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**Number System**

Natural number (N)1,2,3,4,5,………………….

Whole number (W) 0,1,2,3,4……………………

Integers (Z) ………….-3,-2,-1,0,1,2,3………

Rational number 2/3 ,5/10 , 0.001 , 0.6666 or .444444444…………….

Real number all rational number, √2, 1/√5

**Rational number :- **

Numbers that can be expressed in the form p/q, where q is a non-zero integer and p is any integer are called rational numbers.

Examples 0 ,1/2 and 4/5 etc

Every integer is a-rational number but a rational number need not be an integer.

Every fraction is a rational number but a fraction need not be a rational number.

**Irrational number :-**

Numbers that cannot be put in the form of p/q , where q is not equal to 0

When p and q both are integers

Example √2 , √5 and √8 etc

Every fraction is a rational number but a fraction need not be a rational number.

Standard form of rational number :-

A rational number is said to be in the standard form if q is positive integer and the integers p and q have no common isor other than 1.

Example 2/5 , 7/9 etc

We can find as many rational numbers between x and y using the formula (x+y)/2

If x and y are any two rational numbers, then :

Sum or difference of two x and y will always rational

And product or ision of x and y will always rational

If one variable is rational and other is irrational in x and y then

Sum or difference of two x and y will always irrational

And product or ision of x and y will always irrational

**Terminating fractions:-**

Fractions which leaves remainder zero on ision. If Prime factors of the denominator are 2 or 5 or both only. Then the number will terminating

Example 7/10 is terminating as denominator 10 has prime factors 2 and 5 only

Example 3/25 is terminating as denominator 25 has prime factors 5 only

**Recurring fractions:-**

Fractions which never leave a remainder 0 on ision. If Prime factors of the denominator are not 2 , 5 or both only. Then the number will repeating or recurring.

Examples :-

1/15 it will recurring fraction because prime factors of denominator has 3 also other than 2 and 5

There are infinitely many rational numbers between any two rational numbers. We can calculate the rational number between two rational number x and y by (x+y)/2

**Properties of rationalization **

**Properties to solve exponential values **

**Question -1 Is zero a rational number? Can you write it in the form p/q , where p and q are integers and q ≠0?**

**Answer **Yes o is a rational number because it can be written in form of 0/1 , 0/2 and so on

And according to definition of rational number which can be expressed in term form of p/q when q is non zero integer and p is a integer .

And here o is an integer and 1,2 .. are non zero integers

Hence, 0 is a rational number

**Question -2 Find six rational numbers between 3 and 4.Solution:** Since we want six numbers, we write 1 and 2 as rational numbers with denominator 6 + 1 =7

So multiply in numerator and denominator by 7 we get

**Question -3 Find five rational numbers between 3/5 and 4/5.Solution:** Since we want five numbers, we write 3/5 and 4/5 So multiply in numerator and denominator by 5+1 =6 we get

**Question 4 State whether the following statements are true or false. Give reasons for your answers.Solution: (i)Every natural number is a whole number.**

Natural number are all counting number like 1,2,3,4, …….

And whole number has 0 also in natural number so they will 0,1,2,3,4,……..

So here every natural number is a whole number

**(ii)Every integer is a whole number.**

Whole numbers are 0,1,2,3,……….

And integer numbers are ……… - 3, - 2 ,- 1 , 0 , 1 , 2 , 3 ………..

Hence all integers are not whole number

-3,-2,1 are integers but not whole numbers

**(iii)Every rational number is a whole number.**

rational number which can be expressed in term form of p/q when q is non zero integer and p is a integer .

so rational numbers are 2/5 . 0.01 , 10 ………

and here there are not whole number

so all rational number is not whole number

**Question 1. State whether the following statements are true or false. Justify your answers. Solution: (i)Every irrational number is a real number. **True a real number can either rational or irrational

**(ii)Every point on the number line is of the form √m, where m is a natural number.**

False all numbers can lies on number line

**(iii)Every real number is an irrational number**

False, all real numbers are rational number

**Question 2. Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.Solution:** All perfect square number has rational roots

Examples √1 , √4 ,√9 ,√25 ………….

**Question 3 Show how √5 can be represented on the number line. √5 = √(4+1) Solution: **Here 4 and 1 both are perfect square number as √4 = 2 and √1 = 1

So draw right triangle with side 2 and 1

And according to Pythagoras theorem hypotenuse will √(22+12) = √5

Then draw an arc of √5 on number line

**Question 1. Write the following in decimal form and say what kind of decimal expansion each has **

** **

** Solution:** **(i)36/100 = 0.36 and it is terminating.**

**(ii) by long ision method we get **

And

**(iii) 1/11 = 0.090909090………. **

So it is a non terminating number

4.125 is a terminating number

**(iv)**

We are getting 3/13 = 0.230769230769230769…..

There is repetition of 230769 so it is a non terminating number

**(v)**

2/11 = 0.18181818 ……..

There is repartition of 18 so it is not terminating number

**(vi)**

329/400 = 0.8225

So it is terminating number

**Question 2 You know that 1/7 = 0.142857 bar . Can you predict what the decimal expansion of2/7 , 3/7 ,4/7 , 5/7 and 6/ 7are, without actually doing the long ision? If so, how?[Hint:Study the remainders while finding the value of1/7carefully.] Solution:**

**Question 3 Express the following in the form p/q , where p and q are integers and q ≠0. Solution: (i)**

Letx= 0.666…

Multiply by 10 we get

10x= 6.666…

x= 0.666…

subtract them we get

9x = 6 +x

9x= 6

Divide by 9

x=6/9

simplify it by ide 3 we get

x = 2/3 Answer

**(ii) **

Letx= 0.4777…

Multiply by 10 we get

10x= 4.7777…

x = 0.4777…

subtract x we get

9x = 4.3

Divide by 9

X = 4.3/9

To remove decimal add zero in denominator we get

x = 43/90

**(iii)**

Letx= 0.001001…

There are 3 repeating digits so multiply by 10*10*10 = 1000

1000x= 1.001001…

X = 0.001001…

subtract x there we get

999 x = 1

Divide by 999 we get

X = 1/999

**Question 4 . Express 0.99999 .... in the form p/q .Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.Solution:** Letx= 0.9999…

There is only one repeating digit so multiply by 10 we get

10x= 9.9999…

x= 0.9999…

subtract x now we get

9x= 9

Divide by 9 we get

x= 9/9 = 1

**Question 5 What can the maximum number of digits be in the repeating block of digits in the decimal expansion of1/17? Perform the ision to check your answer. Perform the ision to check your answer.**

Question 6 Look at several examples of rational numbers in the formp / q (q≠ 0), wherepandqare integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what propertyq must satisfy? Solution: If the denominator is product of 2 and 5 only then it will always give terminating number For example7/2 = 0.35, 19/10 = 1.9, 12/5 = 2.5, 1/25 = 0.04, 1/ 4 = 0.25and you can write such many examples.

**Question 7. Write three numbers whose decimal expansions are non-terminating non-recurring.**

**Solution:** All irrational number has non terminating and non and recurring decimal expressions

So √2 , √3 ,√5 all will give such decimal expansions

**Question 8: Find three different irrational numbers between the rational numbers5/7and9/11Solution:** Divide the number we get

5/7 = 0.71….

9/11= 0.81…..

So we can write many irrational numbers between both values like

0.720720072000……

0.730730073000……

0.740740074000…….

0.757507500750000………..

**Question 9. Classify the following numbers as rational or irrational: Solution:**

(i) 23 is not a perfect square values so that, it is an irrational number.

(ii)√(225) = √(3*3*5*5)

Pairs comes outside the root and we get

√(3*3*5*5) = 15

That means 225 is a perfect square value so it a rational number

(iii) 0.3796

The decimal expansion of above number is terminating, so that it is a rational number.

(iv) 7.478478 …

The decimal expansion of above number is non-terminating recurring, so that, it is a rational number.

(v) 1.10100100010000 …

The decimal expansion of above number is non-terminating non-repeating, so that , it is an irrational number.

**Question 1. Visualise 3.765 on the number line, using successive magnification. **

**Question 2. **Visualize 4.26 on the number line, up to 4 decimal places. ** **

**Question 1. Classify the following numbers as rational or irrational:**

**Solution:** (i)2 - √5

Here 2 is a rational number and √5 is a irrational number

And difference of rational and irrational is always a irrational number

So that 2 - √5 is a irrational number

(ii) (3 + √23) - √23

Solve the bracket we get

3 + √23 - √23 = 3

and 3 is a rational number so that (3 + √23) - √23 is a rational number

(iii) and 2/7 can be written in form of p/q so it a rational number

(iv) 1/ √2 is an irrational number because 1 is rational and √2 is irrational and product or ision of rational with irrational is always irrational

(v) 2pi it is an irrational here 2 is rational and pi is a irrational number and product or ision of rational with irrational is always irrational

**Question 2: Simplify each of the following expressions:**

**Solution:**

**Question 3: Recall, π is defined as the ratio of the circumference (sayc) of a circle to its diameter (sayd). That is,. This seems to contradict the fact that π is irrational. How will you resolve this contradiction?**

**Solution:** There is no contradiction.. We can never obtain an exact value of c and d so that , either cordis irrational. and product or ision of rational with irrational is always irrational . Therefore, the fractionc/dis irrational. Hence, π is irrational.

**Question 4. Represent√9.3on the number line**

**Solution:**

Draw a line segment AB of length 9.3 units

Extend the line 1 unit more such that BC = 1 units

Find the midpoint of AC

Draw semicircle with center O and radius OC

Draw a line BD perpendicular to AB, intersecting the semicircle at point D

Here BD = √9.3 to represent it in number line draw a arc DE such that BE= BD

Here BE is our required line

**Question 5 Rationalize the denominator of the followings **

**Solution:**

**Question 1. Find (i) 64 ^{1/2} (ii) 32^{1/5} (iii)125^{1/3}**

Solution:

**Question 2. find (i) 9 ^{3/2} (ii)32^{2/5} (iii)16^{3/4} (iv)125^{-1/5}**

Solution:

**Question 3 Simplify : (i) 2 ^{2/3}.2^{1/5} (ii)(1/3^{3})^{7} (iii)11^{1/2}/11^{1/4} (iv)7^{1/2}.8^{1/2}**

Solution:

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