Notes of Ch 1 Number Systems| Class 9th Math

Revision Notes of Ch 1 Number Systems Class 9th Math

Topics in the Chapter

  • Number System
  • Types of Real Numbers
  • Equivalent Rational Number
  • Finding a rational Number between Two Rational Numbers
  • Finding n rational numbers between Two Rational Numbers. 
  • Finding n rational numbers between two Rational Numbers.
  • Locating Irrational Numbers on Number Line
  • Decimal Expansion of Real Numbers
  • Representing Recurring Decimal Expansion
  • Representing √x Geometrically
  • Finding Irrational Numbers between two rational numbers. 
  • Rationalisation 
  • Laws of Exponent 

Number System

Real Numbers

→ The numbers which exist are known as rational numbers.

→ All real numbers can be represented on 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.

Types of Real Numbers

Rational Numbers

→ The numbers in the form of p/q where p and q integers and q≠0 are known as rational numbers.
Examples: -2/3, 4/5, -7, 3/2

Irrational Numbers

→ Real Numbers which cannot be expressed as ratio of two integers.
Examples: π, √3, √5

Equivalent Rational Number

→ All whole numbers, natural numbers, integers are rational numbers.

→ Equivalent Rational Numbers are rational number cannot have unique representation as p/q where p and q are integers and q≠0.

→ For example 1/2 can also be written as


→ Rational number in simplest form is represented in the form of p/q where p and q integers and q≠0 also, p and q are co-prime numbers.

→ 1/2 is simplest form of 2/4, 4/8 as 1 and 2 are co-prime numbers.

Other examples: 3/4=6/10=9/12 is set of equivalent rational numbers. In ¾ , 3 and 4 are co-prime Numbers. ¾ is rational number in simplest form.

Finding a rational Number between Two Rational Numbers

• To find a rational number between two rational numbers a and b 

→ Find d = (a+b)/2

→ d is the rational number between two rational numbers.

Question: Find a rational number between 3/5 and 4/5.

Solution

d=(3/5+4/5)/2 = (7/5)/2 = 7/10

• Note: There can be infinite rational numbers between two rational numbers.

Finding n rational numbers between Two Rational Numbers. 

→ Find d = (b-a)/(n+1)

→ Then the n numbers between a and b are a+d, a+2d,….a+(n-1)d, a+nd.

→ Find all these values.

Question. Find 4 rational numbers between 5 and 5½.


We have to find 4 rational numbers between 5 and 5½ i.e between 5 and 11/2.
Here, n=4 as we have to find 4 rational numbers between a=5 and b=11/2
Find d = (b-a)/(n+1)
Here,
d=(11/2-5)/(4+1)=1/10

So, the four rational numbers are
a+d=5+1/10=51/10
⇒ a+2d=52/10
⇒ a+3d=53/10
⇒ a+4d=54/10

Therefore, the required 4 numbers are 51/10, 52/10, 53/10, 54/10.
In simpler form: 51/10, 26/5, 53/10, 27/5

Finding n rational numbers between two Rational Numbers.

(i) Find d = (b-a)/(n+1)
(ii) Then the n numbers between a and b are a+d, a+2d,….a+(n-1)d, a+nd.
(iii) Find all these values.

We have to find 4 rational numbers between 5 and 5½ i.e between 5 and 11/2.
Here, n=4 as we have to find 4 rational numbers between a=5 and b=11/2
Find d = (b-a)/(n+1)
Here,
d=(11/2 - 5)/(4+1)=1/10
So, the four rational numbers are
a+d=5+1/10=51/10
⇒ a+2d =52/10
⇒ a+3d =53/10
⇒ a+4d =54/10

Therefore, the required 4 numbers are 51/10, 52/10, 53/10, 54/10.
In simpler form: 51/10, 26/5, 53/10, 27/5

Locating Irrational Numbers on Number Line

→ Any real number is called irrational, if it cannot be written in the form p/q, where p and q are integers and q ≠ 0.
Examples: π, √2,√3

Question: Locating √2 in Number Line. 

(i) On Number line, marks point O which is 0, and 1 is marked as A.

(ii) Now a construction is done AB perpendicular to OA, that is number line. AB is equal to OA. Both equal to 1 unit. Since, OA represent 1 on number line.

(iii) Point O is joined by Point B. So, we have right angled triangle, OAB, right angled at A.

(iv) By applying Pythagoras theorem, we get


Question: Locate √3 on Number Line

→ The process is similar to previous case. We have to take base as OP=√2, and from C draw a line perpendicular CD equal to one Unit. By Pythagoras Theorem, we get OD= √3,

Note: In similar way we can fine √n on number line if √(n-1) .

Decimal Expansion of Real Numbers

→ Decimal expansion of any number can be obtained by performing Long division by dividing numerator by denominator.
Example: Decimal Expansion of 2/3 =0.66666……. (6 repeating)



Terminating Decimal Expansion

→ The decimal expansion terminates or ends after a finite number of steps.
Examples: 9/8=1.125

Recurring Decimal Expansion

→ The decimal Expansion do not terminates but a pattern of digits repeats in the decimal expansion.
Examples: 2/3=0.666……
1/7=0.142857142857142857 here 142857 is repeated.

Representing Recurring Decimal Expansion

→ It Can be represented by putting bar over repeating terms.


Non-Terminating Non-recurring Decimal Expansion

→ The decimal expansion, neither repeats, not terminates.
• Examples
π=3.14596…………….
√2=1.414…………….
2.3030030003………

Note: Rational numbers will have either terminating or Non-terminating recurring decimal expansion. 

→ Irrational numbers will always have non-terminating and non-recurring decimal expansion.

Converting Terminating Decimal Number into p/q form



Representing Recurring Decimal Expansion


Steps:

Step 1: Expanding 4-5 area on number line in ten parts.

Step 2: Divide it in 10 parts. Each part is 4.1, 4.2, 4.3, 4.4,4,5,4.6, 4.7.4.8, 4.9

Step 3: Now divide 4.2 to 4.3 part in 10 parts. Each part will be 4. 21, 4,22, 4.23, 4.24. 4.25. 4.26, 4.27, 4.28, 4.29….

Step 4: Again magnifying and dividing 4.261, 4.262, 4.263, 4.264, 4.26,4.27, 4.28. 4.29..

Step 5: 4.26 to 4.27 is divided into 10 parts. We will get 4.26, the point is 4.262626…so, it is just more than 4.2626..

Representing √x Geometrically



Step 1: On a Line we draw AB= x

Step 2: Extend AB to C. Such that BC=1. AC=x+1

Step 3: Take mid-point of AB as O. With OC as arc draw a semi circle.

Step 4: Draw BD perpendicular to AC. BD represents √x . Why this so, you will understand if you go through the proof.

Proof: 

OD = OC = (x+1)/2 Radius of Same Arc

From diagram:



Finding Irrational Numbers between two rational numbers. 

Step 2: First write rational numbers in decimal form.

Step 2:  Write three such decimal numbers between thoese decimal numbers which are having non-terminating and non-repeating decimal expnansion.

Q. Find three Irrational Numbers between 3/5 and 4/5. 
3/5 = 0.6
4/5 =0.8

The three irrational numbers between 0.6 and 0.8 can be
0.61611611161111611111...............
0.67677677767777677777.............
0.707007000700007000007......................

Rationalisation 

⟶ It is the Process of making denominator rational number if it is irrational. It is multiplied by same number in denominator and numerator.


Laws of Exponent 

Watch age fraud in sports in India
Liked NCERT Solutions and Notes, Share this with your friends::
Facebook Comments
0 Comments
© 2019 Study Rankers is a registered trademark.