and Y such that | X + Y | >|X|+|Y|? 5. Are there numbers X and Y such that | X+Y |

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