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CBSE NOTES CLASS 9 MATHEMATICS CHATER 1

NUMBER SYSTEMS

SUMMARY OF BASIC CONCEPTS

N = Natural numbers = {1,2,3, ….}

W = Whole numbers = {0,1,2,3, ….}

Z = Integers = {….,-3,-2,-1,0,1,2,3, ….}

Q = Rational numbers

R = Real numbers,

“A number ‘r’ is called a rational number, if it can be written in the form r = pq, where p and q are co-prime integers and q ≠ 0. (Why do we insist that q ≠ 0?)”

Operations on Real Numbers

Idenities for exponents of real numbers

For positive real numbers a and b,

1.  ab=a b 

2.  ab=ab

3. a+ba-b= a-b

4.  a+ba-b= a2-b

5. a+b2 = a+b+2a b

6. a-b2 = a+b-2a b

Let a > 0 be a real number and p and q be rational numbers. Then

i ap. aq = ap + q

ii apq = apq

iii apaq=ap-q

iv ap bp=(ab)p

v a0 = 1

vi 1an= a-n

vii an=a1/n

viii am/n= anm= amn

To rationalize the denominator of  1a, we multiply 1a by aawhere, a is an integer.

To rationalize the denominator of   1a+b, we multiply 1a+b by   a-ba-b  where a and b are integers.

CBSE NOTES CLASS 9 MATHEMATICS CHATER 1

NUMBER SYSTEMS

DETAILED CHAPTER NOTES

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N = Natural numbers = {1,2,3, ….}

W = Whole numbers = {0,1,2,3, ….}

Z = Integers = {….,-3,-2,-1,0,1,2,3, ….}

Q = Rational numbers

R = Real numbers,

“A number ‘r’ is called a rational number, if it can be written in the form r = pq, where p and q are co-prime integers and q ≠ 0. (Why do we insist that q ≠ 0?)”

Examples, 12, 34, 0.1, 0.4 1, 0, -4, etc.

Co-prime numbers are those numbers which do not have common factors, except 1. 2, 9 are co-prime. 3 and 9 are not co-prime.

Example 1: Are the following statements true or false? Give reasons for your answers.

(i) Every whole number is a natural number.

(ii) Every integer is a rational number.

(iii) Every rational number is an integer.

[NCERT]

Solution:

(i) False, because zero is a whole number but not a natural number.

(ii) True, because every integer m can be expressed in the form m1, and so it is a rational number.

(iii) False, because 35 is not an integer

Number Line

Every real number is represented by a unique point on the number line. Also, every point on the number line represents a unique real number.

This is why we call the number line, the real number line.

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Represent number pq; q > p; on the number line.

Divide the one whole in the q parts equal to denominator and mark the whole number as qq. Each part represents 1q.

For example 134 can be represented as follows,

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For example, find 5 numbers between 14 and 12 Make the denominators of both the numbers equal.

Multiply both numbers by 66, then the numbers will become, 624 and 1224, then the numbers, 724, 824, etc. are rational numbers lying between 14 and 12.

We can also multiply both the numbers by 1010, 100100 etc.

Example 2: Find five rational numbers between 1 and 2.

[NCERT]

Solution:

Since we want five numbers, we write 1 and 2 as rational numbers with denominator 5 + 1,

i.e., 1 =11×66=66 

and 2 =21×66=126

Then 76,86,96,106, and 116 are rational numbers between 1 and 2. 

Example 3: Locate 2 on the number line.

Solution:

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Draw OA = 1 unit. Draw a perpendicular AB at A, such that AB = 1 unit. Now by Pythagoras theorem,

OB =12+ 12=2

Now taking O as centre and OB as radius, draw an arc, which cuts the number line at P. point P now represents 2

Example 4: Locate 3 on the number line.

Solutions:

Construct BD of unit length perpendicular to OB. Then using the Pythagoras theorem, we see that OD = (2)2+12=3

Using a compass, with centre O and radius OD, draw an arc which intersects the number line at the point Q.

Then Q corresponds to 3.

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Decimal Expansion

Example 5: Find the decimal expansions of 103, 78 and 17

Solution: Try yourself using long division method.

Rational Numbers with Repeating Decimal Expansion

We represent repeating digits in a rational number under a bar. For example 1.2727…. can be represented as 1.27̅

Example 6: Show that 3.142678 is a rational number. In other words, express 3.142678 in the form pq, where p and q are integers and q ≠ 0

Solution: 3.142678 =31426781000000

Example 7: Represent 1.27̅ in the form of pq.

Solution:

Let x = 1.27̅ = 1.2727…. (1)

Multiply the number by 10, 100, 1000 etc., depending upon the number of repeating digits.

100x = 127.2727…. (2)

Subtracting equation (1) from (2), we get,

99x = 126 

  x=12699=1411

Example 9: Show that 0.2353535... = 0.235 can be expressed in the form pq, where p and q are integers and q ≠ 0.

Solution:

Let x = 0.235̅  = 0.23535… (1)

Since only two digits are repeating, we multiply x by 100

100 x = 23.53535... (2)

Subtracting equation (1) from (2), we get,

99 x = 23.3

 99x =23310

 x =233990

Example 10: Find an irrational number between 1/7 and 2/7.

Solution:

We know that,

17= 0.142857̅  and 27=  0 285714̅

To find an irrational number between 1/7 and 2/7, we find a number which is non-terminating non-recurring lying between them.

Examples are,

0.150150015000150000...,

0.1601011011101111… and so on.

Representing Real Numbers on the Number Line - Process of Successive Magnification

Operations on Real Numbers

For any given positive real number x , represent x geometrically.

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Idenities for exponents of real numbers

For positive real numbers a and b,

1.  ab=a b 

2.  ab=ab

3. a+ba-b= a-b

4.  a+ba-b= a2-b

5. a+b2 = a+b+2a b

6. a-b2 = a+b-2a b

Let a > 0 be a real number and p and q be rational numbers. Then

i ap. aq = ap + q

ii apq = apq

iii apaq=ap-q

iv ap bp=(ab)p

v a0 = 1

vi 1an= a-n

vii an=a1/n

viii am/n= anm= amn

To rationalise the denominator of 1a, we multiply, 1a by aa 

where a is an integer.

To rationalise the denominator of1a+b, we multiply, 1a+b by

  a-ba-b where a and b are integers.

Take examples of

Rationaisation of denominator,

Exponents and powers.

Find rational and irrational numbers between two numbers.

Represent 26,  29,  37, 13 etc.

Visualise 2.37̅ upto 5 decimal digits etc.