Real Numbers

A number which can be expressed in form of ,

Where p and q are integers and q≠0

[Extra note

Rational numbers include whole numbers, natural numbers, and integers.

In Decimal values either they are terminating or non-terminating and repeating]

Four rational numbers are

Yes 0 is a rational number

As rational numbers are number which can be expressed in form of ,

Where p and q are integers and q≠0

⇒ 0 divides and multiply by any number gives 0

As 0 has infinite number of factors

Here, p & q are integers and q ≠ 0 and p & q have no common factor other than 1

(i) Draw the number line

Place 7 on it

(ii) Draw the number line

Place -4 on it

(iii) Draw the number line

As lies on 0 and 1

Draw number line of 0 to 1 having in between

(iv) Draw the number line

As lies on 4 and 5

Draw number line of 0 to 5 having in between

(v) Draw the number line

As lies on 0 and 1

Draw number line of 0 to 1 having in between

(vi) Draw the number line

As lies on 2 and 3

Draw number line of 0 to 3 having in between

(vii) Draw the number line

As lies on 0 and 1

Draw number line of 0 to 1 having in between

Formula used.

If 2 rational number X and Y

Then, is a rational number lying between X and Y

Solution.

X = 4 and Y = 5

Then number lying between 2 number is

=

Formula used.

If 2 rational number X and Y

Then, is a rational number lying between X and Y

X = 1 and Y = 2

Then number lying between 2 number is

=

Formula used.

If 2 rational number X and Y

Then, is a rational number lying between X and Y

X = and Y =

Then number lying between 2 number is

= =

Formula used.

If 2 rational number X and Y

Then, is a rational number lying between X and Y

X = -1 and Y =

Then number lying between 2 number is

= =

Formula used.

If 2 rational number X and Y

Then, is a rational number lying between X and Y

X = and Y =

Then number lying between 2 number is

= =

Formula used.

If 2 rational number X and Y

Then, is a rational number lying between X and Y

X = -2 and Y = -1

Then number lying between 2 number is

=

Formula used.

If X and Y are 2 rational numbers and X<Y then n rational numbers can be taken on line between X and Y

(X+d),(X+2d),(X+3d)……,(X+nd)

Where d =

X = 4 and Y = 5

n = 3;

d = =

Then the 3 numbers are,

(4+) , (4+2×) , (4+3×)

Formula used.

If X and Y are 2 rational numbers and X<Y then n rational numbers can be taken on line between X and Y

(X+d),(X+2d),(X+3d)……,(X+nd)

Where d =

X = 1 and Y = 2

n = 6;

d = =

Then the 6 numbers are,

(1+) , (1+2×) , (1+3×) , (1+4×) , (1+5×) , (1+6×)

Formula used.

If X and Y are 2 rational numbers and X<Y then n rational numbers can be taken on line between X and Y

(X+d),(X+2d),(X+3d)……,(X+nd)

Where d =

X = and Y =

n = 3;

d = =

Then the 3 numbers are,

(+) , (+2×) , (+3×)

(i) True

Let’s get it by example

Take 2 integers suppose 2 and 3

On adding

We get 2+3 = 5

Which is an integer

On subtracting

We get 2-3 = -1

Which is an integer

On multiplying

We get 2×3 = 6

Which is an integer

∴ On adding, subtracting and multiplying two integers, we get integer

(ii) False

Let’s get it by example

Take 2 integer suppose 2 and 5

On dividing

We get = 0.4

Which is not an integer;

As rational numbers are number which can be expressed in form of ,

Where p and q are integers and q≠0

Let’s take 2 rational number and

On adding 2 rational numbers.

Where result is in form of

As multiplying and adding integers gives an integer

(ys) can’t be 0 because y,s≠0

∴ Adding 2 rational number gives a rational number

On subtracting 2 rational numbers.

Where result is in form of

As multiplying and subtracting integers gives an integer

(ys) can’t be 0 because y,s≠0

∴ Subtracting 2 rational number gives a rational number

On multiplying 2 rational numbers.

Where result is in form of

As multiplying an integers gives an integer

(ys) can’t be 0 because y,s≠0

∴ Multiplying 2 rational number gives a rational number

On Dividing 2 rational numbers.

Where result is in form of

As multiplying an integer gives an integer

(yr) can be 0 because r can be 0

∴ Dividing 2 rational number doesn’t gives a rational number

(i) True

We can explain this with an example.

Let the two rational numbers be and .

On addition, which is also a rational number.

(ii) True

We can explain this with an example.

Let the two irrational numbers be √2 and √3

On addition, √2+√3, which is also an irrational number.

(iii) True

Explanation: We can explain this with an example.

Let the two rational numbers be and .

On addition, which is also a rational number.

(iv) False

We can explain this with an example.

Let the two irrational numbers be √3 and √5

On multiplication, √3 × √5 = √15,

which is an irrational number. So, this is not always true.

(v) True

Since a rational number can be plotted on a number line, therefore, every rational number is a real number.

(vi) False

This can be explained with an example. Let us consider any real number, say 2, 2 is a real number as it can be plotted on a number line.

We can write 2 as , so 2 is a rational number.

∴ The given statement is not always true.

The numbers which cannot be expressed in the form of , where q is not equal to zero are called Irrational Numbers.

√2, since its value 1.414… we cannot write the exact value of √2 and thereby cannot write it in form, as a result it is an irrational number.

Four irrational numbers are: √3, √5, √6 and √7

(i) As we know 9 is a square of 3,

⇒ √9 = 3, which is rational

∴ √9 is a rational number.

(ii) As we know 225 is a square of 15,

⇒ √225 = 15, which is a rational number

∴ √225 is a rational number.

(iii) As value of √7 cannot be written exactly and thereby cannot be written in form, √7 is an irrational number.

(iv) We can write,

√50 = 5√2

As value of √2 cannot be written exactly and thereby cannot be written in form, √2 is an irrational number.

∴ √50 is an irrational number.

(v) As we know 100 is a square of 10,

⇒ √100 = 10, which is a rational number

∴ √100 is a rational number.

(vi) As we know 81 is a square of 9,

⇒ -√100 = -9 , which is a rational number

∴ -√81 is a rational number.

(vii) As value of √42 cannot be written exactly and thereby cannot be written in form, √42 is an irrational number.

(viii) As value of √29 cannot be written exactly and thereby cannot be written in form, √29 is an irrational number.

(ix) We can write,

-√1000 = -10√10

As value of √10 cannot be written exactly and thereby cannot be written in form, √10 is an irrational number.

∴ -√1000 is an irrational number.

Let the point O represent 0 on the number line.

We place a point A on the number line such that OA = 2 units.

Now, we draw AB perpendicular to OA at point A such that AB = 1unit.

By Pythagoras’ Theorem, we know that,

OB² = OA² + AB²

⇒ OB² = 4 + 1 = 5

⇒ OB = √5

Now, taking O as a centre and OB as the radius, an arc is drawn which intersects the number line at P.

∴ OP = √5 units

Hence, Point P represents √5

Let the point O represent 0 on the number line.

We place a point A on the number line such that OA = 1 units.

Now, we draw AB perpendicular to OA at point A such that AB = 1unit.

By Pythagoras’ Theorem, we know that,

OB² = OA² + AB²

⇒ OB² = 1 + 1 = 2

⇒ OB = √2

Now, we draw BC perpendicular to OB at point B such that BC = 1unit.

Again, by Pythagoras’ Theorem, we know that,

OC² = OB² + BC²

⇒ OC² = 2 + 1 = 3

⇒ OC = √3

Now, taking O as a centre and OC as the radius, an arc is drawn which intersects the number line at P.

∴ OP = √3 units

Hence, Point P represents √3

Let the point O represent 0 on the number line.

We place a point A on the number line such that OA = 2 units.

Now, we draw AB perpendicular to OA at point A such that AB = 1unit.

By Pythagoras’ Theorem, we know that,

OB² = OA² + AB²

⇒ OB² = 4 + 1 = 5

⇒ OB = √5

Now, we draw BC perpendicular to OB at point B such that BC = 1unit.

Again, by Pythagoras’ Theorem, we know that,

OC² = OB² + BC²

⇒ OC² = 5 + 1 = 6

⇒ OC = √6

We draw CD perpendicular to OC at point C such that CD = 1unit.

Again, by Pythagoras’ Theorem, OD = √7

Again, we draw ED perpendicular to OD at point D such that ED = 1unit.

Again, by Pythagoras’ Theorem, OE = √8

If we extend ED till F such that EF = 2 units and applying Pythagoras’ theorem on ΔODF, we get OF = √11

Now, taking O as a centre and OB, OC, OD, OE and OF as the radius, the arcs are drawn which intersects the number line at G, H, I, K and L.

∴ OG = √5 units, OH = √6 units, OI = √7 units, OJ = -√6 units, OK = -√8 units and OL = -√11 units