## 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 âˆšx Geometrically
- Finding Irrational Numbers between two rational numbers.
- Rationalisation
- Laws of Exponent

**Number System**

**Real Numbers**

â†’ The numbers which exist are known as rational numbers.

â†’ All real numbers can be represented on Number line.

â†’ 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**

â†’ The numbers in the form of p/q where p and q integers and qâ‰ 0 are known as rational numbers.

Examples: -2/3, 4/5, -7, 3/2

**Irrational Numbers**

â†’ Real Numbers which cannot be expressed as ratio of two integers.

Examples: Ï€, âˆš3, âˆš5

â†’ Equivalent Rational Numbers are rational number cannot have unique representation as p/q where p and q are integers and qâ‰ 0.

â†’ For example 1/2 can also be written as

â†’ Rational number in simplest form is represented in the form of p/q where p and q integers and qâ‰ 0 also, p and q are co-prime numbers.

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

â†’

â†’ Find d = (a+b)/2

â†’ d is the rational number between two rational numbers.

**Equivalent Rational Number****â†’ All whole numbers, natural numbers, integers are rational numbers.**

â†’ Equivalent Rational Numbers are rational number cannot have unique representation as p/q where p and q are integers and qâ‰ 0.

â†’ For example 1/2 can also be written as

â†’ Rational number in simplest form is represented in the form of p/q where p and q integers and qâ‰ 0 also, p and q are co-prime numbers.

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

â†’

**Other examples:**3/4=6/10=9/12 is set of equivalent rational numbers. In Â¾ , 3 and 4 are co-prime Numbers. Â¾ is rational number in simplest form.**Finding a rational Number between Two Rational Numbers**

**â€¢ To find a rational number between two rational numbers a and b**â†’ Find d = (a+b)/2

â†’ 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

â†’ Then the n numbers between a and b are a+d, a+2d,â€¦.a+(n-1)d, a+nd.

â†’ Find all these values.

We have to find 4 rational numbers between 5 and 5Â½ 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

**â€¢ Note:**There can be infinite rational numbers between two rational numbers.**Finding n rational numbers between Two Rational Numbers.****â†’ Find d = (b-a)/(n+1)**

â†’ Then the n numbers between a and b are a+d, a+2d,â€¦.a+(n-1)d, a+nd.

â†’ Find all these values.

Question. Find 4 rational numbers between 5 and 5Â½.Question. Find 4 rational numbers between 5 and 5Â½.

We have to find 4 rational numbers between 5 and 5Â½ 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

â‡’ a+2d=52/10

â‡’ a+3d=53/10

â‡’ 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

Here,

d=(11/2-5)/(4+1)=1/10

So, the four rational numbers are

a+d=5+1/10=51/10

â‡’ a+2d=52/10

â‡’ a+3d=53/10

â‡’ 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,â€¦.a+(n-1)d, a+nd.

(iii) Find all these values.

**We have to find 4 rational numbers between 5 and 5**

**Â½**

**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

â‡’ a+2d =52/10

â‡’ a+3d =53/10

â‡’ 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**

**â†’ Any real number is called irrational, if it cannot be written in the form p/q, where p and q are integers and q â‰ 0.**

Examples: Ï€, âˆš2,âˆš3

**Question: Locating âˆš2 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 âˆš3 on Number Line**

â†’ The process is similar to previous case. We have to take base as OP=âˆš2, and from C draw a line perpendicular CD equal to one Unit. By Pythagoras Theorem, we get OD= âˆš3,

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

**Decimal Expansion of Real Numbers**

**â†’ Decimal expansion of any number can be obtained by performing Long division by dividing numerator by denominator.**

Example: Decimal Expansion of 2/3 =0.66666â€¦â€¦. (6 repeating)

**Terminating Decimal Expansion**
â†’ The decimal expansion terminates or ends after a finite number of steps.

Examples: 9/8=1.125

Examples: 9/8=1.125

**Recurring Decimal Expansion**
â†’ The decimal Expansion do not terminates but a pattern of digits repeats in the decimal expansion.

Examples: 2/3=0.666â€¦â€¦

1/7=0.142857142857142857 here 142857 is repeated.

Examples: 2/3=0.666â€¦â€¦

1/7=0.142857142857142857 here 142857 is repeated.

**Representing Recurring Decimal Expansion**
â†’ It Can be represented by putting bar over repeating terms.

**Non-Terminating Non-recurring Decimal Expansion**
â†’ The decimal expansion, neither repeats, not terminates.

**â€¢ Examples**

Ï€=3.14596â€¦â€¦â€¦â€¦â€¦.

âˆš2=1.414â€¦â€¦â€¦â€¦â€¦.

2.3030030003â€¦â€¦â€¦

âˆš2=1.414â€¦â€¦â€¦â€¦â€¦.

2.3030030003â€¦â€¦â€¦

**Rational numbers will have either terminating or Non-terminating recurring decimal expansion.**__Note:__
â†’ 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, 4.29â€¦.

__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 4.262626â€¦so, it is just more than 4.2626..

**Representing âˆšx**

**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 âˆšx . 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**

**âŸ¶ 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**